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Schedule:
Friday, May 6th (Morning: Pancoe Auditorium, Afternoon: Annenberg G21)
8.30am  9.00am

Breakfast

9.00am  9.45am

Sandra Cerrai (Univ. of Maryland)
Large deviations for the twodimensional stochastic NavierStokes equation with vanishing noise correlation
 Abstract
We are dealing with the validity of a large deviation principle for the twodimensional NavierStokes equation, with periodic boundary conditions, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $\e$ and $\d(\e)$, respectively, with $0<\e,\ \d(\e)<<1$. Depending on the relationship between $\e$ and $\d(\e)$ we will prove the validity of the large deviation principle in different functional spaces

10.00am  10.45am

Rodrigo Banuelos (Purdue University)
HardyLittlewoodSobolev/BurkholderGundy, Doob  Abstract
The HardyLittlewoodSobolev (HLS) inequality (containing the classical Sobolev inequality) has been refined and extended in many directions, including to the setting of symmetric Markovian semigroups. The latter is a 1985 celebrated result of N. Varopoulos which has had many applications. In this talk we present a proof of the HLS inequality given by L. Hedberg in 1972 which extends, with almost no change, to Markovian semigroups. We give a stochastic integral formulation of the HLS inequality and adapt Hedberg’s argument to prove it based on the BurkholderGundy and Doob inequalities. The motivation for this approach comes from efforts to employ probabilistic techniques which have been extremely successful in proving sharp inequalities for various singular integrals, to study (and extend to other geometric settings) the sharp form of the HLS inequality proved by E. Lieb in 1983 and which has been of great interest to many in recent years.

2.00pm  2.45pm

Lea Popovic (Concordia University)
Large deviations for two timescale jumpdiffusions and Markov chain models  Abstract
For a number of processes in biology the appropriate stochastic modelling is done in
terms of multiscale Markov processes with fully dependent slow and fast fluctuating variables.
The most common examples of such multiscale processes are deterministic evolutions, jumpdiffusions,
and state dependent Markov chains. The law of large numbers limit, central limit theorem,
and the corresponding large deviations behaviour of these models are all of interest in applications.
In this talk I will give a proof of the large deviations principle for such multiscale systems,
and give an example of an intracellular reaction model on two timescales for which these results apply.

3.00pm  3.45pm

Nayantara Bhatnagar (Univ. of Delaware)
Decay of Correlations for the Hardcore Model in Random Regular Graphs  Abstract
Gibbs measures on trees play a central role in the theory of spins systems on random graphs. We determine the local weak limit of the hardcore model on random regular graphs upto a density for the largest independent set that is bounded by and goes asymptotically to the condensation threshold. We show that the hardcore measure converges in probability locally in a strong sense to the free boundary condition Gibbs measure on the tree. As a consequence, the reconstruction threshold on the random graph is equal to the reconstruction threshold on the dregular tree.
This is joint work with Allan Sly and Prasad Tetali.

3.45pm  4.15pm

Coffee Break

4.15pm  5.00pm

Davar Khoshnevisan (Univ. of Utah)
Dissipation and Parabolic SPDEs
 Abstract
Physical intermittency of a complex system is frequently associated to the presence of a number of mathematicallydescribable properties of that
system. Those properties typically include the existence of nontrivial moment Lyapunov
exponents and dissipation. The past few years have been witness to a number of results that describe the existence of nontrivial moment Lyapunov exponents or a large family of parabolic SPDEs.
The focus of this talk is the dissipative properties of a family of such parabolic SPDEs.
A number of consequences of dissipation are also discussed.
This talk is based on ongoing work with Kunwoo Kim (Pohang), Carl Mueller (Rochester), and ShangYuan Shiu (Taiwan).

Saturday, May 7th (Swift 107)
10.00am  10.45am

Rick Durrett (Duke)
Spatial evolutionary games with small selection coefficients
 Abstract
The use of evolutionary game theory biology dates to work of MaynardSmith who used it to explain why most fights between animals were of the limited war type. In the mid 1990s Nowak and May demonstrated that a spatial distribution of players can explain the existence of altruism, which would die out in a homogeneously mixing population. Recently, evolutionary games have been used to model cancer, e.g., the fact that stromal cells may cooperate with prostate tumor cells to help cancer spread.
We use results of Cox, Durrett, and Perkins for voter model perturbations to study spatial evolutionary games when the fitness differences between strategies are small. There are two main results: on the ddimensional lattice the effect of space is equivalent to (i) changing the entries of the game matrix and (ii) replacing the replicator ODE by a related PDE. On the torus there are two different weak regimes that lead to a PDE and an ODE respectively. In the second case, our results allow us to prove results of Tarnita and Nowak.
The first result comes from a 2014 paper in Electronic Journal of Probability. The second has not yet been published but can be seen online at Stochastic Processes and Their Applications. Both papers are on my web page.

