We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian in the regime of small noise. We give a new proof of the Eyring-Kramers formula for the spectral gap of the associated generator of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg and of the mean-difference estimate introduced by Chafai and Malrieu. The Eyring-Kramers formula follows as a simple corollary from two main ingredients : The first one shows that the Gibbs measures restricted to a domain of attraction has a "good" Poincaré constant mimicking the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of the mean-difference by a new weighted transportation distance. It contains the main contribution of the spectral gap, resulting from exponential long waiting times of jumps between metastable states of the diffusion. This new approach also allows to derive sharp estimates on the log-Sobolev constant.

I plan to discuss spectral theory-type results for several stochastic interacting particle systems solvable by the coordinate Bethe ansatz. These results include Plancherel type isomorphism theorems which imply completeness and biorthogonality statements for the corresponding Bethe ansatz eigenfunctions. These constructions yield explicit solutions (in terms of multiple contour integrals) for backward and forward Kolmogorov equations with arbitrary initial data. Some of the formulas produced in this way are amenable to asymptotic analysis. In particular, I will discuss the (stochastic) q-Hahn zero-range process introduced recently by Povolotsky, and also the Asymmetric Simple Exclusion Process (ASEP). In particular, the spectral theory provides a new proof of the symmetrization identities of Tracy and Widom (for ASEP with either step or step Bernoulli initial configuration). Another degeneration takes the q-Hahn zero-range process to the stochastic q-Boson particle system dual to q-TASEP studied by Borodin, Corwin et al. Thus, at the spectral theory level we unify two discrete-space regularizations of the Kardar-Parisi-Zhang equation / stochastic heat equation, namely, q-TASEP and ASEP.

We consider the ferromagnetic Ising model on a sequence of graphs $G_n$ converging locally weakly to a rooted random tree. Generalizing [Montanari, Mossel, and Sly (2012)], under an appropriate “continuity" property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with + and - boundary conditions on that tree. Under the extra assumptions that $G_n$ are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization, is the Ising measure with + boundary condition on the limiting tree. The “continuity" property holds except possibly for countably many choices of $\beta$, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton Watson trees. This talk is based on a joint work with Amir Dembo.

9:00 Carl Mueller (Rochester) 10:30 David Herzog (Duke) 1:00 Elchanan Mossel (Berkeley)

In this talk, we discuss stochastic homogeneization on the Hopf fibration. Let us consider Berger's metrics on the three sphere, obtained by shrinking the Hopf circle directions by a factor epsilon. So we think of three spheres as two spheres attached at each point a circle. We consider a particle that is moved by two vector fields: a unit speed vector field, with respect to Berger's metrices, along the Hopf circle; and also a non-zero vector field in` \(S^2\) direction' with speed given by a one dimensional Brownian motion. In the limit of epsilon goes to zero, we obtain a Brownian motion on \(S^2\). The effective motion is obtained by moving a particle along a fast rotating horizontal direction.

I will discuss a series of results concerning the effect of the derivative operator on the locations of the zeros of a random analytic function. Two models are considered. In the first, the zeros are chosen IID from some measure on the complex plane. In the second, the zeros are chosen to be a Poisson point process on the real line. Repeated differentiation results in a nearly deterministic zero set.

The analytic framework for estimating coefficients of a generating function is the same in many variables as in one variable: evaluate Cauchy's integral by manipulating the contour into a "standard" position. That being said, the geometry when dealing with several complex variables can be much more complicated. This talk, drawing on the recent book (with Mark Wilson) of the same title, surveys analytic methods for extracting asymptotics from multivariate generating functions. I will try to give an idea of the main pieces of the puzzle. In particular, I will try to explain in pictures the roles of Morse theory, complex algebraic geometry and hyperbolicity in the asymptotic evaluation of integrals.

A simplicial complex is a collection of vertices, edges, triangles, and simplexes of higher dimensions, and one can think of it as a generalization of a graph. Given a random set of points P in a metric space and a real number r > 0, one can create a simplicial complex by looking at the balls of radius r around the points in P, and adding a k-dimensional face for every subset of k+1 balls that has a nonempty intersection. This construction produces a random topological space known as the Èech complex - C(P,r). We wish to study the homology of this space, more specifically - its Betti numbers - the number of connected components and 'holes' or 'cycles'. In this talk we discuss the limiting behavior of the random Èech complex as the number of points in P goes to infinity and the radius r goes to zero. We show that the limiting behavior exhibits multiple phase transitions at different levels, depending on the rate at which the radius goes to zero. We present the different regimes and phase transitions discovered so far, and observe the nicely ordered fashion in which cycles of different dimensions appear and vanish.

