Mathematics (MA)
Recent Submissions

Weights of highest weight modules over KacMoody algebras
In this dissertation, broadly, we treat the weightsets of arbitrary highest weight modules (uniformly) over all general complex KacMoody Lie algebras $\mathfrak{g}$, achieving the below. We obtain a uniform, explicit, ... 
Quasianalytic Functions, Spherical Means, and Uncertainty Principles on Heisenberg Groups and Symmetric Spaces
This thesis has two parts. The first part revolves around certain theorems related to an uncertainty principle and quasianalyticity. In contrast, the second part reflects a different mathematical theme, focusing on the ... 
Hardy's inequalities for Grushin operator and Hermite multipliers on Modulation spaces
This thesis consists of two broad themes. First one revolves around the Hardy's inequalities for the fractional power of Grushin operator $\G$ which is chased via two different approaches. In the first approach, we first ... 
Homogenization of certain PDEs and associated optimal control problems on various rough domains
This thesis is devoted to the study of the asymptotic behavior of partial differential equations (PDEs) and associated boundary and interior optimal control problems in various oscillatory domains. The present thesis ... 
Finite Element Analysis of Optimal Control Problems Governed by Certain PDEs
The primary goal of this thesis is to study finite element based \textit{a priori} and \textit{a posteriori} error analysis of optimal control problems of various kinds governed by linear elliptic PDEs of second order. ... 
Positivity preservers forbidden to operate on diagonal blocks
The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., ... 
On certain invariant measures for correspondences, their analysis, and an application to recurrence
The aim of this dissertation is to analyse a certain class of dynamically interesting mea sures arising in holomorphic dynamics that goes beyond the classical framework of maps. We study measures associated with semigroups ... 
Hermitian Metrics and Singular Riemann Surface Foliations
The main aim of this thesis is to understand curvature properties of a given hermitian metric by restricting it to the leaves of a suitable singular Riemann surface foliation. We will specifically consider the complete ... 
Nodal sets of random functions
This thesis is devoted to the study of nodal sets of random functions. The random functions and the specific aspect of their nodalset that we study fall into two broad categories: nodal component count of Gaussian Laplace ... 
Role of gamma oscillations in processing of natural stimuli
Gamma oscillations (~3080 Hz) are a prominent signature of electrophysiological signals recorded from the brain with proposed roles in various cognitive functions such as attention and visual perception. Although gamma ... 
On the Geometry and Operator Theory of the Bidisc and the Symmetrized Bidisc
This work is concerned with the geometric and operator theoretic aspects of the bidisc and the symmetrized bidisc. First we have focused on the geometry of these two do mains. The symmetrized bidisc, a nonhomogeneous ... 
Near Optimal Nonmalleable Codes and Leakage Resilient Secret Sharing Schemes
A wellstudied class of attacks on cryptosystems called "sidechannel attacks", stems from the additional access that an adversary can get due to the susceptibility of the hardware on which the cryptosystem (e.g., digital ... 
Structural connectivity correlates of human cognition explored with diffusion MRI and tractography
Intact structural connectivity among brain regions is critical to cognition. Structural connectivity forms the substratum for information flow between brain regions, and its plasticity is a hallmark of learning in the ... 
Algorithmic and Hardness Results for Fundamental FairDivision Problems
The theory of fair division addresses the fundamental problem of dividing a set of resources among the participating agents in a satisfactory or meaningfully fair manner. This thesis examines the key computational challenges ... 
Representations of Special Compact Linear Groups of Order Two : Construction, Representation Growth, Group Algebras and Branching Laws
Let o be the ring of integers of a nonArchimedean local field such that the residue field has characteristic p: Let p be the maximal ideal of o: For Char¹oº = 0; let e be the ramification index of o: i.e., 2o = pe: Let ... 
Some results on Spectral spaces and Spectral sequences
This thesis is made up of two independent parts. We begin the first part by showing that the collection of all prime ideals of a monoid or, in other words, the spectrum of a commutative monoid, endowed with the Zariski ... 
Fourier coeffcients of modular forms and mass of pullbacks of Saito–Kurokawa lifts
In the first part of the talk we would discuss a topic about the Fourier coefficients of modular forms. Namely, we would focus on the question of distinguishing two modular forms by certain ‘arithmetically interesting’ ... 
Characterization of Brain Signals Across Scales Using Temporally Modulated Visual Stimuli
Electrical signals from the brain can be recorded at several different scales, ranging from spiking activity to local field potential (LFP) in animals to scalp electroencephalogram (EEG) in humans. Each signal represents ... 
Correlation Functions in the Finite Toom Model
This thesis is divided into four chapters. In Chapter 1, we give some basic definitions and results that we need throughout the thesis. In Chapter 2, we describe the pn0;n1qToom model and prove some known results about ... 
Duality for Spaces of Holomorphic Functions into a Locally Convex Topological Vector Space
Consider an open subset O of the complex plane and a function f : O ÝÑ C. We understand the concept of holomorphicity of a complexvalued function. Suppose a function f defined on O takes values in a locally convex ...