
Ray, Samya Kumar
Grothendieck published an extraordinary paper entitled ”Resume de la theorie metrique des pro¬duits tensoriels topologiques” in 1953. The main result of this paper is the inequality which is commonly known as Grothendieck Inequality.
Following Kirivine, in this article, we give the proof of Grothendieck Inequality. We refor¬mulate it in different forms. We also investigate the famous Grothendieck constant KG. The Grothendieck constant was achieved by taking supremum over a special class of matrices. But our attempt will be to investigate it, considering a smaller class of matrices, namely only the positive definite matrices in this class. Actually we want to...

Banerjee, Tamal
Recently, the financial world witnessed a series of major defaults by several institutions and investment banks. Therefore, it is not at all surprising that credit risk analysis have turned out to be one of the most important aspect among the finance community. As credit derivatives are long term instruments, it is affected by the changes in the market conditions. Thus, it is a appropriate to take into consideration the effects of the market economy. This thesis addresses some of the important issues in credit risk analysis in a regime switching market. The main contribution in this thesis are the followings:
(1)...

Sanjay, P K
We characterise the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimensionfree bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions.
Next we study the Riesz transforms associated to the Grushin operator G =  Δ  x2@t2 on Rn+1. We prove that both the first order and higher order Riesz...

Balakumar, G P
We deal with two themes that are illustrative of the rigidity and regularity of holomorphic
mappings.
The first one concerns the regularity of continuous CR mappings between smooth pseudo convex, finite type hypersurfaces which is a well studied subject for it is linked with the problem of studying the boundary behaviour of proper holomorphic mappings between domains bounded by such hypersurfaces. More specifically, we study the regularity of Lipschitz CR mappings from an hextendible(or semiregular) hypersurface in Cn .Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from...

Gururaja, H A
This thesis consists of two parts. In the first part, we study certain Ricci flow invariant nonnegative curvature conditions as given by B. Wilking. We begin by proving that any such nonnegative curvature implies nonnegative isotropic curvature in the Riemannian case and nonnegative orthogonal bisectional curvature in the K¨ahler case. For any closed AdSO(n,C) invariant subset S so(n, C) we consider the notion of positive curvature on S, which we call positive S curvature. We show that the class of all such subsets can be naturally divided into two subclasses:
The first subclass consists of those sets S for which...

Fernandes, Jonathan
The study of zeta functions is one of the primary aspects of modern number theory. Hecke was the first to prove that the Dedekind zeta function of any algebraic number field has an analytic continuation over the whole plane and satisfies a simple functional equation. He soon realized that his method would work, not only for Dedekind zeta functions and L–series, but also for a zeta function formed with a new type of ideal character which, for principal ideals depends not only on the residue class of the number(representing the principal ideal) modulo the conductor, but also on the position...

Keshari, Dinesh Kumar
The curvature of a contraction T in the CowenDouglas class is bounded above by the
curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this thesis, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the CowenDouglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples
of operators in the CowenDouglas class.
Secondly, we obtain...

Biswas, Chandan
We wish to study those domains in Cn,for n ≥ 2, the socalled domains of holomorphy, which are in some sense the maximal domains of existence of the holomorphic functions defined on them. We demonstrate that this study is radically different from that of domains in C by discussing some examples of special types of domains in Cn , n ≥2, such that every function holomorphic on them extends to strictly larger domains. Given a domain in Cn , n ≥ 2, we wish to construct the maximal domain of existence for the holomorphic functions defined on the given domain....

Philip, Eliza
The theory of Riemann surfaces is quite old, consequently it is well developed. Riemann surfaces originated in complex analysis as a means of dealing with the problem of multivalued functions. Such multivalued functions occur because the analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function. The theory splits in two parts; the compact and the noncompact case. The function theory developed on these cases are quite dissimilar. The main difficulty one encounters in the compact case is the scarcity of global holomorphic functions, which limits one’s study to meromorphic...

