Conference on New Developments in Probability, jointly hosted by Women in Probability

Schedule:

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 Maynard-Smith 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 d-dimensional 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 well-posedness 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 well-posed 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, k-dimensional 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 Erdos-Renyi 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 equal-sized 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 Erdos-Renyi 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.