My research area is broadly Calculus of Variations, Partial Differential Equations and Geometric Analysis. I am primarily interested in variational problems in a geometric setting, especially in problems where analysis meets geometry and topology. Differential forms have featured quite prominently in my research so far.
There are a number of topics I know a bit and find them interesting enough to seriously learn and plan on working on those areas in near future. A list of such topics is here.If you are wondering what kind of background one needs for research in these areas, see here.
Harmonic, polyharmonic and p-harmonic maps
Yang-Mills connections and higher dimensional gauge theory
Topological invariants below the continuity threshold and Sobolev mappings between manifolds, variational problems for manifold-valued Sobolev maps
Pullback equation for differential forms.
Direct methods of Calculus of variation for differential forms in the nonconvex case
Hodge type systems and other boundary value problems for linear and nonlinear systems of PDEs for differential forms
Transport of differential forms.
Regularity results for nonlinear elliptic systems involving differential forms
Nonlinear potential estimates
Dimension and stratification of singular sets in geometric variational problems.
Gaffney inequality and related Korn inequality, Poincaré-Sobolev and Hardy type inequalities for differential forms and vector fields - their best constants, geometric characterization of 'best' domains and attainability
Gaffney and Hardy type inequalities for differential forms in the borderline spaces L1 and BV. Korn inequalities in BD spaces.
Applications of variational techniques to electromagnetic theory, fluid motion, nonlinear elasticity, mechanics of materials etc.
Other topics in Geometric Analysis
Geometric analysis nowadays has grown into a huge area and many of its subfields have become distinct branches of mathematics in their own right. I am interested in many topics and questions in geometric analysis beyond what I have worked on already. These range from the study of minimal surfaces to questions in Kähler geometry, Hermitian geometry, conformal geometry, spin geometry, calibrated geometry and symplectic geometry and topology, metric geometry, just to name a few. They offer a rich supply of interesting and fascinating PDE and analysis questions and the relevant techniques are not radically different from what I use.
Geometric flows have proved to be an extremely effective tool in many areas of geometric analysis. The PDEs are parabolic systems, which are intimately related to the elliptic systems I am interested in, but have their own flavors.
Geometric inverse problems
These problem involve gathering information about the geometry of a manifold ( in the best of cases, determining it ) by boundary measurements related to an elliptic operator on the manifold. Typical example being determining the Riemannian metric ( upto some obvious invariances ) of a Riemannian manifold with boundary from the Dirichlet to Neumann map of the Laplace-Beltrami operator on the manifold.
Optimal transport in curved geometries is an extremely fascinating area of mathematics where both calculus of variation and geometric analysis come together. Despite an enormous amount of activity, there are still lots of open questions.
The study of fractional Harmonic maps, fractional minimal surfaces and fractional isoperimetric inequalities is an emerging and fast growing field. The non-local nature of the problem often makes the results counterintuitive at first glance.
It is of course impossible to produce an exhaustive list of the entire spectrum of mathematics that is used for research in these areas, since transporting and/or importing tricks, ideas and intuition from other areas of mathematics and even other subjects, like for example physics, is quite common. So in general, the broader one's mathematical 'culture' is, the better. Still some techniques are much more ubiquitous than others. Thus familiriaty with those techniques are rather crucial. A minimal list of such topics is given below.
Elliptic methods is the primary tool in my research area. So a good command in this area is absolutely essential. Fortunately, there are a number of excellent textbooks on the subject.
Gilbarg, David; Trudinger, Neil S. , Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften, 224. Springer-Verlag, Berlin, 1983.
Han, Qing; Lin, Fanghua , Elliptic partial differential equations. Courant Lecture Notes in Mathematics, 1. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997.
Krylov, N. V., Lectures on elliptic and parabolic equations in Hölder spaces. Graduate Studies in Mathematics, 12. American Mathematical Society, Providence, RI, 1996.
Krylov, N. V., Lectures on elliptic and parabolic equations in Sobolev spaces. Graduate Studies in Mathematics, 96. American Mathematical Society, Providence, RI, 2008.
Regularity/singularity in geometric variational problems
Giaquinta, Mariano; Martinazzi, Luca, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Second edition. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 11. Edizioni della Normale, Pisa, 2012.
Simon, Leon,Theorems on regularity and singularity of energy minimizing maps. Based on lecture notes by Norbert Hungerbühler. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996.
Geometric Measure Theory
Evans, Lawrence C.; Gariepy, Ronald F., Measure theory and fine properties of functions. Revised edition. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015.
Simon, Leon, Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, 1983.
Geometry and geometric analysis
Jost, Jürgen, Riemannian geometry and geometric analysis. Sixth edition. Universitext. Springer, Heidelberg, 2011.
Aubin, Thierry, Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.
Tu, Loring W., Differential geometry. Connections, curvature, and characteristic classes. Graduate Texts in Mathematics, 275. Springer, Cham, 2017.
Tu, Loring W., An introduction to manifolds. Second edition. Universitext. Springer, New York, 2011.
Taubes, Clifford Henry, Differential geometry. Bundles, connections, metrics and curvature. Oxford Graduate Texts in Mathematics, 23. Oxford University Press, Oxford, 2011.
Bott, Raoul; Tu, Loring W., Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982.
Bredon, Glen E., Topology and geometry. Corrected third printing of the 1993 original. Graduate Texts in Mathematics, 139. Springer-Verlag, New York, 1997.
Apart from these, there are some specialized topics such as direct methods in calculus of variations, nonlinear potential theory and pullback equations. A reading list for such topics is given below.
Calculus of Variations
Dacorogna, Bernard, Direct methods in the calculus of variations. Second edition. Applied Mathematical Sciences, 78. Springer, New York, 2008.
Giaquinta, Mariano; Modica, Giuseppe; Souček, Jiří, Cartesian currents in the calculus of variations. I. Cartesian currents. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 37. Springer-Verlag, Berlin, 1998.
Giaquinta, Mariano; Modica, Giuseppe; Souček, Jiří, Cartesian currents in the calculus of variations. II. Variational integrals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 38. Springer-Verlag, Berlin, 1998.
Nonlinear potential theory
Beck, Lisa, Elliptic regularity theory. A first course . Lecture Notes of the Unione Matematica Italiana, 19. Springer, Cham; Unione Matematica Italiana, Bologna, 2016.
Malý, Jan; Ziemer, William P., Fine regularity of solutions of elliptic partial differential equations. Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence, RI, 1997.
Heinonen, Juha; Kilpeläinen, Tero; Martio, Olli, Nonlinear potential theory of degenerate elliptic equations . Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993.
Pullback equation and Hodge systems
Csató, Gyula; Dacorogna, Bernard; Kneuss, Olivier, The pullback equation for differential forms . Progress in Nonlinear Differential Equations and their Applications, 83. Birkhäuser/Springer, New York, 2012.
Schwarz, Günter, Hodge decomposition—a method for solving boundary value problems. Lecture Notes in Mathematics, 1607. Springer-Verlag, Berlin, 1995.