Motivated by several biological questions, including insect respiration, we consider ODEs with stochastically switching right-hand sides and PDEs with stochastically switching boundary conditions. In a variety of situations, we prove that the system exhibits surprising behavior. In this talk, we will highlight some of the most interesting results and describe their implications both for the mathematical study of stochastic hybrid systems and for insect respiration.

The graph Laplacian has been of interest in statistics, machine learning, and theoretical computer science in areas from manifold learning to analysis of Markov chains. A common uses of the graph Laplacian has been in spectral clustering and dimension reduction. A theoretical motivation for why spectral clustering works is the Cheeger inequality which relates the eigenvalues of the graph Laplacian to how disconnected the graph is, Betti zero for topology. We ask how the Cheeger inequality extends to higher-order Laplacians, operators on simplicial complexes, and what clustering means for these higher-order operators. This is joint work with John Steenbergen Related to the graph Laplacian is the idea of random walks on graphs. We will define a random walk on simplicial complexes with a stationary distribution that is related to the k-dimensional Laplacian. The stationary distribution reveals (co)homology of the geometry of the random walk. We apply this random walk to the problem of semi-supervised learning, given some labeled observations and many unlabeled observations how does one propagate the labels.

A Dirichlet random probability \(P_t\) on \(\mathbb{R}^d\) of intensity \(t\) and governed by the probability \(\alpha\) is such that for any partition \( (A_0,\ldots,A_n)\) of \(\mathbb{R}^d\) the random variable \( (P_t(A_0),\ldots,P_t(A_n))\) is Dirichlet distributed with parameters \( (t\alpha(A_0),\ldots,t\alpha(A_n).\) If \(\mu(t\alpha)\) is the distribution of \(X_t=\int xP_t(dx),\) the Dirichlet curve is the map \(t\mapsto \mu(t\alpha)\). Its study raises challenging problems and explicit computations are rare. We prove that if \(\lim_{t\to\infty}\mu(t\alpha)\) exists, it is a Cauchy or Dirac distribution on \(\mathbb{R}^d\). If \(\alpha\) has an expectation we prove that \(t\mapsto \int \psi(x)\mu(t\alpha)(dx)\) is decreasing for any positive convex function \(\psi\) on \(\mathbb{R}^d.\) In other terms the Dirichlet curve decreases in the Strassen order. This is joint work with Mauro Piccioni.

The classical result of Bramson gives a precise logarithmic correction to the speed of front propagation in one dimensional branching random walks and Brownian motions. I will discuss several variants of this model where the slowdown term is not classical.

A link to the paper can be found on her web page. In solid tumors, targeted treatments can lead to dramatic regressions, but responses are often short-lived because resistant cancer cells arise. The major strategy proposed for overcoming resistance is combination therapy. We present a mathematical model describing the evolutionary dynamics of lesions in response to treatment. We first studied 20 melanoma patients receiving vemurafenib. We then applied our model to an independent set of pancreatic, colorectal, and melanoma cancer patients with metastatic disease. We find that dual therapy results in long-term disease control for most patients, if there are no single mutations that cause cross-resistance to both drugs; in patients with large disease burden, triple therapy is needed. We also find that simultaneous therapy with two drugs is much more effective than sequential therapy. Our results provide realistic expectations for the efficacy of new drug combinations and inform the design of trials for new cancer therapeutics.

Abstract:One way to convince ourselves that no cooperation can evolve among defectors is via a simple yet one of the most famous games in all of game theory - the Prisoner’s dilemma (PD) game. The players of this game adopt one of the two strategies: a) a cooperator who pays a cost so that another individual can receive a benefit, or b) a defector who can receive benefits, but it has no cost as it does not deal out any benefits at all. As seen from this formulation, no rational individual would opt to be a cooperator. Yet, we can see cooperation everywhere around us and thus (assuming defectors were here first) there must exist at least one mechanism for its evolution. Nowak (2006, 2012) discusses several of such mechanisms, including the kin selection by which cooperation can spread if the benefits go primarily to genetic relatives. In this talk we will introduce a simple PD-like asymmetric matrix game and show how Hamilton’s rule can easily be recovered. We will also introduce a simple PD-like symmetric matrix game to model the evolution of cooperation via greenbeard mechanism, which can be seen as a special case of kin selection.

This is joint work with David Herzog, Kevin McGoff, David Sivakoff and Evan Myers. We develop a dynamic random graph model to capture the heterogeneous structure of adolescent sexual networks. Coupled to a stochastic model of infection with the human papillomavirus (HPV), the network model is used to investigate the effect of different HPV vaccination strategies. The results obtained with the stochastic agent-based model are confirmed and extended by means of a deterministic mean-field model amenable to rigorous analysis. Using parameter values reflecting the current situation in the US, we show that for a large class of cost-benefit measures it is more effective to start implementing male-vaccination than to extend female vaccination further. In view of the stagnating female and low male coverage in the US, our results demonstrate the necessity for empirical assessment of coverage-dependent marginal administration costs of the vaccine.

