MA 200: Multivariable Calculus (3:1)

Prerequisite courses for Undegraduates: UM 204

Functions on \(\R^n\) , directional derivatives, total derivative, Contraction mapping principle, The inverse and implicit function theorem, Maxima, Minima, Saddle points, Lagrange’s Multipliers, higher order derivatives and Taylor series.

Integration on \(\R^n\) , differential forms on \(\R^n\) , closed and exact forms. Green’s theorem, Stokes’ theorem and the Divergence theorem.

Suggested books :

1. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1986.
2. B. V. Limaye and S. Ghorpade, A course in Calculus and Real Analysis, Springer.
3. Spivak, M., Calculus on Manifolds, W.A. Benjamin, co., 1965.
4. Shifrin, Theodore, Multivariable Mathematics- Linear Algebra, Multivariable Calculus and Manifolds
5. Fleming, Wendell, Functions of Several Variables
6. Apostol, Tom M., Calculus, Vol-II

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MA 208: Proofs and Programs (3:1)

This course introduces various aspects of Computer Proofs, both interactive and fully automated. We will consider proofs of mathematical results as well as of correctness of programs. We will primarily use the Lean Theorem Prover 4, which is a formal proof system as well as a programming language. The foundations on which the Lean prover is based, Dependent Type Theory, allow a seamless integration of mathematical objects, theorems, proofs and algorithms.

Topics covered will be among the following.

• Interactively proving mathematical results in the lean theorem prover.
• Programming in lean - functional programming with dependent types.
• Mathematical proofs of correctness of programs.
• Foundations of Mathematics and Computation using Dependent Type Theory
• First-order logic
• Fully automated theorem proving: SAT Solvers, Resolution Theorem Proving etc.
• Use of Machine Learning in Automated and Interactive Theorem Proving.

Suggested books :

1. Jeremy Avigad, Leonardo de Moura, Soonho Kong and Sebastian Ullrich, Theorem Proving in Lean 4, available at https://leanprover.github.io/theorem_proving_in_lean4/.
2. Jeremy Avigad, Marijn Heule, Wojciech Nawrocki, Logic and Mechanical Reasoning, available at https://avigad.github.io/lamr/.
3. Jeremy Avigad, Mathematical Logic and Computation, Cambridge University Press 2022.
4. Homotopy Type Theory: Univalent Foundations of Mathematics, Institute for Advanced Studies, Princeton 2013; available at http://homotopytypetheory.org/book/.

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MA 209: Logic: classical, modal and intuitionistic (3:0)

Pre-requisites :

1. No prior knowledge of logic is assumed.
2. Background in reading and doing mathematical proofs will be assumed.

This course is an introduction to standard material in logic, based on classical first-order logic, after which it ventures into modern treatments of some non-classical logics. Although other proof methods will be discussed, the emphasis will be on proofs using tableaus.

Topics:

• First-order logic: First-order languages, deduction and truth, models, Smullyan-style tableaus, completeness and compactness theorems.
• Modal logics: Kripke frames, characterization of frame conditions, tableaus, completeness, finite model property, decision procedures for validity.
• First-order modal logics: Kripke frames with constant and varying domains, tableaus, rigid and flexible designators, non-designating terms, definite descriptions, ontological arguments.
• Intuitionistic logic: Kripke frames, tableaus, completeness.

Suggested books :

1. I. Chiswell and W. Hodges, Mathematical logic, Oxford Univ Press, 2007.
2. M.C. Fitting, First-order logic and automated theorem proving, Springer, 2nd edition, 1996.
3. M.C. Fitting, Proof methods for modal and intuitionistic logics, Reidel, 1983.
4. M.C. Fitting and R. Mendelsohn, First-order modal logic, Kluwer, 1998.

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MA 210: Logic, Types and Spaces (3:0)

Pre-requisites :

1. No prior knowledge of logic is assumed.
2. Some background in algebra and topology will be assumed.
3. It will be useful to have some familiarity with programming.

This course is an introduction to logic and foundations from both a modern point of view (based on type theory and its relations to topology) as well as in the traditional formulation based on first-order logic.

Topics:

• Basic type theory: terms and types, function types, dependent types, inductive types.
• First order logic: First order languages, deduction and truth, Models, Godel’s completeness and compactness theorems.
• Godel’s incompleteness theorem
• Homotopy Type Theory: propositions as types, the identity type family, topological view of the identity type, foundations of homotopy type theory.
• Most of the material will be developed using the dependently typed language Idris. Connections with programming in functional languages will be explored.

Suggested books :

1. Homotopy Type Theory: Univalent Foundations of Mathematics, Institute for Advanced Studies, Princeton 2013; available at http://homotopytypetheory.org/book/.
2. Manin, Yu. I., A Course in Mathematical Logic for Mathematicians, Second Edition, Graduate Texts in Mathematics, Springer-Verlag, 2010.
3. Srivastava, S. M., A Course on Mathematical Logic, Universitext, Springer-Verlag, 2008

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MA 211: Matrix theory (3:0)

Vector spaces, Bases and dimension, Direct ums, linear transformations, Matrix algebra, Eigenvalues and eigenvectors, Cayley Hamilton Theorem, Jordan canonical form., Orthogonal matrices and rotations, Polar decomposition., Bilinear forms.

Suggested books :

1. Artin, M., Algebra, Prentice-Hall of India, 1994.
2. Hoffman, K and Kunze R., Linear Algebra, Prentice-Hall of India, 1972.
3. Halmos, P.R., Finite dimensional vector spaces, van Nostrand, 1974 .
4. Greub, W.H., Linear algebra, Springer-Verlag, 1967.

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MA 212: Algebra I (3:0)

Prerequisite courses for Undegraduates: UM 203

Part A: Group theory

1. Basic definitions, examples
2. Cyclic groups and its subgroups
3. Homomorphisms, quotient groups, isomorphism theorems
4. Group actions, Sylow’s theorems, simplicity of $A_n$ for $n\geq 5$
5. Direct and semi-direct products
6. Solvable and nilpotent groups
7. Free groups

Part B: Ring theory

1. Basic definitions, examples
2. Ring homomorphisms, quotient rings, properties of ideals
3. Localization, ring of fractions
4. The Chinese remainder theorem
5. Euclidean domains, principal ideal domains, unique factorization domains
6. Polynomial rings over fields, irreducibility criteria

Part C: Module theory

1. Basic definitions and examples
2. Homomorphisms and quotient modules
3. Direct sums and free modules
4. Tensor product of modules
5. Structure theorem of modules over PID’s and consequences
6. Noetherian rings and modules, Hilbert basis theorem

Suggested books :

1. Artin, Algebra, M. Prentice-Hall of India, 1994.
2. Dummit, D. S. and Foote, R. M., Abstract Algebra, McGraw-Hill, 1986.
3. Lang, S., Algebra (3rd Ed.), Springer, 2002.
4. Hungerford, Algebra, Graduate Texts in Mathematics 73, Springer Verlag, 1974.
5. Nathan Jacobson, Basic Algebra I & II, Dover, 2009.
6. Nathan Jacobson, Lectures in Abstract Algebra I, II & III, Graduate Text in Mathematics, Springer Verlag, 1951.

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MA 213: Algebra II (3:1)

Prerequisite courses: MA 212

Part A: Field theory

1. Theory of symmetric polynomials – Newton’s theorem
2. Basic theory of field extensions
3. Algebraic and transcendental extensions (and transcendence degree)
4. Construction with straight edge and compass; Gauss-Wantzel theorem
5. Algebraic closure – Steinitz’s theorem
6. Splitting fields, normal extensions
7. Separable extensions
8. Finite fields: construction, subfields, Frobenius
9. Primitive element theorem
10. Dedekind-Artin linear independence of (semi)group characters

Part B: Galois theory

1. Fundamental theorem of Galois theory (including Normal Basis Theorem)
2. Composite extensions and Galois group
3. Galois group of cyclotomic extensions, finite fields
4. Galois groups of polynomials, Fundamental theorem of Algebra
5. Solvable and radical extensions, insolvability of a quintic

Suggested books :

1. Artin, M., Algebra, Prentice Hall of India, 1994.
2. Dummit, D. S. and Foote, R. M., Abstract Algebra, McGraw-Hill, 1986.
3. Lang, S., Algebra (3rd Ed.), Springer, 2002.
4. Jonathan Alperin and Rowen Bell, Groups and Representations, Graduate Texts in Mathematics 162, Springer Verlag, 1995.
5. Hungerford, Algebra, Graduate Texts in Mathematics 73, Springer Verlag, 1974.
6. Galois Theory, Artin, E., University of Notre Dame Press, 1944.
7. Nathan Jacobson, Basic Algebra I & II, Dover, 2009.
8. Nathan Jacobson, Lectures in Abstract Algebra I, II & III, Graduate Text in Mathematics, Springer Verlag, 1951.

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MA 215: Introduction to Modular Forms (3:0)

Pre-requisites :

1. MA 224 (Complex Analysis) or equivalent

The modular group and its subgroups, the fundamental domain. Modular forms, examples, Eisenstein series, cusp forms. Valence (dimension) formula, Petersson inner product. Hecke operators. L-functios: definition, analytic continution and functional equation.

Suggested books :

1. Serre, J.P., A Course in Arithmetic, Graduate Texts in Mathematics no. 7, Springer-Verlag, 1996.
2. Koblitz, N., Introdution to Modular Forms, Graduate Texts in Mathematics no. 97, Springer-Verlag, 1984.
3. Iwaniec, H., Topics in Classical Automorphic Forms, Graduate Texts in Mathematics 17, AMS, 1997.
4. Diamond, F. and Schurman, J., A First Course in Modular Forms, Graduate Texts in Mathematics no. 228, Springer-Verlag, 2005.

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MA 216: Introduction to Graph Theory (3:0)

Graphs, subgraphs, Eulerian tours, trees, matrix tree theory and Cayley’s formula, connectedness and Menger’s theorem, planarity and Kuratowski’s theorem, chromatic number and chromatic polynomial, Tutte polynomial, the five-colour theorem, matchings, Hall’s theorem, Tutte’s theorem, perfect matchings and Kasteleyn’s theorem, the probabilistic method, basics of algebraic graph theory

No prerequisites are expected, but we will assume a familiarity with linear algebra.

Suggested books :

1. Adrian Bondy and U.S.R. Murty, Graph Theory, Graduate Texts in Mathematics, 244. Springer, New York, 2008, ISBN: 978-1846289699.
2. Reinhard Diestel, Graph theory (Third edition), Graduate Texts in Mathematics, 173. Springer-Verlag, Berlin, 2005. ISBN: 978-3540261827.
3. Douglas B. West, Introduction to graph theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996. ISBN: 0-13-227828-6.

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MA 217: Discrete Mathematics (3:0)

  Combinatorics: Basic counting techniques. Principle of inclusion and exclusion. Recurrence relations and generating functions. Pigeon-hole principle, Ramsey theory. Standard counting numbers, Polya enumeration theorem.

Graph Theory: Elementary notions, Shortest path problems. Eulerian and Hamiltonian graphs, The Chinese postman problem. Matchings, the personal assignment prolem. Colouring or Graphs.

Number Theory: Divisibility Arithmetic functions. Congruences. Diophantine equations. Fermat’s big theorem, Quadratic reciprocity laws. Primitive roots.  

Suggested books :

1. Bondy, J. A. and Muirty, U. S. R., Graph theory with applications, Elsevier-North Holland, 1976.
2. Burton, D., Elementary Number Theory, McGraw Hill, 1997.
3. Clark, J. and Holton, D. A., A first book at Graph Theory, World Scientific Cp., 1991.
4. Polya G. D., Tarjan, R. E. and Woods, D. R., Notes on Introductory Combinations, Springer-Verlag, 1990.

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MA 218: Number Theory (3:0)

Algebraic Number Theory: Algebraic numbers and algebraic integers, Class groups, Groups of units, Quadratic fields, Quadratic reciprocity law, Class number formula.

Analytic Number Theory: Fundamental theorem of arithmetic, Arithmetical functions, Some elementary theorems on the distribution of prime numbers, Congruences, Finite Abelian groups and their characters, Dirichlet theorem on primes in arithmetic progression.

Suggested books :

1. Narasimham, R., Raghavan, S., Rangachari, S. S. and Sunder Lal., Algebraic Number Theory, Lecture Notes in Mathematics, TIFR, 1966.
2. Niven, I. and Zuckerman, H. S., An Introduction to the Theory of numbers, Wiley Eastern Limited, 1989.
3. Apostol, T. M., Introduction to Analytic Number Theory, Springer International Student Edition, 1989.
4. Ireland, K. and Rosen, M., Classical Introduction to Modern Number Theory, Springer-Verlag (GTM), 1990.

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MA 219: Linear Algebra (3:1)

Prerequisite courses for Undegraduates: UM 102

Vector spaces: Definition, Basis and dimension, Direct sums. Linear transformations: Definition, Rank-nullity theorem, Algebra of linear transformations, Dual spaces, Matrices.

Systems of linear equations: Elementary theory of determinants, Cramer’s rule. Eigenvalues and eigenvectors, the characteristic polynomial, the Cayley- Hamilton Theorem, the minimal polynomial, algebraic and geometric multiplicities, Diagonalization, The Jordan canonical form. Symmetry: Group of motions of the plane, Discrete groups of motion, Finite groups of SO(3). Bilinear forms: Symmetric, skew symmetric and Hermitian forms, Sylvester’s law of inertia, Spectral theorem for the Hermitian and normal operators on finite dimensional vector spaces.  

Suggested books :

1. Artin, M., Algebra, Prentice Hall of India, 1994.
2. Halmos, P., Finite dimensional vector spaces, Springer-Verlag (UTM), 1987.
3. Hoffman, K. and Kunze, R., Linear Algebra (2nd Ed.), Prentice-Hall of India, 1992.

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MA 220: Representation theory of Finite groups (3:0)

Prerequisite courses: MA 212 and MA 219

Representation of finite groups, irreducible representations, complete reducibility, Schur’s lemma, characters, orthogonality, class functions, regular representations and induced representation, the group algebra.

Linear groups: Representation of the group $SU(2)$

Suggested books :

1. Etingof Pavel, Golberg Oleg, Hensel Sebastian, Liu Tiankai, Schwendner Alex, Vaintrob Dmitry, Yudovina Elena,, Introduction to representation theory. With historical interludes by Slava Gerovitch, Student Mathematical Library 59. American Mathematical Society. 2011.
2. J. P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics. Vol. 42. Springer-Verlag. New York-Heidelberg. 1977.

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MA 221: Analysis I - Real Analysis (3:0)

Prerequisite courses for Undegraduates: UM 204

Construction of the field of real numbers and the least upper-bound property. Review of sets, countable & uncountable sets. Metric Spaces: topological properties, the topology of Euclidean space. Sequences and series. Continuity: definition and basic theorems, uniform continuity, the Intermediate Value Theorem. Differentiability on the real line: definition, the Mean Value Theorem. The Riemann-Stieltjes integral: definition and examples, the Fundamental Theorem of Calculus. Sequences and series of functions, uniform convergence, the Weierstrass Approximation Theorem. Differentiability in higher dimensions: motivations, the total derivative, and basic theorems. Partial derivatives, characterization of continuously-differentiable functions. The Inverse and Implicit Function Theorems. Higher-order derivatives.  

Suggested books :

1. Rudin, W., Principles of Mathematical Analysis, McGraw-Hill, 1986.
2. Apostol, T. M., Mathematical Analysis, Narosa, 1987.

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MA 222: Analysis II - Measure and Integration (3:1)

Prerequisite courses: MA 221

(Additional )Prerequisite courses for Undegraduates: UM 204

Construction of Lebesgue measure, Measurable functions, Lebesgue integration, Abstract measure and abstract integration, Monotone convergence theorem, Dominated convergence theorem, Fatou’s lemma, Comparison of Riemann integration and Lebesgue integration, Product sigma algebras, Product measures, Sections of measurable functions, Fubini’s theorem, Signed measures and Radon-Nikodym theorem, Lp-spaces, Characterization of continuous linear functionals on Lp - spaces, Change of variables, Complex measures, Riesz representation theorem.  

