Abstract. Given a ring $R$ and a prime ideal $P$ in $R$, the $n$'th symbolic power of $P$, denoted $P^{(n)}$, is defined to be $P^n R_P \cap R$. I will introduce two open problems based on symbolic powers and discuss my results in this direction. Specifically, I will talk about the Eisenbud Mazur conjecture which states that in a regular local ring $(R,\mathfrak{m})$ containing a field of characteristic zero, $P^{(2)}\subseteq \mathfrak{m}P$. Further, I will speak about the question of existence of uniform bounds on the growth of symbolic powers with respect to the ordinary powers of prime ideals. In other words, given a ring $R$, does there exist a positive integer $k$, such that for all prime ideals $P$ in $R$, $P^{(kn)}\subseteq P^n$ for all positive integers $n$.

Abstract. Gorenstein rings are an important class of rings and attractive because they exhibit various kinds of symmetries, leading to many duality properties. They are ubiquitous, arising from very different contexts. In this presentation, I will talk about a recent construction of Gorenstein rings called connected sums, the roots of which can be traced to topology. We will see some current work (joint with E. Celikbas and Z. Yang) where we give some conditions sufficient to decompose a given Gorenstein ring as a connected sum.

Abstract. I will briefly discuss my research on Ising Measures, and Random Matrices. The work on Ising measure is joint with Prof. Amir Dembo (Stanford University). The work on Random Matrices have been done/ongoing with Prof. Arup Bose (ISI, Kolkata) and Prof. Amir Dembo on different problems.
Consider ferromagnetic Ising measure on sparse finite graphs converging locally to limiting tree ${\sf T}$. In case ${\sf T}$ is $d$-regular, it was recently shown by Montanari, Mossel and Sly that these Ising measures converge locally to symmetric mixture of plus and minus (boundary conditions) Ising measures on ${\sf T}$, and for expander graphs, conditioned on positive magnetization these measures converge to plus-boundary condition Ising measure on ${\sf T}$. We explore the case when the limiting tree is more general, and random.
In random matrix theory we identify limiting spectral distribution of different ensembles of random matrices namely, sum of unitary matrices, sum of permutation matrices, sample autocovariance matrices, and band matrices. In the second part of the talk, I will briefly discuss these results.

Abstract. There is increasing interest in broad application areas in defining flexible joint models for data having a variety of measurement scales, while also allowing data of complex types, such as functions, images and documents. We consider a general framework for nonparametric Bayes joint modeling through mixture models that incorporate dependence across data types through a joint mixing measure. The mixing measure is assigned a novel infinite tensor factorization (ITF) prior that allows flexible dependence in cluster allocation across data types. The development encompasses as special cases similar approaches in other subject areas , like non-negative matrix factorization and tensor decomposition in linear algebra, relational models in machine learning, and approximation of graphs from topology and network theory. We show in general how seemingly unrelated developments in different domains can be connected and can lead to major improvements.

Abstract. Page 220 of Ramanujan's Lost Notebook contains a beautiful transformation formula involving the logarithmic derivative of the Gamma function, which is also associated with an integral involving the Riemann $\Xi$-function. This formula can be represented in the form $F(\alpha)=F(\beta)$, where $\alpha\beta=1$. As will be seen in this talk, this is a proto-typical example of efficiently generating modular-type transformation formulas through integrals involving the Riemann $\Xi$-function as well as of obtaining representations for the Riemann zeta function.
Here we discuss several different types of transformation formulas, i.e., $F(z,\alpha)=F(z,\beta)$, $F(z, \alpha,\chi)=F(-z, \beta,\overline{\chi})=F(-z,\alpha,\overline{\chi})=F(z,\beta,\chi)$, where $\chi$ is a primitive Dirichlet character, and $F(z,\alpha)=F(iz,\beta)$. All these generalize the former type $F(\alpha)=F(\beta)$. Specific examples include several new transformation formulas like generalizations of the above-mentioned formula of Ramanujan as well as of the formulas due to G.H. Hardy, W.L. Ferrar and N.S. Koshlyakov. We also give extensions of some well-known formulas in the literature. At the end, we give character analogues as well as a generalization of the Ramanujan-Hardy-Littlewood conjecture involving infinite series of Möbius function.

