ETD at Indian Institute of Science
(2,267 recursos)
Repository of Theses and Dissertations of Indian Institute of Science, Bangalore, India. The repository has been developed to capture, disseminate and preserve research theses of Indian Institute of Science.

Mathematics (math)

Mostrando recursos 1 - 20 de 44

1.
Fourier Analysis On Number Fields And The Global Zeta Functions - 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...

2.
Infinitely Divisible Metrics, Curvature Inequalities And Curvature Formulae - Keshari, Dinesh Kumar
The curvature of a contraction T in the Cowen-Douglas 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 Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples
of operators in the Cowen-Douglas class.
Secondly, we obtain...

3.
Analytic Continuation In Several Complex Variables - Biswas, Chandan
We wish to study those domains in Cn,for n ≥ 2, the so-called 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....

4.
Function Theory On Non-Compact Riemann Surfaces - 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 multi-valued functions. Such multi-valued 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 non-compact 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...

6.
Vector Bundles Over Hypersurfaces Of Projective Varieties - 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 non-singular projective variety, extends as a vector bundle to an open set containing that ample divisor.
Our method is to follow the general Groethendieck-Lefschetz 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...

7.
The Role Of Potential Theory In Complex Dynamics - 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...

9.
Joint Eigenfunctions On The Heisenberg Group And Support Theorems On Rn - 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 K-action on Cn is polar. We prove a Hecke-Bochner 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...

10.
Relative Symplectic Caps, Fibered Knots And 4-Genus - Kulkarni, Dheeraj
The 4-genus of a knot in S3 is an important measure of complexity, related to the unknotting number. A fundamental result used to study the 4-genus and related invariants of homology classes is the Thom conjecture, proved by Kronheimer-Mrowka, and its symplectic extension due to Ozsv´ath-Szab´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 4-manifold with contact type bounday ∂X and Σ is a symplectic surface in X such that ∂Σ is a transverse knot in ∂X. We show...

11.
Curvature Calculations Of The Operators In Cowen-Douglas Class - 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 vice-versa. Fortunately,...

13.
A Study Of The Metric Induced By The Robin Function - 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 sum-of-squares 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 =...

15.
Dilations, Functoinal Model And A Complete Unitary Invariant Of A r-contraction. - 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 r-contraction 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...

16.
An Algorithmic Approach To Crystallographic Coxeter Groups - Malik, Amita
Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. It turns out that the finite Coxeter groups are precisely the finite Euclidean reflection groups. Coxeter studied these groups and classified all finite ones in 1935, however they were known as reflection groups until J. Tits coined the term Coxeter groups for them in the sixties.
Finite crystallographic Coxeter groups, also known as finite Weyl groups, play a prominent role in many branches of mathematics like combinatorics, Lie theory, number theory, and geometry. The computational aspects of these groups are...

18.
Irreducible Representations Of The Symmetric Group And The General Linear Group - Verma, Abhinav
Representation theory is the study of abstract algebraic structures by representing their elements as linear transformations or matrices. It provides a bridge between the abstract symbolic mathematics and its explicit applications in nearly every branch of mathematics. Combinatorial representation theory aims to use combinatorial objects to model representations, thus answering questions in this ﬁeld combinatorially. Combinatorial objects are used to help describe, count and generate representations. This has led to a rich symbiotic relationship where combinatorics has helped answer algebraic questions and algebraic techniques have helped answer combinatorial questions.
In this thesis we discuss the representation theory of the symmetric...