Suggested books :

1. Royden, H. L., Real Analysis, Macmillan, 1988.
2. Folland, G.B., Real Analysis: Modern Techniques and their Applications (2nd Ed.), Wiley.
3. Hewitt, E. and Stromberg, K., Real and Abstract Analysis, Springer, 1969.

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MA 223: Functional Analysis (3:0)

Prerequisite courses: MA 222, MA 224 and MA 219

Basic topological concepts, Metric spaces, Normed linear spaces, Banach spaces, Bounded linear functionals and dual spaces, Hahn-Banach theorem. Bounded linear  operators, open-mapping theorem, closed graph theorem. The Banach- Steinhaus theorem. Hilbert spaces, Riesz representation theorem, orthogonal complements, bounded operators on a Hilbert space up to (and including) the spectral theorem for compact, self-adjoint operators.

Suggested books :

1. Rudin, Functional Anaysis (2nd Ed.), McGraw-Hill, 2006.
2. Yosida, K., Functional Anaysis (4th Edition), Narosa, 1974.
3. Goffman, C. and Pedrick, G., First Course in Functional Analysis, Prentice-Hall of India, 1995.

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MA 224: Complex Analysis (3:1)

Prerequisite courses: MA 221

(Additional )Prerequisite courses for Undegraduates: UM 204

Complex numbers, holomorphic and analytic functions, Cauchy-Riemann equations, Cauchy’s integral formula, Liouville’s theorem and proof of fundamental theorem of algebra, the maximum-modulus principle. Isolated singularities, residue theorem, Argument Principle. Mobius transformations, conformal mappings, Schwarz lemma, automorphisms of the disc and complex plane. Normal families and Montel’s theorem. The Riemann mapping theorem. If time permits - analytic continuation and/or Picard’s theorem.

Suggested books :

1. Ahlfors, L. V., Complex Analysis, McGraw-Hill, 1979.
2. Conway, J. B., Functions of One Complex Variable, Springer-veriag, 1978.

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MA 226: Complex Analysis II (3:0)

Harmonic and subharmonic functions, Green’s function, and the Dirichlet problem for the Laplacian; the Riemann mapping theorem (revisited) and characterizing simple connectedness in the plane; Picard’s theorem; the inhomogeneous Cauchy–Riemann equations and applications; covering spaces and the monodromy theorem.

Suggested books :

1. Narasimhan, R., Complex Analysis in One Variable, 1st ed. or 2nd ed. (with Y. Nievergelt), Birkhauser (2nd ed. is available in Indian reprint, 2004).
2. Greene, R.E. and Krantz, S.G., Functions Theory of One Complex Variable, 2nd ed., AMS 2002 (available in Indian reprint, 2009, 2011).

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MA 229: Calculus on Manifolds (3:0)

Prerequisite courses: MA 221

Functions of several variables, Directional derivatives and continuity, total derivative, mean value theorem for differentiable functions, Taylor’s formula. The inverse function and implicit function theorems, extreme of functions of several variables and Lagrange multipliers. Sard’s theorem. Manifolds: Definitions and examples, vector fields and differential forms on manifolds, Stokes theorem.

Suggested books :

1. Spivak, M., Calculus on Manifolds, W.A. Benjamin, co., 1965.
2. Hirsh, M.W., Differential Topology, Springer-Verlag, 1997.

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MA 231: Topology (3:1)

Prerequisite courses for Undegraduates: UM 204

Point-set topology: Open and closed sets,  continuous functions, Metric topology, Product topology, Connectedness and path-connectedness, Compactness, Countability axioms, Separation axioms, Complete metric spaces, Quotient topology, Topological groups, Orbit spaces.

The fundamental group: Homotopic maps, Construction of the fundamental group, Fundamental group of the circle, Homotopy type, Brouwer’s fixed-point theorem, Separation of the plane.  

Suggested books :

1. Armstrong, M. A., Basic Topology, Springer (India), 2004.
2. Munkres, K. R., Topology, Pearson Education, 2005.
3. Viro, O.Ya., Ivanov, O.A., Netsvetaev, N., and Kharlamov, V.M., Elementary Topology: Problem Textbook, AMS, 2008.

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MA 232: Introduction to algebraic topology (3:0)

Prerequisite courses: MA 231 and MA 212

The fundamental group: Homotopy of maps, multiplication of paths, the fundamental group, induced homomorphisms, the fundamental group of the circle, covering spaces, lifting theorems, the universal covering space, Seifert-van Kampen theorem, applications.

Simplicial Homology: Simplicial complexes, chain complexes, definitions of the simplicial homology groups, properties of homology groups, applications.

Suggested books :

1. Armstrong, M. A., Basic Topology, Springer (India), 2004.
2. Hatcher, A., Algebraic Topology, Cambridge University Press, 2002.
3. Kosniowski, C. A., First Course in Algebraic Topology, Cambridge Univ. Press, 1980.
4. Croom, F. H., Basic Concepts of Algebraic Topology, Springer-Verlag, 1978.

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MA 233: Differential Geometry (3:0)

Curves in Euclidean space: Curves in R3, Tangent vectors, Differential derivations, Principal normal and binomial vectors, Curvature and torsion, Formulae of Frenet.

Surfaces in R3: Surfaces, Charts, Smooth functions, Tangent space, Vector fields, Differential forms, Regular Surfaces, The second fundamental form, Geodesies, Parellel transport, Weingarten map, Curvatures of surfaces, Rules surfaces, Minimal surfaces, Orientation of surfaces.    

Suggested books :

1. do Carmo, M. P., Differential Geometry of curves and surfaces, Prentice-Hall, 1976.
2. Thorpe, J. A., Elementary topics in Differential Geometry, Springer-Verlag (UTM), 1979.
3. O'Neill, B., Elementary Differential Geometry, Academic, 1996.
4. Gray, A., Modern Differential Geometry of Curves and Surfaces, CRC Press, 1993.

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MA 234: Metric Geometry of Spaces and Groups (3:0)

Prerequisite courses: MA 231

Pre-requisites :

1. A first course in Topology (can be taken concurrently)

Metric geometry is the study of geometric properties such as curvature and dimensions in terms of distances, especially in contexts where the methods of calculus are unavailable, An important instance of this is the study of groups viewed as geometric objects, which constitutes the field of geometric group theory. This course will introduce concepts, examples and basic results of Metric Geometry and Geometric Group theory.

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MA 235: Introduction to differentiable manifolds (3:0)

Prerequisite courses: MA 221

A review of continuity and differentiability in more than one variable. The inverse, implicit, and constant rank theorems. Definitions and examples of manifolds, maps between manifolds, regular and critical values, partition of unity, Sard’s theorem and applications. Tangent spaces and the tangent/cotangent bundles, definition of general vector bundles, vector fields and flows, Frobenius’ theorem. Tensors, differential forms, Lie derivative and the exterior derivative, integration on manifolds, Stokes’ theorem. Introduction to de Rham cohomology.

Suggested books :

1. Tu, Loren, An Introduction to Manifolds, Universitext, Springer-Verlag 2011.
2. John Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, Springer-Verlag 2012.
3. Barden, Dennis and Thomas, Charles, An Introduction to Differential Manifolds, World Scientific 2003.
4. Spivak, Michael, Comprehensive Introduction to Differential Geometry, Vol 1, Publish or Perish, 2005.

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MA 237: Introduction to Tilings (3:0)

This course will be an introduction to the mathematical theory of tilings. The first part of the course will concern tilings of the Euclidean plane, and topics covered will include tilings by regular Euclidean polygons, Archimedean tilings, symmetry groups of planar tilings, substitution tilings, aperiodic tilings including the Penrose tiles and the hat tile. The second part of the course will concern tilings of the hyperbolic plane, including triangle groups, existence of weakly aperiodic tiles and semi-regular tilings. In the final part of the course, topics related to tilings on surfaces, conformal tilings, and higher-dimensional tilings (in Euclidean n-space and hyperbolic 3-space) will be discussed. Along the way, the course will cover the basic notions needed from Euclidean and hyperbolic geometry, group theory, topology, and the theory of Riemann surfaces.

Suggested books :

1. Colin Adams, The Tiling Book, American Mathematical Society, 2022.

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MA 241: Ordinary Differential Equations (3:1)

Prerequisite courses: MA 221

(Additional )Prerequisite courses for Undegraduates: UM 204

Basics concepts:Introduction and examples through physical models, First and second order equations, general and particular solutions, linear and nonlinear systems, linear independence, solution techniques. Existence and Uniqueness Theorems :Peano’s and Picard’s theorems, Grownwall’s inequality, Dependence on initial conditions and associated flows. Linear system:The fundamental matrix, stability of equilibrium points, Phase- plane analysis, Sturm-Liouvile theory . Nonlinear system and their stability:Lyapunov’s method, Non-linear Perturbation of linear systems, Periodic solutions and Poincare- Bendixson theorem.

Suggested books :

1. Hartman, Ordinary Differential Equations, P. Birkhaeuser, 1982.
2. Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations, Tata McGraw-Hill, 1972.
3. Perko, L., Differential Equations and Dynamical Systems, Springer-Verlag, 1991.

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MA 242: Partial Differential Equations (3:0)

Prerequisite courses: MA 241

First order partial differential equation and Hamilton-Jacobi equations; Cauchy problem and classification of second order equations, Holmgren’s uniqueness theorem; Laplace equation; Diffusion equation; Wave equation; Some methods of solutions, Variable separable method.

Suggested books :

1. Garabedian, P. R., Partial Differential Equations, John Wiley and Sons, 1964.
2. Prasad. P. and Ravindran, R., Partial Differential Equations, Wiley Eastern, 1985.
3. Renardy, M. and Rogers, R. C., An Introduction to Partial Differential Equations, Springer-Verlag, 1992.
4. Fritz John, Partial Differential Equations, Springer (International Students Edition), 1971.

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MA 246: Mathematical Methods (3:0)

Matrix Algebra: Systems of linear equations, Nullspace, Range, Nullity, Rank, Similarity, Eigenvalues, Eigenvectors, Diagonalization, Jordan Canonical form. Ordinary Differential Equations: Singular points, Series solution Sturm Liouville problem, Linear Systems, Critical points, Fundamental matrix, Classification of critical points, Stability.

Complex Variables: Analytic functions, Cauchy’s integral theorem and integral formula, Taylor and Laurent series, isolated singularities, Residue and Cauchy’s residue theorem chwarz lemma.

Suggested books :

1. Hoffman, K. and Kunze, R., Linear Algebra (2nd Ed.)
2. Herstein, I. N. and Winter, D. J., Matrix Theory and Linear Algebra, Macmillan, 1989.
3. Simmons G. F., Differential Equations, Tata McGraw-Hill, 1985.
4. Churchill, R. V., Complex Variables and Applications, McGraw-Hill, 1960.

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MA 251: Numerical Methods (3:0)

Numerical solution of algebraic and transcendental equations, Iterative algorithms, Convergence, Newton Raphson procedure, Solutions of polynomial and simultaneous linear equations, Gauss method, Relaxation procedure, Error estimates, Numerical integration, Euler-Maclaurin formula. Newton-Cotes formulae, Error estimates, Gaussian quadratures, Extensions to multiple integrals.

Numerical integration of ordinary differential equations: Methods of Euler, Adams, Runge-Kutta and predictor - corrector procedures, Stability of solution. Solution of stiff equations.

Solution of boundary value problems: Shooting method with least square convergence criterion, Quasilinearization method, Parametric differentiation technique and invariant imbedding technique.

Solution of partial differential equations: Finite-difference techniques, Stability and convergence of the solution, Method of characteristics. Finite element and boundary element methods.  

Suggested books :

1. Gupta, A. and Bose, S. C., Introduction to Numerical analysis, Academic Publishers, 1989.
2. Conte, S. D. and Carl de Boor., Elementary Numerical Analysis, McGraw-Hill, 1980.
3. Hildebrand, F. B., Introduction to Numerical Analysis, Tata McGraw-Hill, 1988.
4. Froberg, C. E., Introduction to Numerical Analysis, Wiley, 1965.

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MA 253: Numerical Methods for Partial Differential Equations (3:0)

Finite difference methods for two point boundary value problems, Laplace equation on the square, heat equation and symmetric hyperbolic systems in 1 D. Lax equivalence theorem for abstract initial value problems. Introduction to variational formulation and the Lax-Milgram lemma. Finite element methods for elliptic and parabolic equations.

Suggested books :

1. Smith, G. D., Numerical solution of partial differential equations: Finite Difference Methods, Calarendon Press, 1985.
2. Evans, G. Blackledge, J. and Yardley, P., Numerical methods of partial differential equations, Springer-Verlag, 1999.

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MA 254: Numerical Analysis (3:0)

Introduction: Floating point representation of numbers and roundoff errors, Interpolation Numerical integration.

Linear systems and matrix theory: Various factorizations of inversion of matrices, Condition number and error analysis.

Non-linear systems: Fixed point iteration, Newton-Rapson and other methods, Convergence acceleration.

Numerical methods for ODE: Introduction and analysis of Taylor, Runge-kutta and other methods.

Numerical methods for PDE: Finite difference method for Laplace, Heat and wave equations.  

Suggested books :

1. Faires, J. D. and Burden, R., Numerical Methods, Brooks/Cole Publishing Co., 1998.
2. Conte, S. D. and Carl de Boor., Elementary Numerical Analysis
3. Stoer, J. and Bilrisch, R., Introduction to Numerical Analysis, Springer- Verlag, 1993.
4. Iserlas, A., First course in the numerical analysis of differential equations, Cambridge, 1996.

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MA 261: Probability Models (3:0)

Sample spaces, events, probability, discrete and continuous random variables, Conditioning and independence, Bayes’  formula, moments and moment generating function, characteristic function, laws of large numbers, central limit theorem, theory of estimation, testing of hypotheses, linear models.  

Suggested books :

1. Ross, S.M. , Introduction to Probability Models, Academic Press 1993.
2. Taylor, H.M., and Karlin, S., An Introduction to Stochastic Modelling, Academic Press, 1994.

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MA 262: Introduction to Stochastic Processes (3:0)

1. Discrete parameter Markov Chains: Chapman-Kolmogorov equations, Classification of states, Limit Theorems, Examples: Random Walks, Gambler’s Ruin, Branching processes. Time reversible Markov chains. Simulations and MCMC (16 lectures)
2. Poisson processes, Definitions, and properties: interarrival and waiting time distributions, superposition and thinning, Nonhomogeneous Poisson process, Compound Poisson process. Simulation. (5 lectures)
3. Continuous time Markov Chains: Definition, Birth-Death processes, Kolmogorov backward and forward equations, Limiting probabilities, Time reversibility. Queueing Theory, Simulation. (10 lectures)
4. Renewal Theory. (3 lectures)
5. Brownian Motion. (6 lectures).

Suggested books :

1. Karlin and Taylor, A first course in Stochastic Processes, Academic Press; 2nd edition, 1975.
2. Sheldon Ross, Stochastic Processes, Wiley; 2nd edition, 2008.
3. Bhattacharya and Waymire, Stochastic Processes and Applications, Society for Industrial and Applied Mathematics, 2009.

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MA 263: Stochastic Finance I (3:0)

Financial market. Financial instruments: bonds, stocks, derivatives.  Binomial no-arbitrage pricing model: single period and multi-period models.  Martingale methods for pricing.  American options: the Snell envelope.  Interest rate  dependent assets: binomial models for interest rates, fixed income derivatives, forward measure and future.  Investment portfolio: Markovitz’s diversification.  Capital asset pricing model (CAPM).  Utility theory.

Suggested books :

1. Luenberger, D.V., Investment Science, Oxford University Press, 1998.
2. Shiryaev, A.N., Essentials of Stochastic Finance, World Scientific, 1999.
3. Shreve, S.E., Stochastic Calculus for Finance I:  The Binomial Asset pricing Model, Springer, 2005.

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MA 267: Introduction to Statistical Learning with Applications (3:0)

Exploratory Data Analysis and Descriptive Statistics, with basic introductory programming in R using tidyverse for data visualisation.