Abstract. Chemical reaction networks are used as models of biological systems such as gene regulatory networks and metabolic networks. The existence of multiple steady states is a necessary condition for a reaction network to act as a biomolecular switch, therefore determining which networks permit multiple steady states is an important theoretical problem. We present new conditions to preclude multistationarity based on the network structure. We also give new sufficient conditions for a reaction network to allow multiple steady states for some parameter values by defining certain minimal reaction networks called atoms of multistationarity. Furthermore, we give complete characterization of one reaction atoms of multistationarity. This is joint work with Anne Shiu.

Abstract. The Birkhoff Ergodic Theorem establishes a law of large numbers for measure preserving transformations. However, finer statistical properties can often be studied for such systems. We will give an overview of some of the questions that researchers ask in this field, and will focus on three questions in particular: First, what can be said about the distribution of the distances of orbits from a point in the phase space? Second, what happens if we, instead of iterating a particular map, iterate maps that are topologically close to it? Do we still observe some of the statistical properties that we get by iterating a fixed map? And third, since the techniques used often depend on the dimension of the system under consideration, what can be said about infinite dimensional systems?

Abstract. An affine stratification of a variety is like a cellular complex except built up by affine varieties which are locally closed. I shall explain the topological consequences of affine stratifications. Looijenga showed that the affine stratification number of $M_g$, the moduli space of curves of genus $g$ is $g-2$ for $g\leq 5$. Let $\overline{M}_g$ be the Deligne-Mumford compactification. We explore a natural filtration on $\overline{M}_g = \bigcup_k M_g^k$, and the corresponding stratification on $\overline{H}_g= \bigcup_k H_g^k$, the hyper-eliptic locus. I shall talk about affine stratification of $H_g^k$. To obtain a sharp result we shall have to do some cohomology computations and that is where colored operads, and McDonald's symmetric functions come into the picture.

Abstract. We consider estimating predictive densities for probability forecasting in a time-stationary Gaussian setup. We found that optimal density estimates need diversification of the future risk around efficient location estimates. In unrestricted parametric spaces, future uncertainty can be shared by optimally flattening densities based on the quadratic risk estimate of the corresponding location point estimator. In restricted spaces, particularly under sparsity constraints, new phenomena are witnessed as we exhaustively trace the interplay between sparsity regularization need and uncertainty sharing dynamics.
Diversified predictive strategies can be employed for fabricating log-optimal portfolios over repeated investments in the market. Also, in personalized genomics applications, future probability statements about one's genetic predisposition to a wide range of diseases can be formulated from density estimates based on observing high dimensional genome-wide scans. The uncertainty sharing ideas can be extended to sequential applications in rapidly trend-changing environments.

Abstract. In this presentation I will prove finite generation of the cohomology of quotients of a PBW algebra denoted by A by relating it to the cohomology of quotients of a quantum symmetric algebra denoted by S which is isomorphic to the associated graded algebra of A. The proof uses a spectral sequence argument and a finite generation lemma adapted from Friedlander and Suslin.

Abstract. Estimation of the covariance matrix, especially in higher dimensions ("large $p$ small $n$ but $p \leq n-1$") is a challenging statistical problem which is of great interest in many applications especially in genetics and genomics. The existing methods of estimation (both Bayesian and non-Bayesian) pose a number of challenges even when $p$ is moderately large, primarily since the dimension of the parameter space increases quadratically in $p$. Our approach is to reduce the dimension of the problem and obtain an approximate likelihood on the reduced space. We propose a non-informative prior on the reduced problem and investigate the performance of the corresponding estimator via simulations.