Jana, Indrajit

Tripathi, Amit
In this thesis we study some questions related to vector bundles over hypersurfaces. More precisely, for hypersurfaces of dimension ≥ 2, we study the extension problem of vector bundles. We find some cohomological conditions under which a vector bundle over an ample divisor of nonsingular projective variety, extends as a vector bundle to an open set containing that ample divisor.
Our method is to follow the general GroethendieckLefschetz theory by showing that a vector bundle extension exists over various thickenings of the ample divisor.
For vector bundles of rank > 1, we find two separate cohomological conditions on vector bundles...

Bandyopadhyay, Choiti
Potential theory is the name given to the broad field of analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, Green’s functions, potentials and capacity. In this text, our main goal will be to gain a deeper understanding towards complex dynamics, the study of dynamical systems defined by the iteration of analytic functions, using the tools and techniques of potential theory. We will restrict ourselves to holomorphic polynomials in C.
At first, we will discuss briefly about harmonic and subharmonic functions. In course, potential theory will repay its debt to complex analysis in the form of some...

Garg, Rahul

Samanta, Amit
This work is concerned with two different problems in harmonic analysis, one on the Heisenberg group and other on Rn, as described in the following two paragraphs respectively.
Let Hn be the (2n + 1)dimensional Heisenberg group, and let K be a compact subgroup of U(n), such that (K, Hn) is a Gelfand pair. Also assume that the Kaction on Cn is polar. We prove a HeckeBochner identity associated to the Gelfand pair (K, Hn). For the special case K = U(n), this was proved by Geller, giving a formula for the Weyl transform of a function f of the...

Kulkarni, Dheeraj
The 4genus of a knot in S3 is an important measure of complexity, related to the unknotting number. A fundamental result used to study the 4genus and related invariants of homology classes is the Thom conjecture, proved by KronheimerMrowka, and its symplectic extension due to Ozsv´athSzab´o, which say that closed symplectic surfaces minimize genus.
In this thesis, we prove a relative version of the symplectic capping theorem. More precisely, suppose (X, ω) is a symplectic 4manifold with contact type bounday ∂X and Σ is a symplectic surface in X such that ∂Σ is a transverse knot in ∂X. We show...

Deb, Prahllad
In a foundational paper “Operators Possesing an Open Set of Eigenvalues” written several decades ago, Cowen and Douglas showed that an operator T on a Hilbert space ‘H possessing an open set Ω C of eigenvalues determines a holomorphic Hermitian vector bundle ET . One of the basic theorems they prove states that the unitary equivalence class of the operator T and the equivalence class of the holomorphic Hermitian vector bundle ET are in one to one correspondence. This correspondence appears somewhat mysterious until one detects the invariants for the vector bundle ET in the operator T and viceversa. Fortunately,...

Sen, Suparna

Borah, Diganta
Let D be a smoothly bounded domain in Cn , n> 1. For each point p _ D, we have the Green function G(z, p) associated to the standard sumofsquares Laplacian Δ with pole at p and the Robin constant __
Λ(p) = lim G(z, p) −z − p−2n+2
z→p 
at p. The function p _→ Λ(p) is called the Robin function for D.
Levenberg and Yamaguchi had proved that if D is a C∞smoothly bounded pseudoconvex domain, then the function log(−Λ) is a real analytic, strictly plurisubharmonic exhaustion function for D and thus induces a metric
ds2 =...

Manna, Durga Pada

Pal, Sourav
A pair of commuting bounded operators (S, P) for which the set
r = {(z 1 +z 2,z 1z 2) : z 1 ≤1, z 2 ≤1} C2 is a spectral set, is called a rcontraction in the literature. For a contraction P and a bounded commutant S of P, we seek a solution of the operator equation
S –S*P = (I –P*P)½ X(I –P*P)½
where X is a bounded operator on Ran(I – P*P)½ with numerical radius of X being not
greater than 1. We show the existence and uniqueness of solution to the operator equation above when (S,P) is a...