etd@IISc Collection:
http://hdl.handle.net/2005/41
2016-02-01T18:37:08ZRiesz Transforms Associated With Heisenberg Groups And Grushin Operators
http://hdl.handle.net/2005/2496
Title: Riesz Transforms Associated With Heisenberg Groups And Grushin Operators
Authors: Sanjay, P K
Abstract: We characterise the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free 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 - |x|2@t2 on Rn+1. We prove that both the first order and higher order Riesz transforms are bounded on Lp(Rn+1): We also prove that norms of the first order Riesz transforms are independent of the dimension n.2015-12-07T18:30:00ZRigidity And Regularity Of Holomorphic Mappings
http://hdl.handle.net/2005/2447
Title: Rigidity And Regularity Of Holomorphic Mappings
Authors: Balakumar, G P
Abstract: 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 h-extendible(or semi-regular) 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 strongly pseudo convex domains is also proved.
The second theme dealt with, is the classification upto biholomorphic equivalence of model domains with abelian automorphism group in C3 .It is shown that every model domain i.e.,a hyperbolic rigid polynomial domainin C3 of finite type, with abelian automorphism group is equivalent to a domain that is balanced with respect to some weight.2015-07-15T18:30:00ZRicci Flow And Isotropic Curvature
http://hdl.handle.net/2005/2376
Title: Ricci Flow And Isotropic Curvature
Authors: Gururaja, H A
Abstract: 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 the following holds: If two Riemannian manifolds have positive S- curvature then their connected sum also admits a Riemannian metric of positive S- curvature.
The other subclass consists of those sets for which the normalized Ricci flow on a closed Riemannian manifold with positive S-curvature converges to a metric of constant positive sectional curvature.
In the second part of the thesis, we study the behavior of Ricci flow for a manifold having positive S - curvature, where S is in the first subclass. More specifically, we study the Ricci flow for a special class of metrics on Sp+1 x S1 , p ≥ 4, which have positive isotropic curvature.2014-09-02T18:30:00ZFourier Analysis On Number Fields And The Global Zeta Functions
http://hdl.handle.net/2005/2355
Title: Fourier Analysis On Number Fields And The Global Zeta Functions
Authors: Fernandes, Jonathan
Abstract: 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 of the conjugates of the number in the complex field. He then showed that these “Hecke” zeta functions satisfied the same type of functional equation as the Dedekind zeta function, but with a much more complicated factor.
In his doctoral thesis, John Tate replaced the classical notion of zeta function, as a sum over integral ideals of a certain type of ideal character, by the integral over the idele group of a rather general weight function times an idele character which is trivial on field elements. He derived a Poisson Formula for general functions over the adeles, summed over the discrete subgroup of field elements. This was then used to give an analytic continuation for all of the generalized zeta functions and an elegant functional equation was established for them. The mention of the Poisson Summation Formula immediately reminds one of the Theta function and the proof of the functional equation for the Riemann zeta function. The two proofs share close analogues with the functional equation for the Theta function now replaced by the number theoretic Riemann–Roch Theorem. Translating the results back into classical terms one obtains the Hecke functional equation, together with an interpretation of the complicated factor in it as a product of certain local factors coming form the archimedean primes and the primes of the conductor.
This understanding of Tate’s results in the classical framework essentially boils down to constructing the generalized weight function and idele group characters which are trivial on field elements. This is facilitated by the understanding of the local zeta functions. We explicitly compute in both cases, the local and the global, illustrating the working of the ideas in a concrete setup. I have closely followed Tate’s original thesis in this exposition.2014-08-03T18:30:00Z