On the % of zeros of Riemann zeta-Function on the critical line.

Speaker: 

Nicolas Martinez Robles

Institution: 

Univ. Illinois

Time: 

Tuesday, April 19, 2016 - 11:00am to 11:50am

Host: 

Location: 

RH 306

Abstract: We will review the techniques used to prove that a positive proportion of the zeros of the Riemann zeta-Function lie on the critical line Re(s)=1/2. The famous Riemann hypothesis states that all the zeros lie there. We will then discuss the mollifiers that allow us to show that > 41% of zeros are critical. This is joint work with A. Roy and A. Zaharescu.

Coupling for Brownian Motion with Redistribution

Speaker: 

Iddo Ben-Ari

Institution: 

University of Connecticut

Time: 

Tuesday, April 5, 2016 - 11:00am to 12:00pm

Location: 

RH 306

We consider a model of Brownian motion on a bounded interval which upon exiting the interval is being redistributed back  into the interval according to a probability measure depending on the exit point, then starting afresh, repeating the above mechanism indefinitely.  It is not hard to show that the process is exponentially ergodic, although characterizing the rate of convergence is non-trivial. In this talk, after providing a general overview of the probabilistic method of coupling and its applications,  I’ll show how to study the ergodicity for the model through coupling, how it leads to an  intuitive and geometric explanation for  the rates of convergence previously obtained analytically, other insights, and more questions. The talk will be accessible to general mathematical audience. 

Reminiscing on partition function zeros and the Lee-Yang circle theorem.

Speaker: 

Marek Biskup

Institution: 

UCLA

Time: 

Tuesday, March 29, 2016 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

I will review some (actually quite old) results by C. Borgs, J.T. Chayes, R. Kotecky and myself concerning the partition function zeros of the Ising model. The focus will be on the fact that, for specific boundary conditions, the zeros lie (in a suitable representation) on the unit circle. I will explain (1) the classic proof of the Lee-Yang circle theorem and (2) how one can nail the positions of the zeros up to exponentially small errors in the system size for the periodic boundary conditions. I may find time to explain how one uses this result to prove the so called Griffiths singularities in site-diluted Ising model.

A Boundedness Trichotomy for the Stochastic Heat Equation

Speaker: 

Davar Khoshnevisan

Institution: 

University of Utah

Time: 

Tuesday, January 26, 2016 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

  We consider the stochastic heat equation with a multiplicative space-time white noise forcing term under standard "intermitency conditions.” The main byproduct of this talk is that, under mild regularity hypotheses, the a.s.-boundedness of the solution$x\mapsto u(t\,,x)$ can be characterized generically by the decay rate, at $\pm\infty$, of the initial function $u_0$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $\Lambda:=\lim_{|x|\to\infty} \vert\log u_0(x)\vert/(\log|x|)^{2/3}$.

The spectral gap for random regular graphs

Speaker: 

Tobias Johnson

Institution: 

USC

Time: 

Tuesday, January 12, 2016 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

Expander graphs are useful across mathematics, all the way from number theory to applied computer science. The smaller the second eigenvalue of a regular graph, the better expander it is. Since this connection was discovered in the 1980s, researchers have tried to pinpoint the second eigenvalue of random regular graphs. The most prominent work in this direction was Joel Friedman's proof of Noga Alon's conjecture from 1985 that for a random d-regular graph on n vertices, the second eigenvalue is almost as small as possible, with high probability as n tends to infinity with d held fixed.

We consider the case of denser graphs, where d and n are both growing. Here, the best result (Broder, Frieze, Suen, Upfal 1999) holds only if d = o(n^(1/2)). We extend this to d = O(n^(2/3)). Our result relies on new concentration inequalities for statistics of random regular graphs based on the theory of size biased couplings, an offshoot of Stein's method. The theory we develop should be useful for proving concentration inequalities in a broad range of settings. This is joint with Nicholas Cook and Larry Goldstein.

Diffusive limits for stochastic kinetic equtions

Speaker: 

Arnaud Debussche

Institution: 

Univ. Rennes

Time: 

Tuesday, November 10, 2015 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

In this talk, we consider kinetic equations containing random

terms. The kinetic models contain a small parameter and it is well

known that, after scaling, when this parameter goes to zero the limit

problem is a diffusion equation in the PDE sense, ie a parabolic equation

of second order. A smooth noise is added, accounting for external perturbation.

It scales also with the small parameter. It is expected that the limit

equation is then a stochastic parabolic equation where the noise is in

Stratonovitch form.

Our aim is to justify in this way several SPDEs commonly used.

We first treat linear equations with multiplicative noise. Then show how

to extend the methods to nonlinear equations or to the more physical

case of a random forcing term.

The results have been obtained jointly with S. De Moor and J. Vovelle.

Low Correlation Noise Stability of Euclidean Sets

Speaker: 

Steve Heilman

Institution: 

UCLA

Time: 

Tuesday, November 24, 2015 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

The noise stability of a Euclidean set is a well-studied quantity.  This quantity uses the Ornstein-Uhlenbeck semigroup to generalize the Gaussian perimeter of a set.  The noise stability of a set is large if two correlated Gaussian random vectors have a large probability of both being in the set.  We will first survey old and new results for maximizing the noise stability of a set of fixed Gaussian measure.  We will then discuss some recent results for maximizing the low-correlation noise stability of three sets of fixed Gaussian measures which partition Euclidean space.  Finally, we discuss more recent results for maximizing the low-correlation noise stability of symmetric subsets of Euclidean space of fixed Gaussian measure.  All of these problems are motivated by applications to theoretical computer science.

Geometric properties of eigenfunctions for the fractional Laplacian

Speaker: 

Rodrigo Banuelos

Institution: 

Purdue University

Time: 

Tuesday, October 13, 2015 - 1:00pm to 2:00pm

Location: 

RH 306

 

Abstract

A classical result of H.J Brascamp and E.H. Lieb says that the ground state eigenfunction for the Laplacian in convex regions (and of Schr ̈odinger operators with convex potentials on Rn) is log-concave. A proof can be given (interpreted) in terms of the finite dimensional distributions of Brownian motion. Some years ago the speaker raised similar questions (and made some con- jectures) when the Brownian motion is replaced by other stochastic processes and in particular those with transition probabilities given by the heat kernel of the fractional Laplacian–the rota- tionally symmetric stable processes. These problems (for the most part) remain open even for the unit interval in one dimension. In this talk we elaborate on this topic and outline a proof of a result of M. Kaßmann and L. Silvestre concerning superharmonicity of eigenfunctions for certain fractional powers of the Laplacian. Our proof is joint work with D. DeBlassie. 

Synchronization by Noise

Speaker: 

Michael Scheutzow

Institution: 

Technische Universitat, Berlin

Time: 

Tuesday, September 22, 2015 - 11:00am to 11:50am

Host: 

Location: 

RH 306

 

Whenever a  deterministic system like an ODE or PDE does not possess an

asymptotically stable constant solution but if noise is added then there

exists a random  attractor which consists of a single (random) point,

then we call this phenomenon "synchronization by noise".

 

We first provide some specific examples and then present sufficient

conditions for synchronization to occur. Our results can be applied to

a large class of SDEs and some SPDEs with additive noise and to rather

general order-preserving random dynamical systems.

 

This is joint work with Franco Flandoli (Pisa) and Benjamin Gess (Leipzig).

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