10.45am  11.15am

Coffee Break

11.15am  12.00pm

Gigliola Staffilani (MIT)
The Study of Wave and Dispersive Equations: Random Versus Deterministic Approach  Abstract
The point of this talk is to show how certain wellposedness results that are not available using deterministic techniques involving Fourier and harmonic analysis
can be obtained when introducing randomization in the set of initial data. Along the way I will also prove a certain “probabilistic propagation of regularity” for certain almost sure globally wellposed dispersive equations and I will show how randomization is naturally suitable for nonlinearities in null forms.

2.00pm  2.45pm

Vadim Gorin (MIT)
Lozenge tilings: universal bulk limits, global fluctuations  Abstract
The talk is about random lozenge tilings of large domains on the triangular grid.
We will discuss both global asymptotic behavior of such tilings (analogues of the classical Law
of Large Numbers and Central Limit Theorem), and universal bulk local behavior, where
translation invariant ergodic Gibbs measures on tilings of the plane arise. An important role
in the study is played by tilings of trapezoid domains: these are the most accessible integrable
cases because of their connections to representation theory and symmetric functions.

3.00pm  3.45pm

Elisabeth Werner (Case Western)
Best and random approximation of convex bodies by polytopes  Abstract
How well can a convex body be approximated by a polytope?
This is a fundamental question in convex geometry, also in view of applications
in many other areas of mathematics and related fields.
It often involves side conditions like a prescribed number of vertices, or, more generally,
kdimensional faces and a requirement that the body contains the polytope or vice versa.
Accuracy of approximation is often measured in the symmetric difference metric, but
other metrics can and have been considered.
We will present several results on these issues, mostly related to approximation by
“random polytopes”.

3.45pm  4.15pm

Coffee Break

4.15pm  5.00pm

Roger Jones (DePaul)
Almost Everywhere Convergence and the Work of Alexandra Bellow  Abstract
Alexandra Bellow has written a number of very interesting papers, and posed a number of questions, during her career. She was the first person to show that the ergodic averages along any lacunary sequence will diverge for some function in L1. She, with her coauthor, Viktor Losert, were the first to exhibit a sequence of density zero along which the ergodic averages will converge a.e. for all f ∈ L1. She was the first person to ask about convergence of ergodic averages along the sequence of squares, and was the first person to show that given p1 ≥ 1 and p0 > p1, there is a subsequence such that the ergodic averages converge for all f ∈ Lp0 but diverge for some f ∈ Lp1 . These results, and others, have resulted in subsequent papers by a number of other authors. Thus her work has had a considerable impact on the field of almost everywhere convergence in ergodic theory.
This talk will examine the development of some of her work. In particular we will talk about the paper on moving averages, which came about while trying to find a way to approach the problem about convergence of ergodic averages along the sequence of squares. The proof in the final version of the moving averages paper is very different from the initial proof, and resulted in a stronger theorem. Modifi cations of the initial proof resulted in finding conditions that imply convergence of convolution powers of a measure, as well as a number of other results.

5.10pm  5:30pm

Alexandra Bellow (Northwestern)
"Looking back, looking forward" (A few mathematical reminiscences)

6:00pm  (Harris Hall 108)

Reception

Sunday, May 8th (Swift 107)
10.00am  10.45am

Ioana Dumitriu (U. Washington)
Clustering in two different block stochastic models  Abstract
Clustering and community detection in large networks are important problem with a wide spectrum of applications, from marketing to machine learning and from genomics to social sciences. Given the adjacency matrix of the large block stochastic network with blocks modeled by ErdosRenyi graphs, the problem consists of devising an algorithm that identifies (correctly or approximately) the original vertex partition, with high probability. The problem is difficult enough that the only case that has been completely solved is the two equalsized set case (binary, balanced SBM) through a concerted effort by Mossel, Neeman and Sly, parallelled by Massoulie.
The first part of the talk will introduce and analyze a regular binary SBM, where the ErdosRenyi building block is replaced by uniform regular graphs. The second part will involve recent work on a general SBM model involving recovery regimes (sets of parameters for which correct identification of the partition is possible).

11.15am  12.00pm

Arnab Sen (Minnesota)
Roots of Random Littlewood Polynomials
 Abstract
We consider random polynomials whose coefficients are independent and uniform on {1,1}. We will show that the probability that such a polynomial of degree n has a double root is o(n^{2}) when n+1 is not divisible by 4 and is of the order n^{2} otherwise. We will also discuss extensions to random polynomials with more general coefficient distributions.
This is joint work with Ohad Feldheim, Ron Peled and Ofer Zeitouni.

Organizers
The scientific organizers are Tai Melcher, Kay Kirkpatrick and Kavita Ramanan.
The local organizers are Antonio Auffinger and Elton Hsu.
Acknowledgements
This meeting is partially supported by the National Science Foundation. By Grant NSF DMS 1255574 (PI: Tai Melcher) and by a grant to the probability group at Northwestern University
and by the Northwestern Mathematics Department as part of the 2015/2016 emphasis year in probability theory.