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Series: Stochastics Seminar

Random balls models are collections of Euclidean balls whose centers and radii are generated by a Poisson point process. Such collections model various contexts ranging from imaging to communication network. When the distributions driving the centers and the radii are heavy-tailed, interesting interference phenomena occurs when the model is properly zoomed-out. The talk aims to illustrate such phenomena and to give an overview of the asymptotic behavior of functionals of interest. The limits obtained include in particular stable fields, (fractional) Gaussian fields and Poissonian bridges. Related questions will also be discussed.

Series: Stochastics Seminar

A Markov intertwining relation between two Markov processes X and Y is a weak similitude relation G\Lambda = \Lambda L between their generators L and G, where \Lambda is a transition kernel between the underlying state spaces. This notion is an important tool to deduce quantitative estimates on the speed of convergence to equilibrium of X via strong stationary times when Y is absorbed, as shown by the theory of Diaconis and Fill for finite state spaces. In this talk we will only consider processes Y taking as values some subsets of the state space of X. Our goal is to present extensions of the above method to elliptic
diffusion processes on differentiable manifolds, via stochastic
modifications of mean curvature flows. We will see that Pitman's theorem about the
intertwining relation between the Brownian motion and the Bessel-3
process is curiously ubiquitous in this approach. It even serves as an inspiring guide to construct couplings associated to finite
Markov intertwining relations via random mappings, in the spirit of the
coupling-from-the-past algorithm of Propp and Wilson and of the evolving sets of Morris and Peres.

Wednesday, May 2, 2018 - 14:00 ,
Location: Skiles 006 ,
Hyunki Min ,
Georgia Tech ,
hmin38@gatech.edu ,
Organizer:

Understanding contact structures on
hyperbolic 3-manifolds is one of the major open problems in the area of contact
topology. As a first step, we try to classify tight contact structures on a specific hyperbolic 3-manifold. In this talk, we will review the previous classification
results and classify tight contact structures on the Weeks manifold, which
has the smallest hyperbolic volume. Finally,
we will discuss how to generalize this method to classify tight contact structures
on some other hyperbolic 3-manifolds.

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