
Divakaran, D
Gromov’s compactness theorem for metric spaces, a compactness theorem for the space of compact metric spaces equipped with the GromovHausdorﬀ distance, is a theorem with many applications. In this thesis, we give a generalisation of this landmark result, more precisely, we give a compactness theorem for the space of distance measure spaces equipped with the generalised GromovHausdorﬀLeviProkhorov distance. A distance measure space is a triple (X, d,µ), where (X, d) forms a distance space (a generalisation of a metric space where, we allow the distance between two points to be infinity) and µ is a finite Borel measure.
Using this...

Rao, Balaji R
In this thesis we present a formalization of the combinatorial part of the proof of FeitHigman theorem on generalized polygons. Generalised polygons are abstract geometric structures that generalize ordinary polygons and projective planes. They are closely related to finite groups.
The formalization is carried out in Agda, a dependently typed functional programming language and proof assistant based on the intuitionist type theory by Per MartinLöf.

Porwal, Kamana
The main emphasis of this thesis is to study a posteriori error analysis of discontinuous Galerkin (DG) methods for the elliptic variational inequalities. The DG methods have become very popular in the last two decades due to its nature of handling complex geometries, allowing irregular meshes with hanging nodes and different degrees of polynomial approximation on different elements. Moreover they are high order accurate and stable methods. Adaptive algorithms reﬁne the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main ingredients to steer the adaptive mesh reﬁnement.
The solution of...

Rungta, Satya Prakash
A hallmark of human behaviour is that we can either couple or decouple our thoughts, decision and motor plans from actions. Previous studies have reported evidence of gating of information between intention and action that can happen at different levels in the central nervous system (CNS) involving the motor cortex, subcortical structures such as the basal ganglia and even in the spinal cord. In my research I examine the extent of this gating and its modulation by task context. I will present results obtained by data collected from (a) neck muscles and neural recording from frontal eye field (FEF) in...

Nanda Kishore Reddy, S
In this thesis, we study the exact eigenvalue distribution of product of independent rectangular complex Gaussian matrices and also that of product of independent truncated Haar unitary matrices and inverses of truncated Haar unitary matrices. The eigenvalues of these random matrices form determinantal point processes on the complex plane. We also study the limiting expected empirical distribution of appropriately scaled eigenvalues of those matrices as the size of matrices go to infinity. We give the first example of a random matrix whose eigenvalues form a nonrotation invariant determinantal point process on the plane.
The second theme of this thesis is infinite...

Ruhi, Ankit
Turbulence is an open and challenging problem for mathematical approaches, physical modeling and numerical simulations. Numerical solutions contribute significantly to the understand of the nature and effects of turbulence. The focus of this thesis is the development of appropriate numerical methods for the computer simulation of turbulent flows. Many of the existing approaches to turbulence utilize analogies from kinetic theory. Degond & Lemou (J. Math. Fluid Mech., 4, 257284, 2002) derived a k✏ type turbulence model completely from kinetic theoretic framework. In the first part of this thesis, a numerical method is developed for the computer simulation based on this...

Prathamesh, Turga Venkata Hanumantha
Mechanisation of Mathematics refers to use of computers to generate or check proofs in Mathematics. It involves translation of relevant mathematical theories from one system of logic to another, to render these theories implementable in a computer. This process is termed formalisation of mathematics. Two among the many ways of mechanising are:
1 Generating results using automated theorem provers.
2 Interactive theorem proving in a proof assistant which involves a combination of user intervention and automation.
In the first part of this thesis, we reformulate the question of equivalence of two Links in first order logic using braid groups. This...

Sanki, Bidyut
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs.
A systolic graph has a metric on it, so that all cycles on the graph that correspond to...

Reza, Md. Ramiz

Reza, Md. Ramiz

Bera, Sayani
We use transcendental shiftlike automorphisms of Ck, k > 2 to construct two examples of nondegenerate entire mappings with prescribed ranges. The first example exhibits an entire mapping of Ck, k>2 whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. This generalizes a result of DixonEsterle in C2. The second example shows the existence of a FatouBieberbach domain in Ck,k > 2 that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and RosayRudin.
In the second part we compute the order and type of entire...

Bera, Sayani
We use transcendental shiftlike automorphisms of Ck, k > 2 to construct two examples of nondegenerate entire mappings with prescribed ranges. The first example exhibits an entire mapping of Ck, k>2 whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. This generalizes a result of DixonEsterle in C2. The second example shows the existence of a FatouBieberbach domain in Ck,k > 2 that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and RosayRudin.
In the second part we compute the order and type of entire...