Sampling Distribution and Limit Theorems: Order Statistics, Chi^2, F, Student’s t. Sampling statistics from Normal Population, Law of Large numbers, Central Limit Theorem, Variance Stabilising transformation. Proofs via simulation in R.

Estimation: Method of Moments, Maximum Likelihood Estimate and Confidence intervals.

Hypothesis Testing: Binomial Test for proportion, Normal Test for mean when variance is known/unknown, two sample t-test for equality of means when variance is known.

Linear Models, Normal Equations, Gauss Markov Theorem, Testing of linear hypotheses. One-way and two-way classification models: ANOVA, Random effects. Emphasis on Numerical evaluation. Regularisation and Subset Selection methods.

Basics of Decision trees: Regression Tress, Classification trees and comparison with Linear Models.

Computational Optimal transport.

Applications from Epidemiology, Networks and Optimal transport.

Suggested books :

1. Siva Athreya, Deepayan Sarkar and Steve Tanner, Probability and Statistics with Examples Using R, Institute of Mathematical Statistics, Hayward, CA.
2. Sanford Weisberg, Applied Linear Regression, John Wiley and Sons, New York.
3. Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani, An Introduction to Statistical Learning, Springer-Verlag, New York.

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MA 277: Nonlinear Dynamical Systems and Chaos (3:0)

Conservative Systems: Hamiltonians, canonical transformations, nonlinear pendulum, perturbative methods, standard map, Lyapunov exponents, chaos, KAM theorem, Chirikov criterion.

Dissipative Systems: logistics map, period doubling, chaos, strange attractors, fractal dimensions, Smale horseshoe, coupled maps, synchronization, control of chaos.

Suggested books :

1. Lichtenberg, A. J. and Lieberman, M. A., Regular and Stochastic motion, Springer-Verlag, 1983.
2. Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems and bifurcations of vector fields, Springer-Verlag, 1983.

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MA 278: Introduction to Dynamical Systems Theory (3:0)

Linear stability analysis, attractors, limit cycles, Poincare-Bendixson theorem, relaxation oscillations, elements of bifurcation theory: saddle-node, transcritical, pitchfork, Hopf bifurcations, integrability, Hamiltonian systems, Lotka-Volterra equations, Lyapunov function & direct method for stability, dissipative systems, Lorenz system, chaos & its measures, Lyapunov exponents, strange attractors, simple maps, period-doubling bifurcations, Feigenbaum constants, fractals. Both flows (continuous time systems) & discrete time systems (simple maps) will be discussed. Assignments will include numerical simulations.

Prerequisites, if any: familiarity with linear algebra - matrices, and ordinary differential equations

Desirable: ability to write codes for solving simple problems.

Suggested books :

1. S. Strogatz, Nonlinear Dynamics and Chaos: with Applications to physics, Biology, Chemistry, and Engineering, Westview, 1994.
2. S. Wiggins, Introduction to applied nonlinear dynamics & chaos, Springer-Verlag, 2003.
3. K. Alligood, T. Sauer, & James A.Yorke, Chaos: An Introduction to Dynamical Systems, Springer-Verlag, 1996.
4. M.Tabor, Chaos and Integrability in Non-linear Dynamics, 1989.
5. L. Ya. Adrianova, Introduction to Linear Systems of Differential Equations, AMS 1995.
6. Morris W. Hirsch, Robert L. Devaney, Stephen Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press 2012.

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MA 302: Mechanics (3:0)

Pre-requisites :

1. Calculus on manifolds; rudiments of Lie theory (the equivalent of Chapter 1, Chapter 2, and Section 4.1 of "Foundations of mechanics" by Abraham and Marsden).

This is an introductory course on the foundations of mechanics, focusing mainly on classical mechanics. The laws of classical mechanics are most simply expressed and studied in the language of symplectic geometry. This course can also be viewed as an introduction to symplectic geometry. The role of symmetry in studying mechanical systems will be emphasized.

The core syllabus will consist of Lagrangian mechanics, Hamiltonian mechanics, Hamilton-Jacobi theory, moment maps and symplectic reduction. Additional topics will be drawn from integrable systems, quantum mechanics, hydrodynamics and classical field theory.

Suggested books :

1. Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
2. Vladimir I. Arnol’d, Mathematical methods of classical mechanics, Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989.
3. Ana Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001.
4. Jerrold E. Marsden and Tudor S. Ratiu, Introduction to mechanics and symmetry, second ed., Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1999.

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MA 303: Topics in Operator Theory (3:0)

$C^*$-algebras, Calkin algebra, Compact and Fredholm operators, Index spectral theorem, the Weyl-von Neumann-Berg Theorem and the Brown-Douglas-Fillmore Theorem.

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MA 304: Topics in Harmonic Analysis (3:0)

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MA 305: Lie Groups and Lie Algebras (3:0)

Lie groups, definition and examples, Invariant vector fields and the exponential map, The Lie algebra of a Lie group, Lie subgroups and Lie subalgebras, Correspondence between connected Lie subgroups and Lie subalgebras, Cartan’s theorem, Lie group and Lie algebra homomorphism and their correspondence, Covering space theory of Lie groups, Commutative Lie groups and classification of connected abelian Lie group, Adjoint representation, Normal subgroups and ideals, Lie Group action and Lie transformation Groups, Coset Spaces and homogeneous spaces, Complexification, Classical Lie groups and their examples (Linear groups, Orthogonal Groups, Unitary Groups, Compact symplectic groups, Non-compact symplectic group). Topological properties and fundamental groups of classical Lie groups, The Killing form, Nilpotent and Solvable Lie algebras, Semisimple Lie algebras, Compact Lie algebras

prerequisite: Basic knowledge of Differential Geometry and Algebraic topology

Suggested books :

1. S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press.
2. C. Chevalley, Theory of Lie groups, Dover.
3. F. Warner, Foundations of differentiable manifolds and Lie groups, Springer.
4. S. Kumaresan, A Course in Differential Geometry and Lie Groups, Trim.
5. A. Knapp, Lie groups beyond an Introduction, Birkhaeuser.
6. D. Bump, Lie groups, Springer.

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MA 306: Topics in Morse Theory (3:0)

Prerequisite courses: MA 232, MA 338

Pre-requisites :

1. Manifolds and submanifolds
2. critical points, Sard’s theorem
3. vector fields as differential equations
4. Riemannian metrics, exponential map."

Transversality, Morse functions, stable and unstable manifolds, Morse-Smale moduli spaces, the space of gradient flows, compactification of the moduli spaces of flows, Morse homology, applications.

Suggested books :

1. Michèle Audin, Mihai Damian, Morse Theory and Floer Homology, 2014, Springer-Verlag London.
2. J. Milnor, Morse Theory, Ann. of Math. Stud. 51, Princeton Univ. Press, Princeton, 1963..
3. L. Nicolescu, An invitation to Morse theory, http://www3.nd.edu/~lnicolae/Morse2nd.pdf.
4. M. Schwarz, Morse homology, Birkhäuser, Basel, 1993.
5. R. Cohen, Kevin Iga, Paul Norbury, Topics in morse theory, lecture notes, 2006.

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MA 307: Riemann Surfaces (3:0)

Prerequisite courses: Topology (MA 231), Complex Analysis (MA 224), Introduction to Algebraic Topology (MA 232) or equivalent courses.

Riemann surfaces are one-dimensional complex manifolds, obtained by gluing together pieces of the complex plane by holomorphic maps. This course will be an introduction to the theory of Riemann surfaces, with an emphasis on analytical and topological aspects. After describing examples and constructions of Riemann surfaces, the topics covered would include branched coverings and the Riemann-Hurwitz formula, holomorphic 1-forms and periods, the Weyl’s Lemma and existence theorems, the Hodge decomposition theorem, Riemann’s bilinear relations, Divisors, the Riemann-Roch theorem, theorems of Abel and Jacobi, the Uniformization theorem, Fuchsian groups and hyperbolic surfaces.

Suggested books :

1. H.M. Farkas and I. Kra, Riemann surfaces, Springer GTM 1992.
2. R. Miranda, Algebraic Curves and Riemann Surfaces, AMS Graduate Studies in Mathematics, 1995.
3. W. Schlag, A Course in Complex Analysis and Riemann surfaces, AMS Graduate Studies in Mathematics, 2014.

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MA 308: Basic Algebraic Geometry (3:0)

Prerequisite courses: Algebra II (MA 213).

Pre-requisites :

1. The course will assume that that the student is comfortable with Abstract Algebra at the level of Galois theory.
2. We will develop all the Commutative algebra that we will need.

The material to be covered will include:

• Affine algebraic sets: Zariski topology, irreducible components, Hilbert Nullstellensatz theorem, maps of algebraic sets
• Algebraic varieties: Definitions, affine algebraic varieties, projective varieties, morphisms
• Rational functions and rational maps
• Algebraic curves, B´ezout’s theorem * Reimann-Roch theorem

Suggested books :

1. William D. Fulton, Algebraic curves, available free (and legally) at http://www.math.lsa.umich.edu/ wfulton/CurveBook.pdf.

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MA 310: Algebraic Geometry I (3:0)

• Refresher on Commutative Algebra: localization, local rings, integral closure, Krull dimension.

• Zariski topology, Hochster’s characterization of Zariski topology, spectral spaces.

• Zariski spectrum as a frame

• Refresher on categories : Categories, functors, Yoneda Lemma, equivalence of categories, adjoints.

• Grothendieck sites : Zariski, '{e}tale and Nisnevich sites.

• Presheaves and Sheaves

• Locally ringed spaces and schemes

• Separated schemes, proper schemes, irreducible schemes, reduced schemes, integral schemes, noetherian schemes.

• Morphisms : separated, proper, finite morphisms, finite type morphisms, affine morphisms

• Sheaves of algebras : affine morphisms as sheaves of algebras

• Sheaves of modules over a scheme, Quasi-coherent and coherent sheaves

• Divisors and Line Bundles, Weil divisors, Cartier divisors, Line bundles on Projective spaces, Serre sheaves.

• Projective morphisms, ample and very ample line bundles

• Formal schemes

Suggested books :

1. Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
2. Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A.~Grothendieck, given at Harvard 1963/64. With an appendix by P.~Deligne. Lecture Notes in Mathematics, No. 20 Springer-Verlag, Berlin-New York 1966.

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MA 311: Algebraic Geometry II (3:0)

• Sheaves of differentials.

• Background on homological algebra : resolutions, derived functors, $\delta$-categories.

• Triangulated categories, Derived categories of abelian categories.

• Injective and flasque resolutions.

• Cohomology of sheaves of abelian groups

• Vanishing theorems for cohomology

• Serre’s criterion for affineness

• Čech cohomology

• Cohomology of projective space, twisting by Serre sheaves

• $Ext$ and $Tor$ for sheaves

• Serre duality theorem

• Schemes as functors of points, the idea of stacks

Suggested books :

1. Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
2. Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A.~Grothendieck, given at Harvard 1963/64. With an appendix by P.~Deligne. Lecture Notes in Mathematics, No. 20 Springer-Verlag, Berlin-New York 1966.

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MA 312: Commutative Algebra (3:0)

Noetherian rings and Modules, Localisations, Exact Sequences, Hom, Tensor Products, Hilbert’s Null-stellensatz, Integral dependence, Going-up and Going down theorems, Noether’s normalization lemma , Discrete valuation rings and Dedekind domains.

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MA 313: Algebraic Number Theory (3:0)

Prerequisite courses: MA 213

Number fields and rings of integers, Dedekind domains; prime factorization, ideal class group, finiteness of class number, Dirichlet’s unit theorem, cyclotomic fields, theory of valuations, local fields.

Suggested books :

1. Jurgen Neukirch, Algebraic Number theory, Springer, 1999.
2. Daniel A. Marcus, Number fields, Springer Universitext, 2018.
3. J.P Serre, Local fields, Springer GTM 67, 1979.

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MA 314: Topics in Commutative Algebra (3:0)

Abstract relations and Dickson’s Lemma; Hilbert Basis theorem, Buchberger Criterion for Grobner Bases and Elimination Theorem; Field Extensions and the Hilbert Nullstellensatz; Decomposition, Radical, and Zeroes of Ideals; Syzygies, Grobner Bases for Modules, Computation of Hom, Free Resolutions; Universal Grobner Bases and Toric Ideals.

Suggested books :

1. T. Becker and V. Weispfenning, Grobner Bases–a Computational Approach to Commutative Algebra, Springer 1993.
2. W.W. Adams and P. Loustaunau, An Introduction to Grobner Bases, Graduate Studies in Mathematics, Vol. 3, American Mathematical Society, 1994.
3. B. Sturmfels, Grobner bases and convex polytopes, American Mathematical Society 1996.

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MA 315: Lie Algebras and their Representations (3:0)

LIE ALGEBRAS AND THEIR REPRESENTATIONS

Finite dimensional Lie algebras, Ideals, Homomorphisms, Solvable and Nilpotent Lie algebras, Semisimple Lie algebras, Jordan decomposition, Kiling form, root space decomposition, root systems, classification of complex semisimple Lie algebras Representations Complete reducibility, weight spaces, Weyl character formula, Kostant, steinberg and Freudenthal formulas

Suggested books :

1. J E Humphreys, Introduction to Lie algebras and Representation theory, Springer-Verlag, 1972.
2. J P Serre, Complex Semisimple Lie Algebras, Springer, 2001.
3. Fulton. W., and Harris J., Representation theory, Springer-Verlag. 1991.

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MA 316: Introduction to Homological Algebra (3:0)

Polynomial ring, Projective modules, injective modules, flat modules, additive category, abelian category, exact functor, adjoint functors, (co)limits, category of complexes, snake lemma, derived functor, resolutions, Tor and Ext, dimension, local cohomology,group (co)homology, sheaf cohomology, Cech cohomology, Grothendieck spectral sequence, Leray spectral sequence.

Suggested books :

1. Cartan and Eilenberg, Homological Algebra
2. Weibel, Introduction to Homological Algebra
3. Rotman, Introduction to Homological Algebra

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MA 317: Introduction to Analytic Number Theory (3:0)

Pre-requisites :

1. MA 224 (complex Analysis) or equivalent
2. An introductory course in Number Theory, or Consent of instructor

Review of arithmetical functions, averages of arithmetical functions, elementary results on the distribution of prime numbers, Dirichlet characters, Dirichlet’s theorem on primes in arithmetic functions, Dirichlet series and Euler products, the Riemann zeta function and related objects, the prime number theorem. (Time permitting: advanced topics like sieves, bounds on exponential sums, zeros of functions. the circle method.)

Suggested books :

1. Apostol, T.M., Introduction to Analytic Number Theory, Springer-Verlag, 1976.
2. Davenport, H., Multiplicative Number theory, 3rd edition, Springer, 2000.

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MA 318: Combinatorics (3:0)

Pre-requisites :

1. Calculus, Linear algebra and some exposure to proofs and abstract mathematics.
2. Programming in Sage will be a part of every lecture. Students will need to bring a laptop with access to the IISc WLAN.

Counting problems in sets, multisets, permutations, partitions, trees, tableaux; ordinary and exponential generating functions; posets and principle of inclusion-exclusion, the transfer matrix method; the exponential formula, Polya theory; bijections, combinatorial identities and the WZ method.

Suggested books :

1. Herbert Wilf, Generatingfunctionology, ISBN-13 - 978-1568812793; Freely downloadable from http://www.math.upenn.edu/~wilf/DownldGF.html.
2. Richard P. Stanley, Enumerative Combinatorics: Volume 1 (Second Edition), ISBN-13 - 978-1107602625 Older version freely downloadable from http://www-math.mit.edu/~rstan/ec/ec1/.

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MA 319: Algebraic Combinatorics (3:0)

The algebra of symmetric functions, Schur functions, RSK algorithm, Murnaghan- Nakayama Rule, Hillman-Grassl correspondence, Knuth equivalence, jeu de taquim, promotion and evacuation, Littlewood-Richardson rules.

No prior knowledge of combinatorics is expected, but a familiarity with linear algebra and finite groubs will be assumed.