Abstract. Chow-Witt theory and Euler class theory were developed to obtain precise invariants for splitting projective modules. This talk will study these theories.

Abstract. Mechanistic models in toxicology and several other areas use systems of differential equation as a modeling technique. In toxicology researchers are often interested in analyzing the mechanism of action of a chemical from it's kinetics within the human physiology. Systems of ordinary differential equations are used to describe the different processes a chemical undergoes, such as absorption, distribution, metabolism and excretion. These mathematical models, known as physiologically based pharmacokinetic (PBPK) models provide valuable insight into the adverse effects caused by exposure to these chemicals. This work focuses on developing a functional data analysis based methodology for statistical inference in this general class of models, in the context of PBPK models. Using basis function expansions, the methodology avoids solving the system of differential equations while taking into account the effect of covariates along with intra and inter-individual variability, commonly present in multi subject toxicological studies. It provides a general statistical framework and also reduces the computational burden usually encountered in the existing approaches. It has been illustrated on simulated and real data examples studying the kinetics of benzene in human subjects through inhalation. The developed methodology has potential application in a number of areas such as viral dynamics, immunology, gene regulatory networks and infectious diseases, due to the use of differential equations to model these complex dynamic phenomena.

Abstract. The loop space of a topological space is a fundamental object in mathematics, appearing in differential geometry, homotopy theory and topology. The study of algebraic topology of (free) loop spaces and the natural algebraic structures that are present, often goes by the name of string topology. We show that techniques of string topology can be used, for instance, to distinguish spaces which are homotopy equivalent but not homeomorphic.

Abstract. India is still one of the worst effected countries of the world affected by Tuberculosis. According to the 2009 report of WHO there is about 9.4 million cases and 1.7 million deaths. In a recently published work "A tuberculosis (TB) model with undetected compartment : An application to China", various factors have been investigated to detect the still persistence of the disease in China. There was not many research been made on the prevalence of the disease in India. My goal is to extend this work in China and to investigate the causes of the disease in India and to indicate the main reasons for its existence when the disease is extinct in the western countries of the world.

Abstract. The subject of my PhD thesis is probabilistic divide-and-conquer, which is a new method for exact simulation that can be used to randomly sample from combinatorial structures. I applied this novel technique to the problem of simulating a uniformly-at-random integer partition of a fixed size $n$, and obtained a practical algorithm that runs in time within a constant factor of the lower entropy limit.
For us, the random objects $S$ whose simulation might benefit from a divide-and-conquer approach are those that can be described as $S=(A,B)$, where there is some ability to simulate $A$ and $B$ separately. Specifically, we require that $A \in \mathcal{A}$, $B \in \mathcal{B}$, and that there is a function $h: \mathcal{A} \times \mathcal{B} \to \{0,1 \}$, so that for $a \in \mathcal{A},b \in \mathcal{B}$, $h(a,b)$ is the indicator that "$a$ and $b$ fit together to form an acceptable $s$." Furthermore, we require that $A$ and $B$ be independent, and that the desired $S$ be equal in distribution to $( \ (A,B) | h(A,B)=1)$. This description, independent objects conditional on some event, may seem restrictive, but very broad classes --- combinatorial assemblies, multisets, and selections --- fit into this setup.

Abstract. Chern classes play an important role in the study of vector bundles. Their Chern-Weil representatives give more refined information about the same. We computed the Chern and Bott-Chern forms for trivial bundles with a certain kind of a non-diagonal metric and proved that every d-dbar exact form is represented by a virtual bundle. We also studied the general question of representing a given form by a Chern-Weil form of a metric connection on a holomorphic vector bundle. The former could (potentially) be applied to a Sullivan-Simons kind of a construction of a differential K-theory which might be useful in Arakelov geometry. The latter is motivated by considerations of local index theory and is also interesting in its own right, as it presents a new Monge-Ampere PDE.