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Series: Dissertation Defense

We provide a new definition of a local walk dimension beta that depends only on the metric. Moreover, we study the local Hausdorff dimension and prove that any variable Ahlfors regular measure of variable dimension Q is strongly equivalent to the local Hausdorff measure with Q the local Hausdorff dimension, generalizing the constant dimensional case. Additionally, we provide constructions of several variable dimensional spaces, including a new example of a variable dimensional Sierpinski carpet. We use the local exponent beta in time-scale renormalization of discrete time random walks, that are approximate at a given scale in the sense that the expected jump size is the order of the space scale. We consider the condition that the expected time to leave a ball scales like the radius of the ball to the power beta of the center. We then study the Gamma and Mosco convergence of the resulting continuous time approximate walks as the space scale goes to zero. We prove that a non-trivial Dirichlet form with Dirichlet boundary conditions on a ball exists as a Mosco limit of approximate forms. We also prove tightness of the associated continuous time processes.

Series: Geometry Topology Seminar

The h-principle is a powerful tool in differential topology which is used to study spaces of functionswith certain distinguished properties (immersions, submersions, k-mersions, embeddings, free maps, etc.). Iwill discuss some examples of the h-principle and give a neat proof of a special case of the Smale-HirschTheorem, using the "removal of singularities" h-principle technique due to Eliashberg and Gromov. Finally, I willdefine and discuss totally convex immersions and discuss some h-principle statements in this context.

Friday, April 27, 2018 - 15:05 ,
Location: Skiles 271 ,
Bhanu Kumar ,
GTMath ,
Organizer: Jiaqi Yang

This talk follows Chapter 4 of the well known text by Guckenheimer and Holmes. It is intended to present the theorems on averaging for systems with periodic perturbation, but slow evolution of the solution. Also, a discussion of Melnikov’s method for finding persistence of homoclinic orbits and periodic orbits will also be given. Time permitting, an application to the circular restricted three body problem may also be included.

Series: Math Physics Seminar

Electrons possess both spin and charge. In one dimension, quantum theory predicts that systems of interacting electrons may behave as though their charge and spin are transported at different speeds.We discuss examples of how such many-particle effects may be simulated using neutral atoms and radiation fields. Joint work with Xiao-Feng Shi

Series: Combinatorics Seminar

Given a collection of finite sets, Kneser-type problems aim to partition this collection into parts with well-understood intersection pattern, such as in each part any two sets intersect. Since Lovász' solution of Kneser's conjecture, concerning intersections of all k-subsets of an n-set, topological methods have been a central tool in understanding intersection patterns of finite sets. We will develop a method that in addition to using topological machinery takes the topology of the collection of finite sets into account via a translation to a problem in Euclidean geometry. This leads to simple proofs of old and new results.

Series: PDE Seminar

We prove an abstract theorem giving a $t^\epsilon$ bound for any $\epsilon> 0$ on the growth of the Sobolev norms in some abstract linear Schrödinger equations. The abstract theorem is applied to nonresonant Harmonic oscillators in R^d. The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator. Finally, time permitting, we will show how to construct a perturbation of the harmonic oscillator which provokes growth of Sobolev norms.

Series: Analysis Seminar

Abstract: I will state a version of Voiculescu's noncommutative
Weyl-von Neumann theorem for operators on l^p that I obtained.
This allows certain classical results concerning unitary
equivalence of operators on l^2 to be generalized to operators on
l^p if we relax unitary equivalence to similarity. For example,
the unilateral shift on l^p, 1

Series: PDE Seminar

(Due to a flight cancellation, this talk will be moved to Thursday (Apr 26) 3pm at Skiles 257). We prove an abstract theorem giving a $t^\epsilon$ bound for any $\epsilon> 0$ on the growth of the Sobolev norms in some abstract linear Schrödinger equations. The abstract theorem is applied to nonresonant Harmonic oscillators in R^d. The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator. Finally, time permitting, we will show how to construct a perturbation of the harmonic oscillator which provokes growth of Sobolev norms.

Series: Algebra Seminar

The talk reports on joint work with Wayne Raskind and concerns the conjectural definition of a new type of regulator map into a quotient of an algebraic torus by a discrete subgroup, that should fit in "refined" Beilinson type conjectures, exteding special cases considered by Gross and Mazur-Tate.The construction applies to a smooth complete variety over a p-adic field K which has totally degenerate reduction, a technical term roughly saying that cycles acount for the entire etale cohomology of each component of the special fiber. The regulator is constructed out of the l-adic regulators for all primes l simulateously. I will explain the construction, the special case of the Tate elliptic curve where the regulator on cycles is the identity map, and the case of K_2 of Mumford curves, where the regulator turns out to be a map constructed by Pal. Time permitting I will also say something about the relation with syntomic regulators.

Series: Geometry Topology Seminar

Novikov homology was introduced by Novikov in the
early 1980s motivated by problems in hydrodynamics. The Novikov
inequalities in the Novikov homology theory
give lower bounds for the number of critical points of a Morse closed 1-form on a compact
differentiable manifold M. In the first part of my talk I shall
survey the Novikov homology theory in finite dimensional setting
and its further developments in infinite dimensional setting with applications in the theory of symplectic fixed points and Lagrangian intersection/embedding problems. In the second part of my
talk
I shall report on my recent joint work with Jean-Francois Barraud and Agnes Gadbled on construction of the Novikov fundamental group
associated to a cohomology
class of a closed 1-form on M and its application to obtaining
new lower bounds for the number of
critical points of a Morse 1-form.