Gaussian processes are employed to model spatially varying errors in various stochastic systems. In this talk, we consider the analysis of the extreme behaviors and the rare-event simulation problems for such systems. In particular, the topic covers various nonlinear functionals of Gaussian processes including the supremum norm and integral of convex functions. We present the asymptotic results and the efficient simulation algorithms for the associated rare-event probabilities.

Dvoretzky's theorem tells us that if we put an arbitrary norm on n-dimensional Euclidean space, no matter what that normed space is like, if we pass to subspaces of dimension about log(n), the space looks pretty much Euclidean. A related measure-theoretic phenomenon has long been observed: the (one-dimensional) marginals of many natural high-dimensional probability distributions look about Gaussian. A question which had received little attention until recently is whether this phenomenon persists for k-dimensional marginals for k growing with n, and if so, for how large a k? In this talk I will discuss recent work showing that the phenomenon does indeed persist if k less than 2log(n)/log(log(n)), and that this bound is sharp (even the 2!).

Formation of crystals, spread of infections, and flow of fluids through porous rocks are modeled mathematically as systems consisting of many particles that behave randomly. We will use fluctuations to quantify the randomness, and measure its decay as the number of particles increase. Then we will study the opposite problem: growth of randomness. It turns out that situations exist where it is beneficial to increase chaos. As one example, we will study methods to anonymously distribute files over the internet in such a way that nobody can trace the senders.

Tempered stable distributions were introduced in Rosinski 2007 as models that look like infinite variance stable distributions in some central region, but they have lighter (i.e. tempered) tails. We extend this class of models to allow for more variety in the tails. While some cases no longer correspond to stable distributions they serve to make the class more flexible and in certain subclasses they have been shown to provide a good fit to data. To characterize the possible tails we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs. We then characterize the weak limits of sequences of tempered stable distributions. We will conclude by discussing a mechanism by which distributions that are stable-like in some central region but with lighter tails show up in applications.

In this talk I will discuss a framework for transforming one variant of the Schramm–Loewner evolution (SLE) into another. The main tool in this approach is the Brownian loop measure. A simple case is to relate the reversal of radial SLE to whole-plane SLE, which looks the same locally. Writing the formula one might naïvely expect fails, because the loop measure term is infinite. In joint work with Greg Lawler, we show that there is a finite normalized version of the loop measure term, and that with this change, the naïve formula relating the two SLEs becomes correct.

The idea behind excited random walks (ERWs), roughly speaking, is to take a well-known underlying process (such as, for example, simple symmetric random walk (SSRW)) and modify its transition probabilities for the "first few" visits to every site of the state space. These modifications can be deterministic or random. The resulting process is not markovian, and its properties can be very different from those of the underlying process. I shall give a short review of some of the known results for ERW (with SSRW as underlying process) on the d-dimensional integer lattice and then concentrate on a specific model for d=1. For this model we can give a complete picture including functional limit theorems.

In this talk, I will discuss several finite-horizon Markov decision problems (MDPs) in which the goal is to gather distributional information regarding the total reward that one obtains when implementing a policy that maximizes total expected rewards. I will begin by studying the optimal policy for the sequential selection of an alternating subsequence from a sequence of n independent observations from a continuous distribution, and I will prove a central limit theorem for the number of selections made by that policy. Then, I will discuss a simple version of a sequential knapsack problem, and I will use its structure to characterize a class of MDPs in which the optimal total reward has variance that can be bounded in terms of its mean. Surprisingly, such characterization turns out to be common in several examples of MDPs from operations research, financial engineering and combinatorial optimization. (The talk is based on joint work with Robert W. Chen, Noah Gans, Larry Shepp, and J. Michael Steele.)

I will discuss a stochastic model of mutation accumulation and spread in a spatially-structured population. This situation arises in a variety of ecological and biological problems, including the process of cancer initiation from healthy tissue. Cancer arises via the accumulation of mutations to the genetic code. Although many mathematical models of cancer initiation have assumed `perfect mixing' or spatial homogeneity, solid tumors often initiate from tissues with well-regulated spatial architecture and dynamics. Here, we study a stochastic model to investigate the temporal dynamics and patterns of mutation accumulation (i.e. how they depend on system parameters such as mutation rate, population size, and selective fitness advantage of mutations). Joint work with R. Durrett (Duke) and K. Leder (Minnesota).

The goal of this talk is to give a self-contained introduction to some aspects of the thermodynamic formalism in ergodic theory that should be accessible to probabilists. In particular, the talk will focus on equilibrium states and Gibbs measures on the Z^d lattice. We'll present some basic examples in the theory, as well as some recent results that are joint with Christopher Hoffman.