Suggested books :

1. Stanley, R., Enumerative Combinatorics, volume 2, Cambridge University Press, 2001.
2. Sagan, B., The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Graduate Texts in Mathematics vol. 203, Springer-Verlag, 2001.
3. Prasad, A., Representation Theory : A Combinational Viewpoint, Cambridge Studies in Advanced Mathematics vol. 147, 2014.
4. Stanley, R., Lecture notes on Topics in Algebraic Combinatorics

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MA 320: Representation theory of compact Lie groups (3:0)

Prerequisite courses: MA 223

Lie groups, Lie algebras, matrix groups , representations, Schur’s orthogonality relations, Peter-Weyl theorem, structure of compact semisimple Lie groups, maximal tori, roots and rootspaces, classification of fundamental systems Weyl group, Highest weight theorem, Weyl integration formula, Weyl’s character formula.

Suggested books :

1. V. S. Varadarajan, Lie groups, Lie algebras and their representations, Sringer 1984.
2. A. C. Hall, Lie groups, Lie algebras and representations, Springer 2003.
3. Barry Simon, Representations of finite and compact groups, AMS 1996.
4. A. W. Knapp, Representation theory of semismiple lie groups. An overview based on examples, Princeton university press 2002.

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MA 321: Analysis III (3:0)

  Theory of Distributions: Introduction, Topology of test functions, Convolutions, Schwartz Space, Tempered distributions.

Fourier transform and Sobolev-spaces:Definitions, Extension operators, Continuum and Compact imbeddings, Trace results.

Elliptic boundary value problems: Variational formulation, Weak solutions, Maximum Principle, Regularity results.

Suggested books :

1. Barros-Nato, An Introduction to the Theory of Distributions, Marcel Dekker Inc., New York, 1873.
2. Kesavan, S., Topics in Functional Analysis and Applications, Wiley Eastern Ltd., 1989.
3. Evans, L. C., Partial Differential Equations, Univ. of California, Berkeley, 1998.
4. Schwartz, L. Hermann, Theorie des Distributions, 1966.

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MA 322: Harmonic Analysis (3:0)

Harmonic Analysis on the Poincare disc-Fourier transform, Spherical functions, Jacobi transform, Paley-Wiener theorem, Heat kernels, Hardy’s theorem etc.,    

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MA 323: Operator Theory (3:0)

Review of basic notions from Banach and Hilbert space theory.

Bounded linear operators: Spectral theory of compact, self adjoint, and normal operators. Sturm-Liouville problems, Green’s function, Fredholm integral operators.

Unbound linear operators on Hilbert spaces: Symmetric and self adjoint operators, Spectral theory, Banach algebras, Gelfand representation theorem, $C^*$-algebras, Gelfand-Naimark-Segal construction.

Suggested books :

1. Conway, J. B., A course in Functional Analysis, Springer-Verlag, 1990.
2. Rudin, W., Functional Analysis, Tata Mcgraw-Hill, 1974.
3. Berberian, S. K., Lectures in Functional Analysis, Frederic Ungar, 1955.

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MA 324: Topics in Complex Analysis (3:0)

The general theory of holomorphic mappings between bounded domains, automorphisms of bounded domains, discussions on the non-existence of a classical Riemann Mapping Theorem in several variables, discussion of the various forms of the one-variable Riemann Mapping Theorem, the Rosay-Wong Theorem, other Riemann-Rosay-Wong-type results (e.g., the work of Pinchuk) to the extent that time permits.

Suggested books :

1. Krantz, S. G., Geometric analysis and function spaces, CBMS Regional Conference Series in Mathematics, 81 (A M S, Providence, USA).
2. Rudin, W., Function theory in the unit ball of $\mathbb{C^n}$, Grundlehren der Mathematischen Wissenschaften (Springer-Verlag, New York-Berlin, 1980).
3. Krantz., S. G., Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001.

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MA 325: Operator Theory II (3:0)

Sz.-Nagy Foias theory: Dilation of contractions on a Hilbert space, minimal isometric dilation, unitary dilation. Von Neumann’s inequality.

Ando’s theorem: simultaneous dilation of a pair of commuting contractions. Parrott’s example of a triple of contractions which cannot be dilated simultaneously. Creation operators on the full Fock space and the symmetric Fock space.

Operators spaces. Completely positive and completely bounded maps. Endomorphisms. Towards dilation of completely positive maps. Unbounded operators: Basic theory of unbounded self-adjoint operators.

Suggested books :

1. John B. Conway, A course in Functional Analysis, Springer, 1985.
2. Vern Paulson, Completely Bounded Maps and Dilations, Pitman Research Notes, 1986.

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MA 326: Fourier Analysis (3:0)

Prerequisite courses: MA 223

Introduction to Fourier Series; Plancherel theorem, basis approximation theorems, Dini’s Condition etc. Introduction to Fourier transform; Plancherel theorem, Wiener-Tauberian theorems, Interpolation of operators, Maximal functions, Lebesgue differentiation theorem, Poisson representation of harmonic functions, introduction to singular integral operators.

Suggested books :

1. Dym, H. and Mckean, H.P., Fourier Series and Integrals, 1972.
2. Stein, E.M., Singular Integrals and Differentiability Properties of Functions, 1970.
3. Stein, E.M., and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, 1975.
4. Sadosky, C., Interpolation of Operators and Singular integrals, 1979.

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MA 327: Topics in Analysis (3:0)

Pre-requisites :

1. Real analysis
2. Complex analysis
3. Basic probability
4. Linear algebra
5. Groups
6. It would help to know or to concurrently take a course in measure theory and /or functional analysis.

In this course we begin by stating many wonderful theorems in analysis and proceed to prove them one by one. In contrast to usual courses (where we learn techniques and see results as “applications of those techniques). We take a somewhat experimental approach in stating the results and then exploring the techniques to prove them. The theorems themselves have the common feature that the statements are easy to understand but the proofs are non-trivial and instructive. And the techniques involve analysis.

We intend to cover a subset of the following theoremes: Isoperimetric inequality, infinitude of primes in arithmetic progressions, Weyl’s equidistribution theorem on the circle, Shannon’s source coding theorem, uncertainty, principles including Heisenberg’s Wigner’s law for eigenvalue of a random matrix, Picard’s theorem on the range of an entire function, principal component analysis to reduce dimensionality of data.  

Suggested books :

1. Korner, I. T. W., Fourier Analysis (1st Ed.), Cambridge Univ., Press, 1988.
2. Robert Ash., Information Theory, Dover Special Priced, 2008.
3. Serre, J. P., A course in Arithmetic, Springer-Verlag, 1973.
4. Thangavelu, S., An Introduction to the Uncertainity Principle, Birkhauser, 2003.
5. Rudin W., Real and Complex Analysis (3rd Edition), Tata McGraw Hill Education, 2007.

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MA 328: Introduction to Several Complex Variables (3:0)

Preliminaries: Holomorphic functions in $C^n$ : definition , the generalized Cauchy integral formula, holomorphic functions: power series development(s), circular and Reinhardt domains, analytic continuation : basic theory and comparisons with the one- variable theory.

Convexity theory: Analytic continuation: the role of convexity, holomorphic convexity, plurisub-harmonic functions, the Levi problem and the role of the d-bar equation.

The d- bar equation: Review of distribution theory, Hormander’s solution and estimates for the d-bar operator.

Suggested books :

1. Lars Hormander, An Introduction to Complex Analysis in Several Variables, 3rd edition, North-Holland Mathematical Library, North-Holland, 1989.
2. Function Theory of Several Complex Variables, 2nd edition, Wadsworth & Brooks/Cole, 1992.
3. Raghavan Narasimhan, Several Complex Variables, Chicago Lectures in Mathematics Series, The University of Chicago Press, 1971.

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MA 329: Topics in Several Complex Variables (3:0)

Pre-requisites :

1. A first course in complex analysis at the level of MA 224 (i.e., our first course in complex analysis).
2. Students who are unsure of the contents of MA 224 (e.g., students who completed their M.Sc. elsewhere) and are interested in this course are encouraged to speak/write to the instructor.

This topics course is being run as an experiment in approaching the properties of holomorphic maps in several complex variables (SCV) in a self-contained manner (i.e., without requiring any prior exposure to SCV).

The course will begin with a complete and rigorous introduction to holomorphic functions in several variables and their basic properties. This will pave the way to motivating and studying some objects that are, perhaps, entirely indigenous to SCV: e.g., plurisubharmonic functions and invariant metrics. This will allow us to discuss the inequivalence of the (Euclidean) ball and the polydisc in higher dimensions, and to discuss appropriate analogues of the one-variable Riemann Mapping Theorem in higher dimensions.

Next, we shall study the properties of the Kobayashi metric (which is one of the invariant metrics mentioned above) and the Kobayashi distance. This will be used to study the behaviour of automorphisms of bounded domains and refinements of some of the results hinted at above – to the extent that time permits.

Suggested books :

1. L. Hormander, Complex Analysis in Several Variables, 3rd edition, North-Holland Publishing Co. Amsterdam, 1990.
2. M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, de Gruyter Expositions in Mathematics, No. 9, Walter de Gruyter, Berlin, 1993.

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MA 330: Topology - II (3:0)

Prerequisite courses: MA 231

TOPOLOGY - II

Point Set Topology: Continuous functions, metric topology, connectedness, path connectedness, compactness, countability axioms, separation axioms, complete metric spaces,  function  spaces, quotient  topology, topological groups, orbit

The fundamental group:  Homotopy  of  maps, multiplication of paths, the fundamental group, induced homomorphisms, the  fundamental group of the circle,  covering spaces, lifting theorems, the universal covering space, Seifert-Van Kampen theorem, applications.

Suggested books :

1. Armstrong, M. A., Basic Topology, Springer (India), 2004.
2. Hatcher, A., Algebraic Topology, Cambridge Univ. Press,  2002.
3. Janich, K., Topology, Springer-Verlag (UTM), 1984.
4. Kosniowski, C., A First Course in Algebraic Topology, Cambridge Univ. Press, 1980.
5. Munkres,  K. R., Topology, Pearson Education, 2005.

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MA 331: Topology and Geometry (3:0)

Prerequisite courses: MA 231

Manifolds: Differentiable manifolds, differentiable maps and tangent  spaces,  regular values and Sard’s theorem, vector fields, submersions  and immersions, Lie  groups,  the  Lie algebra of a  Lie  group.

Fundamental Groups: Homotopy of maps, multiplication of paths, the fundamental group, induced homomorphisms, the fundamental group of the circle, covering spaces, lifting theorems, the universal covering space, Seifert-Van Kampen theorem, applications.  

Suggested books :

1. Brickell, F. and Clark, R. S., Differentiable Manifolds, Van Nostrand Reinhold Co., London, 1970.
2. Guillemin, V. and Pollack, A., Differential Topology, Prentice Hall, 1974.
3. Kosniowski, C., A, First Course in Algebraic Topology, Cambridge Univ. Press, 1980.
4. Milnor, John W., Topology from the Differentiable Viewpoint, Princeton Landmarks in Mathematics, Princeton Univ. Press, 1997.
5. Munkres, J. R., Elements of Algebraic Topology, Addison-Wesley, 1984.

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MA 332: Algebraic Topology (3:0)

Prerequisite courses: MA 232

Homology : Singular homology, excision, Mayer-Vietoris theorem, acyclic models, CW-complexes, simplicial and cellular homology, homology with coefficients.

Cohomology : Comology groups, relative cohomology,cup products, Kunneth formula, cap product, orientation on manifolds, Poincare duality.

Suggested books :

1. Hatcher, A., Algebraic Topology, Cambridge Univ. Press, 2002 (Indian edition is available).
2. Rotman, J, An Introduction to Algebraic Topology, Graduate Texts in Mathematics, 119, Springer-Verlag, 1988.
3. Munkres, I. R., Elements of Algebraic Topology, Addison-Wesiley, 1984.

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MA 333: Riemannian Geometry (3:0)

Review of differentiable manifolds and tensors, Riemannian metrics, Levi-Civita connection, geodesics, exponential map, curvature tensor, first and second variation formulas, Jacobi fields, conjugate points and cut locus, Cartan-Hadamard and Bonnet Myers theorems. Special topics - Comparison geometry (theorems of Rauch, Toponogov, Bishop-Gromov), and Bochner techniques.

Suggested books :

1. Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine, Riemannian geometry, Third edition., Universitext. Springer-Verlag, Berlin, 2004.
2. Peter Petersen, Riemannian geometry, Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998.
3. John Lee, Riemannian Geometry - An introduction to curvature, Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997.

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MA 334: Introduction to Homotopy Type Theory (3:0)

Prerequisite courses: MA 210, MA 232, MA 332

Pre-requisites :

1. Algebraic Topology, Dependent Type Theory

This course introduces homotopy type theory, which provides alternative foundations for mathematics based on deep connections between type theory, from logic and computer science, and homotopy theory, from topology. This connection is based on interpreting types as spaces, terms as points and equalities as paths. Many homotopical notions have type-theoretic counterparts which are very useful for foundations.

Such foundations are far closer to actual mathematics than the traditional ones based on set theory and logic, and are very well suited for use in computer-based proof systems, especially formal verification systems.

The course will also include background material in Algebraic Topology (beyond a second course in Algebraic Topology).

Suggested books :

1. Homotopy Type Theory: Univalent Foundations of Mathematics, Institute for Advanced Studies, Princeton 2013.
2. Hatcher, A., Algebraic topology, Cambridge University Press, Cambridge, 2002.
3. Tutorial for the Lean Theorem Prover

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MA 335: Introduction to Hyperbolic Manifolds (3:0)

Prerequisite courses: MA 231

Pre-requisites :

1. Topology (MA 231)
2. Introduction to Algebraic Topology (MA 232) or equivalent.

This is an introduction to hyperbolic surfaces and 3-manifolds, which played a key role in the development of geometric topology in the preceding few decades. Topics that shall be discussed will be from the following list: Basic notions of Riemannian geometry, Models of hyperbolic space, Fuchsian groups, Thick-thin decomposition, Teichmüller space, The Nielsen Realisation problem, Kleinian groups, The boundary at infinity, Mostow rigidity theorem, 3-manifold topology and the JSJ-decomposition, Statement of Thurston’s Geometrization Conjecture (proved by Perelman)

Suggested books :

1. Ratcliffe, Foundations of Hyperbolic Manifolds
2. Benedetti-Petronio, Lectures on Hyperbolic Geometry
3. Martelli, Introduction to Geometric Topology

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MA 336: Topics in Riemannian Geometry (3:0)

Prerequisite courses: MA 333 - Riemannian Geometry

Bochner formula, Laplace comparison, Volume comparison, Heat kernel estimates, Cheng-Yau gradient estimates, Cheeger-Gromoll splitting theorem, Gromov-Haudorff convergence, epsilon regularity, almost rigidity, quantitative structure theory of Riemannian manifolds with Ricci curvature bounds. If time permits, we will discuss the proof of the co-dimension four conjecture due to Cheeger and Naber.

Suggested books :

1. Peter Petersen, Riemannian geometry, Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998.
2. Richard Schoen and ST Yau, Lectures of Differential Geometry, International Press, 1997.
3. Jeff Cheeger, Degenerations of Riemannian metrics under Ricci curvature bounds, Publications of the Scuola Normale Superiore, Birkhauser, 2001.

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MA 337: Computer Assisted Topology and Geometry (1:3)

Prerequisite courses: MA 232

Pre-requisites :

1. familiarity with constructing proofs (e.g., having taken an Algebra/Linear Algebra/Analysis course in the mathematics department)
2. familiarity with programming, ideally in a functional language (such as Scala, Haskell, OCaml or Idris).

The goal of this course is to use computers to address various questions in Topology and Geometry, with an emphasis on arriving at rigorous proofs. The course will consist primarily of projects which will be contributions to open source software written in the scala programming language.

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MA 338: Differentiable manifolds and Lie groups (3:0)

Pre-requisites :

1. Point set topology. A first course in algebraic topology is helpful but not necessary.
2. Real analysis in more than one variable.
3. Linear algebra.

Differentiable manifolds, differentiable maps, regular values and Sard’s theorem, submersions and immersions, tangent and cotangent bundles as examples of vector bundles, vector fields and flows, exponential map, Frobenius theorem, Lie groups and Lie algebras, exponential map , tensors and differential forms, exterior algebra, Lie derivative, Orientable manifolds, integration on manifolds and Stokes Theorem . Covariant differentiation, Riemannian metrics, Levi-Civita connection, Curvature and parallel transport, spaces of constant curvature.