Boggarapu, Pradeep
This thesis is divided into three parts. In the first part we study mixed norm estimates for Riesz transforms associated with various differential operators. First we prove the mixed norm estimates for the Riesz transforms associated with Dunkl harmonic oscillator by means of vector valued inequalities for sequences of operators defined in terms of Laguerre function expansions. In certain cases, the result can be deduced from the corresponding result for Hermite Riesz transforms, for which we give a simple and an independent proof. The mixed norm estimates for Riesz transforms associated with other operators, namely the subLaplacian on Heisenberg group,...

Boggarapu, Pradeep
This thesis is divided into three parts. In the first part we study mixed norm estimates for Riesz transforms associated with various differential operators. First we prove the mixed norm estimates for the Riesz transforms associated with Dunkl harmonic oscillator by means of vector valued inequalities for sequences of operators defined in terms of Laguerre function expansions. In certain cases, the result can be deduced from the corresponding result for Hermite Riesz transforms, for which we give a simple and an independent proof. The mixed norm estimates for Riesz transforms associated with other operators, namely the subLaplacian on Heisenberg group,...

Kumar, Manish
We start with the recollection of basic results about differential manifolds and Lie groups. We also recall some preliminary terminologies in Lie algebra. Then we define the Lie algebra corresponding to a Lie group. In the next section, we define a strongly continuous representation of a Lie group on a Banach space. We further define the smooth, analytic and entire vectors for a given representation. Then, we move on to develop some necessary and sufficient criteria to characterize smooth, analytic and entire vectors. We, in particular, take into account of some specific representations of Lie groups like the regular representation...

Kumar, Manish
We start with the recollection of basic results about differential manifolds and Lie groups. We also recall some preliminary terminologies in Lie algebra. Then we define the Lie algebra corresponding to a Lie group. In the next section, we define a strongly continuous representation of a Lie group on a Banach space. We further define the smooth, analytic and entire vectors for a given representation. Then, we move on to develop some necessary and sufficient criteria to characterize smooth, analytic and entire vectors. We, in particular, take into account of some specific representations of Lie groups like the regular representation...

Ghosh, Satadal
Guidance is defined as the determination of a strategy for following a nominal path in the presence of onominal conditions, disturbances and uncertainties, and the strategy employed is called a guidance law. Variants of Proportional Navigation (PN), such as True Proportional Navigation (TPN) and Pure Proportional Navigation (PPN), have been studied extensively in the literature on tactical missile guidance. In the absence of target maneuvers, in a linear interceptor guidance problem, TPN was shown to be optimal. However, the standard PN class of guidance laws per se does not show good performance against maneuvering targets, and was found to be...

Sarkar, Santanu
In this thesis we have investigated two diﬀerent types of problems in multivariable operator theory. The ﬁrst one deals with the defect sequence for contractive tuples and maximal contractive tuples. These condone deals with the wandering subspaces of the Bergman space and the Dirichlet space over the polydisc. These are described in thefollowing two sections.
(I) The Defect Sequence for ContractiveTuples
LetT=(T1,...,Td)beadtuple of bounded linear operators on some Hilbert space
H. We say that T is a row contraction, or, acontractive tuplei f the row operator
(Pl refer the abstract pdf file)

Sarkar, Santanu
In this thesis we have investigated two diﬀerent types of problems in multivariable operator theory. The ﬁrst one deals with the defect sequence for contractive tuples and maximal contractive tuples. These condone deals with the wandering subspaces of the Bergman space and the Dirichlet space over the polydisc. These are described in thefollowing two sections.
(I) The Defect Sequence for ContractiveTuples
LetT=(T1,...,Td)beadtuple of bounded linear operators on some Hilbert space
H. We say that T is a row contraction, or, acontractive tuplei f the row operator
(Pl refer the abstract pdf file)

Ram Mohan, Devang S
In the first chapter of this report, our aim is to introduce harmonic maps between Riemann surfaces using the Energy integral of a map. Once we have the desired prerequisites, we move on to show how to continuously deform a given map to a harmonic map (i.e., find a harmonic map in its homotopy class). We follow J¨urgen Jost’s approach using classical potential theory techniques. Subsequently, we analyze the additional conditions needed to ensure a certain uniqueness property of harmonic maps within a given homotopy class. In conclusion, we look at a couple of applications of what we have shown...