In this thesis, we study some important classification problems related to affine Kac--Moody Lie algebras. First, we address the combinatorial problem of classifying symmetric real-closed subsets of affine root systems ...

This thesis focuses on the study of specialized characters of irreducible polynomial representations of the complex classical Lie groups of types A, B, C and D. We study various specializations where the characters are ...

This thesis concerns the construction of harmonic maps from certain non-compact surfaces into hyperbolic 3-space H3 with prescribed asymptotic behavior and has two parts. The focus of the first part is when the domain ...

This thesis consists of two parts. In both the parts we study two new notions of canonical K\"ahler metrics on compact K\"ahler manifolds which were introduced by Pingali, viz. `higher extremal K\"ahler metric' and ...

In this thesis, we study certain aspects of the geometry of the Kobayashi (pseudo)distance and the Kobayashi (pseudo)metric for domains in $\mathbb{C}^n$. We focus on the following themes: on the interaction between Kobayashi ...

This thesis comprises two main parts. The details of the two parts are as follows: The first part of the thesis deals with the monopole-dimer model. The dimer (resp. monomer-dimer) model deals with weighted enumeration of ...

This thesis focuses on studying proper maps between two non-compact surfaces with a particular emphasis on the topological rigidity and the Hopfian property. Topological rigidity is the property that every homotopy equivalence ...

The broad aim of this thesis is to study certain tuples of bounded operators that satisfy an operator inequality arising from a complete Nevanlinna-Pick kernel. Specifically, we focus on two invariants of these operator ...

The problem of filter design is ubiquitous. Frequency selective filters are used in speech/audio processing, image analysis, convolutional neural networks for tasks such as denoising, deblurring/deconvolution, enhancement, ...

In this dissertation, broadly, we treat the weight-sets of arbitrary highest weight modules (uniformly) over all general complex Kac-Moody Lie algebras $\mathfrak{g}$, achieving the below. We obtain a uniform, explicit, ...