Suggested books :

1. Spivak M., A comprehensive introduction to differential geometry (Vol. 1) (3rd Ed.), Publish or Perish, Inc., Houston, Texas, 2005.
2. Kumaresan S., A course in differential geometry and Lie groups, Texts and Readings in Mathematics, 22. Hindustan Book Agency, New Delhi, 2002.
3. Warner F., Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
4. Lee J., Introduction to smooth manifolds, Graduate Texts in Mathematics, 218., Springer, New York, 2013.

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MA 339: Geometric Analysis (3:0)

Pre-requisites :

1. A first course on manifolds (MA 338 should do).
2. Analysis (multivariable calculus, some measure theory, function spaces).
3. Functional analysis (The Hahn-Banach theorem, Riesz representation theorem, Open mapping theorem. Ideally, the spectral theory of compact self-adjoint operators too, but we will recall the statement if not the proof)

Basics of Riemannian geometry (Metrics, Levi-Civita connection, curvature, Geodesics, Normal coordinates, Riemannian Volume form), The Laplace equation on compact manifolds (Existence, Uniqueness, Sobolev spaces, Schauder estimates), Hodge theory, more general elliptic equations (Fredholmness etc), Uniformization theorem.

Suggested books :

1. Do Carmo, Riemannian Geometry
2. Griffiths and Harris, Principles of Algebraic Geometry
3. S. Donaldson, Lecture Notes for TCC Course “Geometric Analysis”
4. J. Kazdan, Applications of Partial Differential Equations To Problems in Geometry
5. L. Nicolaescu, Lectures on the Geometry of Manifolds
6. T. Aubin, Some nonlinear problems in geometry
7. C. Evans, Partial differential equations
8. Gilbarg and Trudinger, Elliptic partial differential equations of the second order
9. G. Szekelyhidi, Extremal Kahler metrics

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MA 340: Advanced Functional Analysis (3:0)

Prerequisite courses: MA 223

Banach algebras, Gelfand theory, $C^{*}$-algebras the GNS construction, spectral theorem for normal operators, Fredholm operators. The L-infinity functional calculus for normal operators.

Suggested books :

1. Conway, J.B., A Course in Functional Analysis, Springer, 1985.
2. Douglas, R. G., Banach Algebra Techniques in Operator Theory, Academic Press, 1972.

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MA 341: Matrix Analysis and Positivity (3:0)

Prerequisite courses: MA 219

Pre-requisites :

1. A course in linear algebra, and a course in calculus/real analysis.

This course explores matrix positivity and operations that preserve it. These involve fundamental questions that have been extensively studied over the past century, and are still being studied in the mathematics literature, including with additional motivation from modern applications to high-dimensional covariance estimation. The course will bring together techniques from different areas: analysis, linear algebra, combinatorics, and symmetric functions.

List of topics (time permitting):

1. The cone of positive semidefinite matrices. Totally positive/non-negative matrices. Examples of PSD and TP/TN matrices (Gram, Hankel, Toeplitz, Vandermonde, $\mathbb{P}_G$). Matrix identities (Cauchy-Binet, Andreief). Generalized Rayleigh quotients and spectral radius. Schur complements.

2. Positivity preservers. Schur product theorem. Polya-Szego observation. Schoenberg’s theorem. Positive definite functions to correlation matrices. Rudin’s (stronger) theorem. Herz, Christensen-Ressel. 

3. Fixed-dimension problem. Introduction and modern motivations. H.L. Vasudeva’s theorem and simplifications. Roger Horn’s theorem and simplifications.

4. Proof of Schoenberg’s theorem. Characterization of (Hankel total) positivity preservers in the dimension-free setting.

5. Analytic/polynomial preservers – I. Which coefficients can be negative? Bounded and unbounded domains: Horn-type necessary conditions.

6. Schur polynomials. Two definitions and properties. Specialization over fields and for real powers. First-order approximation.

7. Analytic/polynomial preservers – II. Sign patterns: The Horn-type necessary conditions are best possible. Sharp quantitative bound. Extension principle I: dimension increase.

8. Entrywise maps preserving total positivity. Extension principle II: Hankel TN matrices. Variants for all TP matrices and for symmetric TP matrices. Matrix completion problems.

9. Entrywise powers preserving positivity. Application of Extension principle I. Low-rank counterexamples. Tanvi Jain’s result.

10. Characterizations for functions preserving $\mathbb{P}_G$. Extension principle III: pendant edges. The case of trees. Chordal graphs and their properties. Functions and powers preserving $\mathbb{P}_G$ for $G$ chordal. Non-chordal graphs.

11. Cayley-Menger matrices. Connections to Gram matrices, GPS trilateration, simplex volumes, and Heron’s theorem.

Suggested books :

1. Rajendra Bhatia, Matrix Analysis, vol. 169 of Graduate Texts in Mathematics, Springer, 1997.
2. Rajendra Bhatia, Positive definite matrices, Princeton Series in Applied Mathematics, 2007.
3. Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, 1990.
4. Roger A. Horn and Charles R. Johnson, Topics in matrix analysis, Cambridge University Press, 1991.
5. Samuel Karlin, Total positivity, Stanford University Press, 1968.
6. Apoorva Khare, Matrix analysis and entrywise positivity preservers, Cambridge University Press + TRIM Series, 2022.

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MA 342: Partial Differential Equations II (3:0)

Introduction to distribution theory and Sobolev spaces, Fundamental solutions for Laplace, heat and wave operations.

Second order elliptic equations: Boundary value problems, Regularity of weak solutions, Maximum principle, Eigenvalues.

Semi group theory:Hille-Yosida theorem, Applications to heat, Schroedinger and wave equations.

System of first order hyperbolic equations: Bicharacteristics, Shocks, Ray theory, symmetric hyperbolic systems.

Suggested books :

1. Evans, L. C., Partial Differential Equations, AMS, 1998.
2. Kesavan, S., Topics in Functional Analysis and Applications, Wiley Eastern, 1988.
3. Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.
4. Prasad, P. and Ravindran, R., Partial Differential Equations, Willey Eastern, 1985.
5. Treves, J. E., Basic Linear Partial Differential Equations, Academic Press, 1975.

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MA 343: Complex analytic techniques in operator theory (3:0)

Prerequisite courses: MA 223

Ando dilation of a commuting pair of contractions, Distinguished varieties of the bidisc, Description of all distinguished varieties, Construction of a distinguished variety corresponding to a pair of commuting matrices, Sharpening of Ando’s inequality, Extending the sharpened Ando inequality to operators with finite dimensional defect spaces, The extension property, Holomorphic retracts.

Suggested books :

1. T. Ando, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963) 88–90..
2. Agler, Jim and McCarthy, John E., Distinguished varieties., Acta Math. 194 (2005), no. 2, 133–153..
3. Das, B. Krishna and Sarkar, Jaydeb, Ando dilations, von Neumann inequality, and distinguished varieties., J. Funct. Anal. 272 (2017), no. 5, 2114–2131..

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MA 344: Homogenization of Partial Differential Equations (3:0)

Review of Distributions, Sobolev spaces and Variational formulation. Introduction to Homogenization. Homogenization of elliptic PDEs. Specific Cases: Periodic structures and layered materials. Convergence Results: Energy method, Two-scale multi-scale methods, H-Convergence, Bloch wave method. General Variational convergence: G -convergence and G- convergence, Compensated compactness. Study of specific examples and applications  

Suggested books :

1. A. Bensoussan, J. L., Lions and G., Papanicolaon., Asymptotic Analysis for Periodic Structures, North Holland (1978).
2. G. Dal Maso, An introduction to $\\Gamma$ convergence, Birkauser (1993).,
3. V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer (1991).
4. E. Sanchez Palencia, Non homogeneous Media and Vibration Theory, Springer lecture Notes in Physics, 127 (1980).

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MA 345: Complex Manifolds (3:0)

Prerequisite courses: MA 224: Complex Analysis, MA 235: Introduction to Differentiable Manifolds

Pre-requisites :

1. Ideal to have some knowledge of Riemannian geometry.

Basic definitions and examples, Line bundles and divisors, sheaves and Cech cohomology, de Rham’s theorem, Kahler condition and consequences, Hodge Theorem, L^2 methods in complex geometry, Kodaira embedding theorem.

Suggested books :

1. Huybrechts, Daniel, Complex geometry. An introduction., Springer-Verlag, Berlin, 2005.
2. Griffiths, Phillip; Harris, Joseph, Principles of algebraic geometry. Reprint of the 1978 original., Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994.
3. Morrow, James; Kodaira, Kunihiko, Complex manifolds. Reprint of the 1971 edition with errata., AMS Chelsea Publishing, Providence, RI, 1994.

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MA 346: Ergodic theory (3:0)

Pre-requisites :

1. Measure theory
2. Elements of Functional analysis.
3. It would help to have some familiarity with Fourier analysis and some probability theory.

Measure preserving systems, Poincare recurrence, von Neumann ergodic theorem, Khintichine’s theorem, spectral theorem and applications to combinatorics, ergodicity, Birkhoff ergodic theorem, mixing, unique ergodicity, disintegration of measures, Furstenberg correspondence principle, Furstenberg-Sarkozy theorem, Jacobs-de Leevuw-Glicksberg decomposition theorem and application to Roth’s theorem, The Kronecker Factor. (Additional material: Bhattcharya’s proof of the periodic tiling conjecture in $\Z$^2)

Suggested books :

1. Peter Walters, An Introduction to Ergodic Theory, Springer-Verlag (1982).
2. Manfred Einsiedler and Thomas Ward, Ergodic Theory with a view Towards Number Theory, Springer-Verlag (2012).
3. Karl E. Petersen, Ergodic Theory, Cambridge university press (1983).

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MA 346a: Homogeneous dynamics (3:0)

Pre-requisites :

1. Measure theory (equivalent to MA222) will be assumed.
2. Familiarity with basic notions of differential geometry (covered in MA235) will be helpful.

The course is about the ergodic theory of actions by (subgroups of) semisimple Lie groups which arise as groups of isometries of non-compact symmetric spaces. Some of the main topics include Howe-Moore’s theorem on vanishing of matrix coefficients at infinity for unitary actions on Hilbert spaces, Moore’s ergodicity theorem, ergodic aspects of the geodesic flow, the horocycle flow and classification of ergodic invariant measures of the horocycle flow. Dani-Margulis’ proof of a stronger version of Oppenheim’s conjecture will be discussed at the end of the course as an application of topics covered. Topics from the theory of non-compact semisimple Lie groups including Cartan involution, restricted root spaces, Weyl chambers, Iwasawa decomposition, Cartan decomposition and Bruhat decomposition will be discussed in some detail. Basic topics from ergodic theory like ergodicity, strong mixing and the pointwise ergodic theorem will also be recalled.

Suggested books :

1. M. B. Bekka, M. Mayer, Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces, Cambridge University Press 2013.
2. Einsiedler, Ward, Homogeneous dynamics and applications

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MA 347: Advanced Partial Differential Equations and Finite Element Method (3:0)

Distribution Theory - Introduction, Topology of Test functions, Convolutions, Schwartz Space, Tempered Distributions, Fourier Transform;

Sobolev Spaces - Definitions, Extension Operators, Continuous and Compact Imbeddings, Trace results; Weak Solutions - Variational formulation of Elliptic Boundary Value Problems, Weak solutions, Maximum Principle, Regularity results;

Finite Element Method (FEM) - Introduction to FEM, Finite element solution of Elliptic boundary value problems.  

Suggested books :

1. L. Schwartz, Theorie des Distributions, Hermann, (1966).
2. S. Kesavan, Topics in Functional Analysis and applications, John Wiley & Sons (1989).
3. P. G. Ciarlet, Lectures on Finite Element Method, TIFR Lecture Notes Series, Bombay (1975).
4. J. T. Marti, Introduction to Finite Element Method and Finite Element Solution of Elliptic Boundary Value Problems, Academic Press (1986).

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MA 348: Topics in function theoretic operator Theory (3:0)

Prerequisite courses: MA 219, MA 222, MA 223 and MA 224

Banach algebras – Gelfand theory, L-infinity functional calculus for bounded normal operators, Pick - Nevanlinna and Caratheodory Interpolation problems, Distinguished varieties in the bidisc.

Suggested books :

1. Douglas, Ronald G., Banach algebra techniques in operator theory. Second edition, Graduate Texts in Mathematics, 179. Springer-Verlag, New York, 1998.
2. Conway, John B., A course in functional analysis. Second edition, Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990.
3. Agler, Jim; McCarthy, John Edward; Young, Nicholas, Operator analysis—Hilbert space methods in complex analysis, Cambridge Tracts in Mathematics, 219. Cambridge University Press, Cambridge, 2020.
4. Bhattacharyya, Tirthankar; Kumar, Poornendu; Sau, Haripada, Distinguished varieties through the Berger-Coburn-Lebow theorem., Anal. PDE 15 (2022), no. 2, 477–506.

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MA 349: Topics around the Grothendieck inequality (3:0)

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MA 350: Topics in Analytic Number Theory (3:0)

Pre-requisites :

1. Basics of number theory
2. Complex analysis
3. Preferably some familiarity with MA 352 (=Introduction to Analytic number theory)

Arithmetical functions, Primes in Arithmetic Progressions, Prime number theorem for arithmetic progressions and zeros of Dirichlet L-functions, Bombieri-Vinogradov theorem, Equidistribution, circle method and applications (ternary Goldbach in mind), the Large Sieve and applications, Brun’s theorem on twin primes.

(Further topics if time permits: more on sieves, automorphic forms and L-functions, Hecke’s L-functions for number fields, bounds on exponential sums etc.)

Suggested books :

1. H. Davenport, Multiplicative Number Theory, Springer GTM 74.
2. M. Ram Murty, Problems in Analytic Number Theory, Springer GTM 206.
3. H. Iwaniec and E. Kowalski., Analytic Number Theory, AMS Colloquium Publ. 53.

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MA 351: Semigroup Theory and Evolution Equations (3:0)

Semigroup Theory: Introduction, Continuous and Contraction Semigroups, Generators, Hille-Yoshida and Lumer-Philips Theorems

Evolution Equations: Semigroup Approach to Heat, Wave and Schrodinger Equations

Suggested books :

1. A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer Verlag (1983).
2. S. Kesavan, Topics in Functional Analysis and Application, Wiley Eastern (1989).
3. A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston (1969).
4. H. Brezis, Analyse Fonctionnelle, Theorie et Applications, Mason, Paris (1983).

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MA 352: Introduction to Analytic Number Theory (3:0)

Pre-requisites :

1. Introductory courses in basic number theory
2. Complex analysis

Review of arithmetical functions, Averages of arithmetical functions, Elementary results on the distribution of prime numbers, Dirichlet characters, Dirichlet’s theorem on primes in arithmetic functions, Dirichlet series and Euler products, Riemann zeta function and related objects, The prime number theorem.

(Time permitting: More advanced topics like Sieves, bounds on exponential sums, zeros of zeta functions, circle method etc.)

Suggested books :

1. H. Davenport., Multiplicative Number Theory, Springer GTM 74 (third ed.) 2000.
2. Tom. M. Apostol., Introduction to Analytic Number Theory, Springer-Verlag, 1976.

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MA 353: Elliptic Curves (3:0)

Pre-requisites :

1. Algebra I and II
2. a working knowledge of basic algebraic number theory

Elliptic curves are smooth projective curves of genus 1 with a marked point. Over a field of characteristic zero they are given by an equation of the form $y^2 = x^3+ax+b$. They are at the boundary of our (conjectural) understanding of rational points on varieties and are subject of many famous conjectures as well as celebrated results. They play an important role in number theory.

The course will begin with an introduction to algebraic curves. We will then study elliptic curves over complex number, over finite fields, over local fields of characteristic zero and finally over number fields. Our goal will be to prove the Mordell-Weil theorem.

Suggested books :

1. Joseph Silverman, The arithmetic of elliptic curves, Springer GTM 106, 2009.
2. Joseph Silverman and John Tate, Rational points on elliptic curves, Springer UTM, 1992.
3. J.W.S. Cassels, Lectures on elliptic curves, Cambridge University Press, 2012.

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MA 354: Topics in Number Theory (3:0)

Pre-requisites :

1. a good background in commutative algebra (inverse limits, $I$-adic completion, Galois theory, possibly some familiarity with Dedekind domains),
2. some previous knowledge of algebraic number theory should be useful.

The goal is to give an introduction to adeles and some of their uses in modern number theory, discussing also some topics which are not too common in textbooks.

Topics to be covered: absolute values and Ostrowski’s Theorem; classification of locally compact fields; definition of adeles and some applications (finiteness of class number and of the generators of the group of S-units; structure of modules over Dedekind domains; applications to the geometry of curves); an introduction to the Strong Approximation Theorem; adelic points of varieties and schemes; possibly other topics (depending on time left and interests of the audience; for example Tate’s thesis, quasi-characters of the idele class group and p-adic L-functions).

Suggested books :

1. J. W. S. Cassels and A. Fröhlich (editors), Algebraic Number Theory, Papers from the conference held at the University of Sussex, Brighton, September 1–17, 1965.
2. A. Weil, Basic Number Theory, Classics in Mathematics, Springer 1974.
3. B. Conrad, Weil and Grothendieck approaches to adelic points, Enseign. Math. (2) 58 (2012), no. 1-2, 61–97.

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MA 355: Topics in Geometric Topology: Geometric structures (3:0)

Pre-requisites :

1. Topology (MA 231)
2. Introduction to Algebraic Topology (MA 232) or equivalent
3. preferably MA 335 (Introduction to Hyperbolic Manifolds) or equivalent

This course would be a survey of fundamental results as well as current research. Topics will be related to the following areas: geometric structures on surfaces, hyperbolic 3-manifolds, Riemann surfaces and Teichmüller theory, and will focus on the various interactions between these fields. Students will be encouraged to explore open-ended questions and/or write related computer programs. The following is the course plan:

Part I

1. A review of hyperbolic structures on surfaces
2. A review of Teichmüller spaces and mapping class groups
3. The topology of the PSL(2,R) representation variety
4. The notion of a geometric structure or (G,X)-structure on a manifold

Part II

5. Translation structures on a surface
6. Holomorphic 1-forms and their periods
7. An introduction to Teichmüller dynamics

Part III

8. Complex projective structures on a surface
9. Surface group representations into PSL(2,C)
10. The Schwarzian derivative and holomorphic quadratic differentials
11. Measured laminations and Thurston’s grafting theorem

Part IV

12. Other geometric structures, including affine structures and real projective structures
13. The case of open surfaces.

Suggested books :

1. W. P. Thurston, Three-dimensional Geometry and Topology, Princeton University Press, 1997.
2. B. Martelli, An Introduction to Geometric Topology, CreateSpace Publishing, 2016.

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MA 356: Class field theory: a course in arithmetic geometry (3:0)

Pre-requisites :

1. Basic algebra, commutative rings, Noetherian rings, basic number theory,
2. basic knowledge of Galois theory of fields.
3. Number fields and finite fields.

Syllabus:

1. Review of Dedekind domains and rings of integers in number fields.
2. Topology of discrete valuation fields.
3. Group cohomology and Galois cohomology.
4. Brauer group.
5. Brauer group of local fields.
6. Hasse principle for Brauer group.
7. Norm subgroups and their openness.
8. Class field theory for local and global fields.
9. Class field theory for compact curves over finite fields.
10. Class field theory for open curves over finite fields.

Suggested books :

1. J. P. Serre, Local fields, Graduate Texts in Mathematics. Vol. 67. Springer-Verlag. New York-Heidelberg. 1979.
2. J. Milne, Lecture notes on class field theory

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MA 357: Topics in Representation Theory (3:0)

Pre-requisites :

1. working knowledge of local fields
2. Linear Algebraic groups (as covered in MA 379)
3. generalities of root systems and Coxeter groups

This course will be an introduction to Bruhat-Tits theory. Given a connected, reductive group $G$ over a non-archimedean local field $F$, the theory constructs a contractible topological space $B(G)$, called the Bruhat-Tits building of $G(F)$. This space has the structure of a poly-simplicial complex and the topological group $G(F)$ acts on the building via automorphisms that preserve this poly-simplicial structure. To each point $x$ in $B(G)$, one can associate various subgroups of $G(F)$, the most obvious one being the stabilizer of the point $x$. The building serves the purpose of organizing the various compact open subgroups of $G(F)$ and these subgroups play a tremendous role in the study of representations of $p$-adic groups.

Organization: The first part of the course will be on affine root systems, Tits’ systems, and the Tits building. Then, we will construct the Bruhat-Tits building and various associated objects for two examples: The group $SL(2)$ and the quasi-split group $SU(3)$. Finally, after a review of the theory of reductive groups over general fields, we will embark on the construction of the building of a connected, reductive group over a non-archimedean local field, first by doing it for quasi-split groups, and then “descending this construction” to the general case.

Suggested books :

1. F. Bruhat and J. Tits, Groupes reductifs sur un corps local., Publ. Math. IHES 41 (1972).
2. F. Bruhat and J. Tits, Groupes reductifs sur un corps local II., Publ. Math. IHES 60 (1984).
3. E. Landvogt, A compactification of the Bruhat-Tits building., Lecture Notes in Math., 1619 Springer-Verlag, Berlin, viii+152 pp. (1996).
4. T. Kaletha and G. Prasad, Bruhat-Tits theory - a new approach, New Math. Monogr. 44, Cambridge University Press, Cambridge, xxx+718 pp. (2023).

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MA 358: Topics in Number Theory 4 (p-adic L-functions) (3:0)

Pre-requisites :

1. a solid background in Algebraic Number Theory
2. some familiarity with elliptic curves and modular forms.
3. a good background in commutative algebra
4. some knowledge of L-functions (analytic number theory) will be preferable.

We plan to cover (possibly a subset of) the following topics:

1. Kubota—Leopoldt p-adic L-functions
2. P-adic measures
3. Leopoldt’s formula for L_p(1,chi)
4. P-adic L-functions of totally real fields (following Deligne—Ribet)
5. P-adic L-functions of ordinary elliptic curves and modular forms

Suggested books :

1. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions
2. Iwasawa, Lectures on P-Adic L-Functions
3. Washington, Introduction to cyclotomic fields
4. Lang, Cyclotomic fields
5. Diamond--Shurman, A First Course in Modular Forms
6. Manin, Parabolic points and Zeta functions of modular curves
7. Deligne—Ribet, Values of abelian L-functions at negative integers over totally real fields

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MA 358A: Topics in Number Theory 3 (Iwasawa theory) (3:0)

Pre-requisites :

1. MA 213 : Algebra 2
2. MA 313 : Algebraic Number Theory
3. This course will use material from MA 358 : Topics in Number Theory 2 ($p$-adic $L$-functions) and should be taken concurrently.

This course is an introduction to classical Iwasawa theory, up to the proof of the Iwasawa main conjecture following Mazur and Wiles. We will begin with a review of results from algebraic number theory, class field theory etc. This will be followed by a study of $\mathbb{Z}_p$ extensions of number fields. We will then concentrate on the cyclotomic $\mathbb{Z}_p$ extensions of number fields. This will be followed by formulation of the Iwasawa main conjecture. For this part we need knowledge of $p$-adic $L$-functions. If time permits we will see Wiles’s proof of the Iwasawa main conjecture.

Suggested books :

1. K. Iwasawa, On $\mathbb{Z}_l$-Extensions of Algebraic Number Fields, Annals of Mathematics Vol. 98, No. 2 (1973).
2. R. Greenberg, Iwasawa Theory -- Past and Present, Advanced Studies in Pure Mathematics 30 (2001).
3. P. Deligne, K. Ribet, Values of Abelian L-functions at Negative Integers over Totally Real Fields, Inventiones Mathematicae Vol. 59 (1980).
4. A. Wiles, The Iwasawa Conjecture for Totally Real Fields, Annals of Mathematics Vol. 131, No. 3 (1990).

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MA 359: Stochastic Dynamic Optimization (3:0)

Prerequisite courses: MA 361, MA 221, MA 231

The topic covered will be the control of discrete-time infinite state-space Markovian systems. These techniques appear frequently in the analysis and optimization of stochastic systems e.g. control of queues, resource allocation problems in networks, machine learning, reinforcement learning, operations research, etc. The course is aimed at students who work in applied probability, stochastic control, machine learning, networking. Course is divided into the following three parts:

• Control of Markovian systems that have countably infinite state-space.
• Continuous State-Space Systems: Measurability Questions, Control under Monotonicity Assumption, Control under Contraction Assumption. Borel Models, Borel Spaces, Analytic Sets, Imperfect State Observations Model.
• General (Irreducible) Markov Chains: Kernels, Transience and Recurrence, Embedded Renewal Processes, Positive and Null Recurrence, Control of Harris Chains.

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MA 360: Random Matrix Theory (3:0)

• Wigner’s semicircle law: (a) combinatorial method, (b) Stieltjes’ transform method, (c) Chatterjee’s invariance principle method.

• Gaussian unitary and orthogonal ensembles: (a) Exact density of eigenvalues. (b) Orthogonal polynomials and determinantal formulas leading to another proof of Wigner’s semicircle law.

• Tridiagonal reduction for GUE and GOE: (a) Another derivation of eigenvalue density. (b) Another proof of Wigner’s semicircle law. (c) Matrix models for Beta ensembles. (d) Selberg’s integral.

• Other models of random matrices - Wishart and Jacobi ensembles.

• Free probability: (a) Noncommutative probability space and free independence. (b) Combinatorial approach to freeness. (c) Limiting spectra of sums of random matrices.

• Non-hemitian random matrices: (a) Ginibre ensemble. (b) Circular law for matrices with i.i.d entries.

• Fluctuation behaviour of eigenvalues (if time permits).

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MA 361: Probability Theory (3:0)

Prerequisite courses: MA 222

Probability measures and randown variables, pi and lambda systems, expectation, the moment generating function, the characteristic function, laws  of large numbers, limit theorems, conditional contribution and expectation, martingales, infinitely  divisible laws and stable laws.

Suggested books :

1. Durrett, R., Probability: Theory and Examples (4th Ed.), Cambridge University Press, 2010.
2. Billingsley, P., Probability and Measure (3rd Ed.), Wiley India, 2008.
3. Kallenberg, O., Foundations of Modern Probability (2nd Ed.), Springer-Verlag, 2002.
4. Walsh, J., Knowing the Odds: An Introduction to Probability, AMS, 2012.

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MA 362: Stochastic Processes (3:0)

Prerequisite courses: MA361

First Construction of Brownian Motion, convergence in $C[0,\infty)$, $D[0,\infty)$, Donsker’s invariance principle, Properties of the Brownian motion, continuous-time martingales, optional sampling theorem, Doob-Meyer decomposition, stochastic integration, Ito’s formula, martingale representation theorem, Girsanov’s theorem, Brownian motion and the heat equation, Feynman- Kac formula, diffusion processes and stochastic differential equations, strong and weak solutions, martingale problem.

Suggested books :

1. P. Billingsley, Convergence of probability measures
2. Karatzas and Shreve, Brownian motion and stochastic calculus
3. Revuz and Yor, Continuous martingales and Brownian motion
4. A. Oksendal, Introduction to stochastic differential equations

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MA 363: Probability in higher dimensions (3:0)

Pre-requisites :

1. This is a graduate level topics course in probability theory.
2. Graduate level measure theoretic probability will be useful, but not a requirement.
3. Students are expected to be familiar with basic probability theory and linear algebra.
4. The course will be accessible to advanced undergraduates who have had sufficient exposure to probability and linear algebra.

This course will be aimed at understanding the behavior of random geometric objects in high dimensional spaces such as random vectors, random graphs, random matrices, and random subspaces, as well. Topics will include the concentration of measure phenomenon, non-asymptotic random matrix theory, chaining and Gaussian processes, empirical processes, and some related topics from geometric functional analysis and convex geometry. Towards the latter half of the course, a few applications of the topics covered in the first half will be considered such as community detection, covariance estimation, randomized dimension reduction, and sparse recovery problems.

Suggested books :

1. Roman Vershynin, High-dimensional probability: An introduction with Applications in Data Science, Cambridge Series in Statistical and Probabilistic Mathematics (Series Number 47), 2018.
2. Roman Vershynin, Introduction to the non-asymptotic analysis of random matrices, Compressed sensing, 210-268, Cambridge University Press, 2012.
3. Stéphane Boucheron, Gábor Lugosi, and Pascal Massart, Concentration Inequalities: A nonasymptotic theory of independence, Oxford University Press, 2013.
4. Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Springer Science & Business Media, 2013.
5. Avrim Blum, John Hopcroft, and Ravindran Kannan, Foundations of Data Science, Cambridge University Press, 2020.
6. Joel Tropp, An Introduction to Matrix Concentration Inequalities, Foundations and Trends in Machine Learning, Vol. 8, No. 1-2, pp 1-230, 2015..

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MA 364: Linear and Non-linear Time Series Analysis (3:0)

Linear time series analysis - modelling time series using stochastic processes, stationarity, autocovariance, auto correlation, multivariate analysis - AR, MA, ARMA, AIC criterion for order selection;

Spectral analysis - deterministic processes, concentration problem, stochastic spectral analysis, nonparametric spectral estimation (periodogram, tapering, windowing), multitaper spectral estimation; parametric spectral estimation (Yule-Walker equations, Levinson Durbin)(recursions);

Multivariate analysis - coherence, causality relations; bootstrap techniques for estimation of parameters;

Nonlinear time series analysis - Lyapunov exponents, correlation dimension, embedding methods, surrogate data analysis.     

Suggested books :

1. Box, G. E. P. and jenkins, G. M., Time series analysis, Holden-Day, 1976.
2. Jenkins, G. M. and Watts, D. G., Spectral analysis and its applications, Holden-Day, 1986.
3. Efron, B., The Jackknife, the bootstrap and other resampling plans, SIAM, 1982.
4. Parker, T. S. and Chua, L. O., Practical numerical algorithms for chaotic systems, Springer-Verlag, 1989.

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MA 365: Topics in Gaussian Processes (3:0)

Prerequisite courses: MA 361

A course in Gaussian processes. At first we shall study basic facts about Gaussian processes - isoperimetric inequality and concentration, comparison inequalities, boundedness and continuity of Gaussian processes, Gaussian series of functions, etc. Later we specialize to smooth Gaussian processes and their nodal sets , in particular expected length and number of nodal sets, persistence probability and other such results from recent papers of many authors.

Suggested books :

1. Robert Adler and Jonathan Taylor, Gaussian Random Fields, Springer, New York, 2007.
2. Svante Janson, Gaussian Hilbert Spaces, Cambridge University Press, Cambridge, 1997.
3. A. I. Bogachev, Gaussian Measures, American Mathematical Society, Providence, RI, 1998.
4. Michel Ledoux and Michel Talagrand, Probability in Banach spaces. Isoperimetry and processes, Springer-Verlag, Berlin, 2011.
5. Michel Ledoux, Isoperimetry and Gaussian analysis, St. Flour lecture notes-1994.

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MA 366: Stochastic Finance II (3:0)

Trading in continuous time : geometric Brownian motion model.  Option pricing : Black-Scholes-Merton theory.  Hedging in continuous time :  the Greeks. American options.  Exotic options.  Market imperfections.  Term-structure models.  Vasicek, Hull-White and CIR models.  HJM model. LIBOR model.  Introduction to credit Rsik Models:  structural  and intensity models.  Credit derivatives.

Suggested books :

1. Amman, M., Credit Risk Valuation, Second Edition, Springer, 2001.
2. Brigo, D and Mercurio, F., Interest Rate Models Theory and Practice, Second Edition, Springer, 2007 .
3. Shiryaev, A.N., Essentials of Stochastic Finance, World Scientific, 1999.
4. Shreve, S.E., Stochastic Calculus for Finance II : The continous Time Models, Springer, 2004.

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MA 367: Brownian Motion (3:0)

Prerequisite courses: MA 361

• Levy’s construction of Brownian motion.
• Invariance properties : Under scaling, rotation, time-reversal, conformal maps (dim=2), shifts (Markov property), random shifts (strong Markov property).
• Blumenthal’s and Kolmogorov’s zero-one law, Law of large numbers, Strassen’s law of iterated logarithm.
• Continuity properties: law of iterated logarithm, Levy’s theorem on modulus of Continuity of BM, Nowhere Holder continuity of order greater than 1/2.
• Hausdorff and Minkowski dimensions. Dimension computation of certain random fractals derived from Brownian motion (range, graph and zero set).
• Random walks and discrete harmonic functions. Skorokhod and Dubins embedding of random walks in Brownian motion, Donsker’s invariance principle. Brownian motion and harmonic functions.
• Recurrence and transience. What sets does Brownian motion hit? (Polar sets and Capacity).
• Stochastic integral and Ito’s formula. Martingales. Levy’s characterization of Brownian motion. Tanaka’s formula for Local time.
• Brownian motion in the plane : Conformal invariance, Winding number. Davis’ proof of Picard’s theorem for entire functions using Brownian motion. Distribution of the filling of Brownian motion in a simply connected domain (Virag’s lemma).
• Gaussian free field : Definition and basic properties. A synopsis of some recent advances due to Scott Sheffield and others involving the GFF.

Suggested books :

1. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1991.
2. A. Kallenberg, Foundation of Modern Prability Theory, Second Edition, Springer.

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MA 368: Topics in Probability and Stochastic Processes (3:0)

Prerequisite courses: MA361

Discrete parameter martingales: Conditional expectation. Optional sampling theorems. Doob’s inequalities. Martingale convergence theorems. Applications.

Brownian motion. Construction. Continuity properties. Markov and strong Markov property and applications. Donsker’s invariance principle. Further sample path properties.  

Suggested books :

1. Rick Durrett, Probability: theory and examples., Cambridge University Press, 2010..
2. David Williams, Probability with Martingales, Cambridge Univ., Press, 1991.
3. Peter Mörters and Yuval Peres, Brownian motion, Cambridge University Press, 2010..
4. Olav Kallenberg, Foundations of modern Probability. Second Edition, Springer-Verlag, 2002..
5. John Walsh, Knowing the Odds: An Introduction to Probability, AMS, 2012..

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MA 368a: Exclusion processes (3:0)

Prerequisite courses: MA261

Review of discrete and continuous time Markov chains, review of equilibrium and nonequilibrium statistical mechanics, Ising model in one dimension, Glauber dynamics, Bethe ansatz, Yang-Baxter equation, asymmetric simple exclusion processes with periodic and open boundary conditions, multispecies exclusion processes, zero range processes, Schur and Macdonald processes

Suggested books :

1. Rodney J. Baxter, Exactly solved models in statistical mechanics, Integrable Systems in Statistical Mechanics. May 1985, 5–63.
2. Bernard Derrida, An exactly soluble non-equilibrium system: The asymmetric simple exclusion process, Physics Reports 301 (1998), 65–83.
3. Arvind Ayyer, Exclusion processes with drift, LPS 2017 lecture notes.

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MA 369: Quantum Mechanics (3:0)

Origins, states, observables, interference, symmetries, uncertainty, wave and matrix mechanics, Measurement, scattering theory in 1 dimension, quantum computation and information, Prerequisites are analysis and linear algebra.

Suggested books :

1. Srinivas, M.D., Measurements and Quantum Probabilities, University Press, Hyderabad (2001).
2. John von Neumann and Robert T Beyer, Mathematical Foundations of Quantum Mechanics, Princeton Univ. Press (1996).
3. Leonard Schiff, Quantum Mechanics, McGraw Hill (Education) 2010.
4. Gerad Tesch, Mathematical Methods in Quantum Mechanics with applications to Schrodinger operators, Graduate Studies in Mathematics, 99 AMS, Providence, 2009.
5. Parthasarathy, K.R., Lectures on Quantum Computation, Quantum Error Correcting Codes and Information Theory, Narosa Publishers, 2006.
6. Parthasarathy, K.R., Mathematical Foundations of Quantum, Hindustan Book Agency, New Delhi.

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MA 370: Hermitian Analysis (3:0)

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MA 371: Control and Homogenization (3:0)

Pre-requisites :

1. Sobolev spaces
2. Elliptic boundary value problems
3. Heat and wave equations
4. Variational formulation and semigroup theory

Optimal Control of PDE:Optimal control problems governed by elliptic equations and linear parabolic and hyperbolic equations with distributed and boundary controls, Computational methods. Homogenization:Examples of periodic composites and layered materials. Various methods of homogenization. Applications and Extensions:Control in coefficients of elliptic equations, Controllability and Stabilization of Infinite Dimensional Systems, Hamilton- Jacobi-Bellman equations and Riccati equations, Optimal control and stabilization of flow related models.

Suggested books :

1. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley, 1968.
2. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1991.
3. L. Lions, Controlabilite exact et Stabilisation des systemes distribues, Vol. 1, 2 Masson, Paris, 1988.
4. Bardi, I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, 1997.
5. Kesavan, Topics in Functional Analysis and Applications, Wiley-Eastern, New Delhi, 1989.
6. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhauser, 1993.

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MA 372: Abstract Harmonic Analysis (3:0)

Prerequisite courses: MA 223

Topological groups, locally compact groups, Haar measure, Modular function, Convolutions, homogeneous spaces, unitary representations, Gelfand-Raikov Theorem. Functions of positive type, GNS construction, Potrjagin duality, Bochner’s theorem, Induced representations, Mackey’s impritivity theorem.

Suggested books :

1. Folland, G. B., A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, 1995.
2. Hewitt, E and Ross, K., Abstract Harmonic Analysis, Vol. 1, Springer 1979.
3. Gaal, S.A., Linear Analysis and Representation Theory, Dover, 2010.

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MA 373: Optimal Control of Distributed Systems (3:0)

Pre-requisites :

1. Functional analysis
2. PDE and advanced PDE

Introduction and examples, optimal control problems governed by elliptic and parabolic systems, adjoint systems, optimality conditions, optimal control and optimality systems for other PDEs like Stokes systems.

Suggested books :

1. Fredi Troltzsch, Optimal control of partial differential equations; Theory, Methods and Applications, Graduate Studies in Mathematics, Volume 112, AMS (2010).
2. J. L. Lions, Optimal control of systems governed by partial differential equations, Springer Verlag, 1971.
3. J. L. Lions, Controlabilite exacte, perturbations et stabilisation de systems distribues, Tome 1 and 2, Research in applied mathematics, Vol. 8 and 9, Masson, Paris, 1988.
4. V. Barbu, Mathematical methods in optimization of differential systems, Mathematics and its applications, Vol. 310 Kluwer academic Publishers, 1994.

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MA 374: Introduction to the Calculus of Variations (3:0)

Prerequisite courses: MA 221, MA 222, MA 223

Course Objective To provide a gentle introduction to the direct methods in Calculus of Variations concerning minimizations problems, excluding minmax methods. The focus will be on illustrating the main methods using important prototype examples and not on proving the most general or the sharpest results.

Target audience This course is primarily intended for students of Mathematics with interests in Analysis, PDE and/or differential Geometry and geometric analysis, especially minimal surfaces. However, students of physics and different branches of engineering ( especially mechanical engineering ) and economics would still probably find a portion of the course useful for them.

Course contents and outline Our goal is to cover the following topics:

• Classical Methods: Euler-Lagrange equations, Lagrangian and Hamiltonian formulations, Hamilton-Jacobi equations, constrained problems and Lagrange multipliers, An illustration of the methods: Geodesic curves.

• Direct Methods: Dirichlet integral and $p$-Dirichlet Integral: Existence of minimizers: Existence theorem for convex functional with lower order terms, examples and counterexamples, weak form of the Euler-Lagrange equations, Dirichlet Principle, weak continuity of determinants. Regularity questions.

• Plateau’s problem and minimal surfaces: Parametric Plateau’s problem: Douglas-Courant-Tonelli method, Regularity, uniqueness and nonuniqueness, Nonparametric minimal surfaces, Isoperimetric inequality.

Suggested books :

1. Dacorogna, B., Introduction to the calculus of variations, third ed., Imperial College Press, London, 2015.
2. Jost, J., and Li-Jost, X., Calculus of variations, vol.64 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1998.
3. Struwe, M., Plateau's problem and the calculus of variations, vol.35 of Mathematical Notes, Princeton University Press, Princeton, NJ, 1988.

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MA 375: Algebraic Graph Theory (3:0)

Basic definitions in graph theory, line graphs, some matrices related to graphs and their spectral properties, the Perron-Frobenius theorem, Cauchy’s interlacing theorem, strongly regular graphs, the Laplacian matrix, cuts and flows.

Suggested books :

1. Chris Godsil and Gordon Royle, Algebraic graph theory, Graduate Texts in Mathematics, 207. Springer-Verlag, New York, 2001.
2. R. B. Bapat, Graphs and matrices, Hindustan Book Agency TRIM 58, New Delhi, 2014..

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MA 376: Extremal Combinatorics (3:0)

This course will explain ideas for solving the problem of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects that satisfies certain requirements. For example, how many edges can a graph on n vertices have if it does not contain a triangle?

Topics: Double counting, pigeonhole principle, Erdos-Szekeres theorem, Mantel’s theorem, Turan’s theorem, Dirichlet’s theorem. Ramsey theorem for graphs: bounds on Ramsey numbers. Extremal set theory: intersecting families, Erdos-Ko-Rado theorem, maximal intersecting families, Furedi’s theorem. Chains and antichains: Dilworth’s theorem, Sperner’s theorem, Bollobas’ theorem.

Algebraic Methods: Even-odd town problem, Fisher’s inequality, 2-distance sets in $\mathbb{R}^n$, bounds on the number of sets with restricted pairwise intersections. Probabilistic methods: lower bounds for Ramsey numbers, tournaments, dominating sets, sum-free sets of integers.

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MA 379: Linear Algebraic Groups (3:0)

Pre-requisites :

1. Commutative algebra
2. Some familiarity with basic algebraic geometry and Lie algebras will be helpful, but it will be covered in the course as required.

Basic notions of linear algebraic groups (connected components, orbits, Jordan decomposition), Lie algebras, algebraic tori, solvable and unipotent groups, parabolic and Borel subgroups, representations of linear algebraic groups, reductive and semi-simple groups, the Weyl group, root systems and root datum, classification of connected reductive groups over an algebraically closed field.

Suggested books :

1. T. A. Springer, Linear Algebraic Groups, Modern Birkhaeuser Classics, 2nd edition, 1998.
2. Armand Borel, Linear Algebraic Groups, Springer-Verlag GTM 126, 2nd edition, 1991.
3. James Humphreys, Linear Algebraic Groups, Springer-Verlag GTM 21, 1975.

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MA 380: Introduction to Complex Dynamics (3:0)

The dynamics alluded to by the title of the course refers to dynamical systems that arise from iterating a holomorphic self-map of a complex manifold. In this course, the manifolds underlying these dynamical systems will be of complex dimension 1. The foundations of complex dynamics are best introduced in the setting of compact spaces. Iterative dynamical systems on compact Riemann surfaces other than the Riemann sphere – viewed here as the one-point compactification of the complex plane – are relatively simple. We shall study what this means. Thereafter, the focus will shift to rational functions: these are the holomorphic self-maps of the Riemann sphere. Along the way, some of the local theory of fixed points will be presented. In the case of rational maps, some ergodic-theoretic properties of the orbits under iteration will be studied. The development of the latter will be self-contained. The properties/ theory coverd will depend on the time available and on the audience’s interest.

Suggested books :

1. J. Milnor, Dynamics in One Complex Variable, Annals of Mathematics Studies no. 160, Princeton University Press, 2006.
2. A.F. Beardon, Iteration of Rational Functions: Complex Analytic Dynamical Systems, Graduate Texts in Mathematics no. 132, Springer-Verlag, 1991.

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MA 381: Topics in Several Complex Variables - II (3:0)

Prerequisite courses: MA 224 (Complex Analysis).

Pre-requisites :

1. Familiarity with the following concepts: differentiable manifolds, tangent and cotangent bundles, and systems of (first order) PDEs.
2. Although this is a Topics in Several Complex Variables course, MA 328 (Introduction to SCV) is not a prerequisite. All the necessary concepts from SCV will be rigorously introduced along the way.

The aim of this course is to provide an introduction to CR (Cauchy Riemann/Complex Real) geometry, which is broadly the study of the structure(s) inherited by real submanifolds in complex spaces. We will first give a parallel introduction to the fundamental objects of SCV and CR geometry. These include holomorphic functions in several variables, CR manifolds (embedded and abstract) and CR functions. Next, we will cover some examples, results, and techniques from the following range of topics.

a) embeddability of abstract CR structures;

b) holomorphic extendability of CR functions;

c) CR singularities.

Wherever possible (and time permitting), we will highlight the connections of this field to other areas of analysis and geometry. For instance, abstract CR structures will be discussed in the broader context of involutive structures on smooth manifolds.

Suggested books :

1. A. Boggess, CR Manifolds and the tangential Cauchy-Riemann complex, CRC Press, Boca Raton, FL (1991).
2. M. S. Baouendi, P. Ebenfelt, and L. P. Rothschild, Real Submanifolds in Complex Space and their Mappings, Princeton Math. Series., Princeton Univ. Press (1999).

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MA 382: Special Topics in Operator Theory (3:0)

Pre-requisites :

1. MA 233

Quick  review  of  the  theory  of  bounded  operators  on  a  Hilbert  space  compactoperators,  Fredholm  operators,  spectral  theory  of  compact  self-adjoint  operators.

Spectral  theorem  and  functional  calculus  for  a  bounded  self-adjoint  operator.

Unbounded  operators  -  examples,  spectral  theorem  and  functional  calculus  for an  unbounded  self-adjoint  operator.

Schatten  p-classes- interpolation.

Krein’s  spectral  shift  function  

Suggested books :

1. Conway,  J.  B., A  course  in  functional  analysis, Springer.
2. Amrein,  W. O., Jauch,  J.  M.,  and  Sinha,  K.  B., Scattering  theory  in  quantum  mechanics, W. A.  Benjamin  Inc.

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MA 383: Introduction to Minimal Surfaces (3:0)

• Serre-Frenet formula for curves, Parametric surfaces, Isothermal parameters,

• Gauss Map, Gaussian Curvature, Mean curvature, Area functional etc.

• Surfaces that locally minimise area in Euclidean space (minimal surfaces).

• Harmonic coordinates in isothermal parameters. Examples of minimal surfaces.

• Minimal surfaces with boundary: Plateau’s problem.

• The gauss map for minimal surfaces with some examples.

• The Weierstrass-Enneper representation of minimal surfaces. Many more examples of minimal surfaces.

• Conjugate minimal surfaces. One parameter family of isometric minimal surfaces. the Bjorling problem and Schwartz’s solution to it.

If time permits:

• Surfaces that locally maximise area in Lorenztian space (maximal surfaces). A lot of examples and analogous results, as in minimal surface theory, for maximal surfaces.

• Connection betwen minimal and maximal surfaces and Born Infeld solitions.
• Constant mean curvature surfaces of non-zero mean curvature (the optimization problem they solve)  

Suggested books :

1. Dierkes, Hildebrandt, Kuster, Wohlrab, Minimal Surfaces I
2. Manfredo Do Carmo, Differential Geometry of curves and surfaces
3. Robert Osserman, A survey of minimal surfaces
4. Yi Fang, Lectures on Minimal Surfaces in $\R^n$
5. K. Kenmotsu, Surfaces of constant mean curvature

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MA 384: Mathematical Physics (3:0)

The purpose of this course will be to understand (to an extent) and appreciate the symbiotic relationship that exists between mathematics and physics. Topics to be covered can vary but those in this edition include: a brisk introduction to basic notions of differential geometry (manifolds, vector fields, metrics, geodesics, curvature, Lie groups and such), classical mechanics (Hamiltonian and Lagrangian formulations, n-body problems with special emphasis on the n=3 case) and time permitting, an introduction to integrable systems.

Suggested books :

1. Abraham and Marsden, Foundations of Mechanics, AMS Chelsea.
2. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, Graduate texts in mathematics 60.
3. T. Frankel, The geometry of physics, Cambridge Univ Press 2012.
4. H. Goldstein, Classical Mechanics, Addison-Wesley.
5. Hitchin, Segal and Ward, Integrable systems, Oxford Univ Press.

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MA 385: Classical Groups (3:0)

Prerequisite courses: MA 220

This course will focus on the structure as well as on finite dimensional complex representations of the following classical groups: General and special Linear groups, Symplectic groups, Orthogonal and Unitary groups.

Suggested books :

1. L. C. Grove, Classical Groups and Geometric Algebra, Graduate Studies in Mathematics 39, American Mathematical Society, 2002.
2. A. Artin, Geometric Algebra, John Wiley & sons, 1988.
3. Herman Weyl, The Classical Groups, Princeton University Press, Princeton, 1946.
4. J. A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80 (1955), 402-447.

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MA 386: Coxeter Groups (3:0)

Reflection groups and their generalisations, Coxeter systems, permutation representations, reduced words, Bruhat order, Kazhdan-Lusztig theory, Chevalley’s theorem, Poincare series, root systems, classification of finite and affine Coxeter groups

No prior knowledge of combinatorics or algebra is expected, but we will assume a familiarity with linear algebra and basics of group theory.

Suggested books :

1. Anders Bjorner & Francesco Brenti, Combinatorics of Coxeter Groups, Springer GTM, 2005.
2. James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990.
3. Michael W. Davis, The Geometry and Topology of Coxeter Groups, Princeton University Press, 2008.
4. Nicolas Bourbaki, Elements Of Mathematics: Lie Groups and Lie Algebras: Chapters 4-6, Springer 2002.

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MA 387: Representation theory of affine Lie algebras (3:0)

Prerequisite courses: MA 219, MA 212, MA 213, MA 315

Loop algebras, central extensions, untwisted affine Lie algebras, root systems, and Weyl groups of untwisted affine Lie algebras. Graph automorphisms of untwisted affine Lie algebras, twisted affine Lie algebras, root systems and Weyl groups of twisted affine Lie algebras. Representations of affine Lie algebras, weight space decomposition, the Category O, Verma modules, integrable modules in Category O. The generalized Casimir operator, Weyl-Kac Character formula, Weyl-Kac denominator identities and Macdonald identities.

Suggested books :

1. Carter, R. W., Lie algebras of finite and affine type, Cambridge Studies in Advanced Mathematics, 96. Cambridge University Press, Cambridge, 2005.
2. Kac, Victor G., Infinite-dimensional Lie algebras. Third edition, Cambridge University Press, Cambridge, 1990.
3. Wakimoto, Minoru, Lectures on infinite-dimensional Lie algebra, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.

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MA 388: Topics in Non-linear Functional Analysis (3:0)

In this course, our main aim is to develop abstract variational techniques which can be employed to study the existence of solutions of various Semi-linear elliptic Partial Differential Equations. The main fundamental results, that will be covered in this course, are functional analytic in nature and can be used in many other situations. A basic outline of the course is as follows:

1. The Pohozaev identity and non-existence of solutions.
2. Calculus in normed linear space: Fréchet and Gâteaux differentiability, Notion of integral, Existence and uniqueness theorem for ODE in Banach space.
3. Dirichlet’s principle, Basics of Sobolev spaces, Connection between critical points and solutions of PDE
4. Direct Methods in Calculus of Variations:Existence of extrema, Ekeland’s Variational Principle, Constrained critical points (method of Lagrange Multiplier).
5. Deformation and the Palais-Smale condition, Saddle points and min-max methods: The mountain pass theorem and its application, The concentration compactness lemmas and their application.
6. Additional topics (to be covered if time permits): Linking Theorem,Index Theory, The Brezis-Nirenberg Problem.

Suggested books :

1. Ambrosetti, A, and Malchiodi, A, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2007.
2. Struwe, Michael, Variational methods, Applications to nonlinear partial differential equations and Hamiltonian systems, Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 34. Springer-Verlag, Berlin 1996.

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MA 390: Percolation and Random Graphs (3:0)

Prerequisite courses: UM 201 or MA 261

Discrete parameter martingales, branching process, percolation on graphs, random graphs, random walks on graphs, interacting particle systems.

Suggested books :

1. Geoffrey Grimmett, Probability on Graphs, Cambridge University Press.
2. Rick Durrett, Random Graph Dynamics, Cambridge University Press.
3. Bollobas, Random Graphs, Cambridge University Press.
4. Geoffrey Grimmett, Percolation, Springer.

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MA 391: Spectral Algorithms (3:0)

Part I - Applications of Spectral Algotihms: Best-Fit Subspaces, Mixture models, Probabilistic Clustering,Recursive Clustering, Optimization via low-rank approximation.

Part II - Algorithms: Matrix Approximation via Random Sampling, Adaptive Sampling Methods, Extensions of SVD to tensors.

Suggested books :

1. Ravindran Kannan and Santosh Vempala, Spectral Algorithms, Foundations and Trends in Theoretical Computer Science, 4:3-4, now Publishers.

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MA 392: Random Graphs and interacting particle systems (3:0)

Prerequisite courses: MA361

Erdos - Renyi random graphs, graphs with power law degree distributions, Ising Potts and contact process, voter model, epidemic models.

Suggested books :

1. Rick Durrett, Random Graph Dynamics, Cambridge University Press, 2001.
2. Bollobas, Random Graphs, Cambridge University Press, 2006.
3. Janson, S., Luczak, T and Rucinski, Random Graphs, Wiley, 2000.

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MA 393: Topics in random discrete structures (3:0)

Pre-requisites :

1. Sufficient exposure to probability.
2. Familiarity with basic properties of Brownian motion.

Real trees, the Brownian continuum random tree, phase transition in random graphs, scaling limits of discrete combinatorial structures, random maps, the Brownian map and its geometry

Suggested books :

1. Jim Pitman, Combinatorial stochastic processes, Lecture Notes in Mathematics, vol. 1875, Springer-Verlag, Berlin (2006)..
2. Jean-François Le Gall, Random trees and applications, Probability Surveys (2005)..
3. Grégory Miermont, Aspects of random maps, Saint-Flour lecture notes (2014)..

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MA 394: Techniques in discrete probability (3:0)

Pre-requisites :

1. This course is aimed at Ph.D. students from different fields who expect to use discrete probability in their research. Graduate level measure theoretic probability will be useful, but not a requirement. I expect the course will be accessible to advanced undergraduates who have had sufficient exposure to probability.

We shall illustrate some important techniques in studying discrete random structures through a number of examples. The techniques we shall focus on will include (if time permits)

1. the probabilistic method;
2. first and second moment methods, martingale techniques for concentration inequalities;
3. coupling techniques, monotone coupling and censoring techniques;
4. correlation inequalities, FKG and BK inequalities;
5. isoperimetric inequalities, spectral gap, Poincare inequality;
6. Fourier analysis on hypercube, Hypercontractivity, noise sensitivity and sharp threshold phenomenon;
7. Stein’s method;
8. entropy and information theoretic techniques.

We shall discuss applications of these techniques in various fields such as Markov chains, percolation, interacting particle systems and random graphs.

Suggested books :

1. Noga Alon and Joel Spencer, The Probabilistic Method, Wiley, 2008.
2. Geoffrey Grimmett, Probability on Graphs, Cambridge University Press, 2010.
3. Ryan O'Donnell, Analysis of Boolean Functions, Cambridge University Press, 2014.

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MA 395: Topics in Stochastic Finance (3:0)

Financial market. Financial instruments: bonds, stocks, derivatives. Binomial no- arbitrage pricing model: single period and multi-period models. Martingale methods for pricing. American options: the Snell envelope. Investment portfolio: Markovitz’s diversification. Capital asset pricing model(CAPM). Utility theory.

Trading in continuous time: geometric Brownian motion model. Option pricing: Black-Scholes-Merton theory. Hedging in continuous time: the Greeks. American options. Exotic options. Market imperfections. Term-Structure models: Vasicek, Hull-White and CIR models. HJM model. Forward LIBOR model.

Suggested books :

1. Luenberger, D. V., , Oxford University Press, 1998.
2. Roman, S., Introduction to the Mathematics of Finance, Springer, 2004.
3. Shiryaev, A. N., Essentials of Stochastic Finance, World Scientific, 1999.
4. Shreve, S. E., Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer, 2004.
5. Shreve, S. E., Stochastic Calculus for Finance II: The Continuous Time Models, Springer, 2004.

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MA 396: Large Deviations (3:0)

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MA 397: Topics in Rigorous Statistical Mechanics (3:0)

We shall cover a selection of topics in probability theory coming from statistical physics models on the Euclidean lattice. A few possible examples of the models include: Ising model, O(N) model, Gaussian free field, contact process, voter model and exclusion processes.

Pre-requisites: This course will be aimed at Int-Ph.D. and PhD students working in probability theory and related areas. A course in graduate probability theory is useful, but not absolutely necessary. A student with a strong undergraduate background in probability (i.e., without measure theory) might also find this course accessible.

Suggested books :

1. Frideli and Velenik, Statistical Mechanics of Lattice Systems, Cambridge University Press, 2017.
2. Liggett, Interacting Particle Systems, Springer, 1985.

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MA 399: Seminar on Topics in Mathematics (2:0)

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UM 203: Elementary Algebra and Number Theory (3:1)

Note: This course has been replaced by UM 205.

Divisibility and Euclid’s algorithm; Fundamental theorem of arithmetic; Infinitude of primes; Congruences; (Reduced) residue systems, Application to sums of squares; Chinese Remainder Theorem; Solutions of polynomial congruences, Hensel’s lemma; A few arithmetic functions (in particular, discussion of the floor function); the Mobius inversion formula; Recurrence relations; Basic combinatorial number theory (pigeonhole principle, inclusion-exclusion, etc.); Primitive roots and power residues, Quadratic residues and the quadratic reciprocity law, the Jacobi symbol; Some Diophantine equations, Pythagorean triples, Fermat’s descent, examples; Definitions of groups, rings and fields, motivations, examples and basic properties; polynomial rings over fields, factorisation of polynomials, content of a polynomial and Gauss’ lemma, Eisenstein’s irreducibility criterion; Elementary symmetric polynomials, the fundamental theorem on Symmetric polynomials; Algebraic and transcendental numbers (an introduction).

Suggested books :

1. Burton, D. M., Elementary Number Theory, McGraw Hill.
2. Niven, Zuckerman, H. S. and Montgomery, H. L., An Introduction to the Theory of Numbers, 5th edition, Wiley Student Editions.
3. Fraleigh, G., A First Course in Abstract Algebra, 7th edition, Pearson.

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UM 204: Introduction to Basic Analysis (3:1)

Basic notions from set theory, countable and uncountable sets. Metric spaces: definition and examples, basic topological notions. The topology of $\R^n$: topology induced by norms, the Heine-Borel theorem, connected sets. Sequences and series: essential definitions, absolute versus conditional convergence of series, some tests of convergence of series. Continuous functions: properties, the sequential and the open- set characterizations of continuity, uniform continuity. Differentiation in one variable. The Riemann integral: formal definitions and properties, continuous functions and integration, the Fundamental Theorem of Calculus. Uniform convergence: definition, motivations and examples, uniform convergence and integration, the Weierstrass Approximation Theorem.

Suggested books :

1. Tao, T. 2014., Analysis I, 3rd edition, Texts and Readings in Mathematics, vol. 37, Hindustan Book Agency.
2. Tao, T. 2014., Analysis II, 3rd edition, Texts and Readings in Mathematics, vol. 38, Hindustan Book Agency.
3. Apostol, T. M., Mathematical Analysis, 2nd edition, Narosa.

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UM 205: Introduction to algebraic structures (3:1)

1. Set theory: equivalence classes, partitions, posets, axiom of choice/Zorn’s lemma, countable and uncountable sets.
2. Combinatorics: induction, pigeonhole principle, inclusion-exclusion, Möbius inversion formula, recurrence relations.
3. Number theory: Divisibility and Euclids algorithm, Pythagorean triples, solving cubics, Infinitude of primes, arithmetic functions, Fun- damental theorem of arithmetic, Congruences, Fermat’s little theorem and Euler’s theorem, ring of integers modulo n, factorisation of poly- nomials, algebraic and transcendental numbers.
4. Graph theory: Basic definitions, trees, Eulerian tours, matchings, matrices associated to graphs.
5. Algebra: groups, permutations, group actions, Cayley’s theorem, di- hedral groups, introduction to rings and fields.

Suggested books :

1. L. Childs, A Concrete Introduction to Higher Algebra, 3rd edition, Springer-Verlag.
2. M. A. Armstrong, Groups and Symmetry, Springer-Verlag.
3. Miklos Bona, A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, World Scientific.
4. D. M. Burton., Elementary Number Theory, McGraw Hill.
5. Niven, Zuckerman, H. S. and Montgomery, H. L., An Introduction to the Theory of Numbers, 5th edition, Wiley Student Editions.
6. Fraleigh, G., A First Course in Abstract Algebra, 7th edition, Pearson.

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UM 400: Masters project A (0:13)

Mandatory project for undergraduate mathematics majors in their fourth year, second semester.

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UM 501: Masters project A (0:6)

Optional project for undergraduate mathematics majors in their fifth year, first semester.

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UM 502: Masters project B (0:6)

Optional project for undergraduate mathematics majors in their fifth year, second semester.

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UMA 101: Analysis and Linear Algebra I (4:0)

One-variable Calculus: Real and Complex numbers; Convergence of sequences and series; Continuity, intermediate value theorem, existence of maxima and minima; Differentiation, mean value theorem,Taylor series; Integration, fundamental theorem of Calculus, improper integrals. Linear Algebra: Vector spaces (over real and complex numbers), basis and dimension; Linear transformations and matrices.

Suggested books :

1. Apostol, T. M., Calculus, Volume I, 2nd edition, Wiley, India, 2007.
2. Strang, G., Linear Algebra and its Applications, 4th Edition, Brooks/Cole, 2006.

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UMA 102: Analysis and Linear Algebra II (4:0)

Linear Algebra continued: Inner products and Orthogonality; Determinants; Eigenvalues and Eigenvectors; Diagonalisation of symmetric matrices. Multivariable calculus: Functions on $\R^n$, partial and total derivatives; Chain rule; Maxima, minima and saddles; Lagrange multipliers; Integration in $\R^n$, change of variables, Fubini’s theorem; Gradient, Divergence and Curl; Line and Surface integrals in $\R^2$ and $\R^3$; Stokes, Green’s and Divergence theorems. Introduction to Ordinary Differential Equations; Linear ODEs and Canonical forms for linear transformations.

Suggested books :

1. Apostol, T. M., Calculus, Volume II, 2nd edition, Wiley, India, 2007.
2. Strang, G., Linear Algebra and its Applications, 4th Edition, Brooks/Cole, 2006.
3. Artin, M., Algebra, Prentice Hall of India.
4. Hirsch, M., Smale, S. and Devaney, R. L., Differential Equations, Dynamical Systems, and an Introduction to Chaos, 2nd edition, Academic Press, 2004.

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UMA 201: Probability and Statistics (4:0)

Basic notions of probability, conditional probability and independence, Bayes’ theorem, random variables and distributions, expectation and variance, conditional expectation, moment generating functions, limit theorems. Samples and sampling distributions, estimation of parameters, testing of hypotheses, regression, correlation and analysis of variance.

Suggested books :

1. Ross, S., Introduction to Probability and Statistics for Engineers and Scientists, Academic Press; 4th ed. (2009),
2. Freedman, Pisani and Purves, Statistics, Viva Books; 4th ed. (2011).
3. Feller, W., An Introduction to Probability Theory and its Applications - Vol. 1, Wiley; 3rd ed. (2008).
4. Ross, S., A First Course in Probability, Pearson Education; 9th ed. (2013).
5. Athreya, S., Sarkar, D. and Tanner, S., Probability and Statistics (with Examples using R), Unfinished book.

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