First passage percolation is the study of a random metric space generated by replacing each edge in a graph by an edge of a random length. The distance between two vertices u and v is the length of the shortest path connecting u and v. An infinite path P is a geodesic if for any two vertices u and v on P the shortest path between them in the random graph is along P. It is easy to show that in the nearest neighbor graph with vertices Z^2 that there exists at least one (one sided) infinite geodesic starting at any given vertex. It is widely expected that there are infinitely many such one sided infinite geodesics that begin at the origin, with (at least) one in every direction. But it turns out to be very difficult to prove that there are even two with positive probability. We will discuss some recent results which get closer to proving this widely held belief.

In 1971, Schelling introduced a model in which individuals move if they have too many neighbors of the opposite type. In this paper we will consider a metapopulation version of the model in which a city is divided into N neighborhoods each of which has L houses. There are ρ NL red indivdiuals and an equal number of blue individuals. Individuals are happy if the fraction of individuals of the opposite type in their neighborhood, is ≤ ρ_cand move to vacant houses at rates that depend on their state and that of their destination. Our goal is to show that if L is large then as ρ passes through ρ_c the system goes from a homogeneous state in which all neighborhoods have \approx ρL of each color to a segregated state in which 1/2 of the neighborhoods have ρ₁L reds and ρ₂L blues and 1/2 with the opposite composition.

I will present results on constructing an estimator of a function on a compact manifold for the purpose of recovering its "topology". What this means will be explained in detail. The talk will conclude with an application to brain imaging.

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.)

We introduce some recent advances in the study of nonlinear stochastic heat equations, and related stochastic PDEs. Special attention will be paid to the local structure of the solution. In particular, we show that, frequently, the solution exhibits a form of intermittency. Time permitting, we discuss related connections to classical potential theory and mathematical physics as well.

We study the long time behaviour of the heat kernel on Abelian covers of compact Riemannian manifolds. For manifolds without boundary work of Lott and Kotani-Sunada establishes precise long time asymptotics. Extending these results to manifolds with boundary reduces to a cute eigenvalue minimization problem, which we resolve for a Dirichlet and Neumann boundary conditions. We will show how these results can be applied to studying the ``winding'' / ``entanglement'' of Brownian trajectories in Riemannian manifolds.

We give results on the properties of Markov kernels that approximate another Markov kernel. The basic idea is that when the original kernel satisfies a contraction condition in some metric, the long-time dynamics of the two chains -- as well as the invariant measures, when they exist -- will be close in that metric, so long as the approximating kernel satisfies a suitable approximation error condition. We focus on weighted total variation and Wasserstein metrics, and motivate the results with applications to scalable Markov chain Monte Carlo algorithms. This is joint work with Jonathan Mattingly.

Our primary goal is to describe a strong quantitative form of the Shannon-McMillan-Breiman theorem for log-concave probability measures on linear spaces, even in the absence of stationarity. The main technical result is a concentration of measure inequality for the ``information content'' of certain random vectors. We will also briefly discuss implications. In particular, by combining this concentration result with ideas from information theory and convex geometry, we obtain a reverse entropy power inequality for convex measures that generalizes the reverse Brunn-Minkowski inequality of V. Milman. Along the way, we also develop a new information-theoretic formulation of Bourgain's hyperplane conjecture, as well as some Gaussian comparison inequalities for the entropy of log-concave probability measures. This is joint work with Sergey Bobkov (Minnesota).

We consider the heat equation on an interval with heat being introduced according to a random mechanism. When the random mechanism is space-time white noise, this equation has been much studied. We look at the case where the white noise is modified by a function A(u)(x) of the current temperatures u and where A is H\"older continuous as a function of u. Unlike other work along these lines, A(u)(x) can depend on the temperatures throughout the interval. Our method involves looking at the Fourier coefficients, which leads to an infinite dimensional system of stochastic differential equations. This is joint work with Ed Perkins.

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.

It is well known that for a regular stable potential of pair interaction and a small value of activity one can define the corresponding Gibbs field (a measure on the space of configurations of points in $\mathbb{Z}^d$ or $\mathbb{R}^d$). We consider a converse problem. Namely, we show that for a sufficiently small constant $\overline{\rho}_1$ and a sufficiently small function $\overline{\rho}_2(x)$, $x \in \mathbb{Z}^d$ or $\mathbb{R}^d$, there exist a hard core pair potential, and a value of activity, such that $\overline{\rho}_1$ is the density and $\overline{\rho}_2$ is the pair correlation function of the corresponding Gibbs field.

It is anticipated that chaotic regimes (e.g., strange attractors) arise in a wide variety of dynamical systems, including those arising from the study of ensembles of gas particles and fluid mechanics. However, in most cases the problem of rigorously verifying asymptotic chaotic regimes is notoriously difficult. For volume-preserving systems (e.g., incompressible fluid flow or Hamiltonian systems), these issues are exemplified by coexistence phenomena: even in quite simple models which should be chaotic, e.g. the Chirikov standard map, completely opposite dynamical regimes (elliptic islands vs. hyperbolic sets) can be tangled together in phase space in a convoluted way. Recent developments have indicated, however, that verifying chaos is tractable for systems subjected to a small amount of noise— from the perspective of modeling, this is not so unnatural, as the real world is inherently noisy. In this talk, I will discuss two recent results: (1) a large positive Lyapunov exponent for (extremely small) random perturbations of the Chirikov standard map, and (2) a positive Lyapunov exponent for the Lagrangian flow corresponding to various incompressible stochastic fluids models, including stochastic 2D Navier-Stokes and 3D hyperviscous Navier-Stokes on the periodic box. The work in this talk is joint with Jacob Bedrossian, Samuel Punshon-Smith, Jinxin Xue and Lai-Sang Young.

In the evolving voter model we choose oriented edges (x,y) at random. If the two individuals have the same opinion, nothing happens. If not, x imitates y with probability 1-á, and otherwise severs the connection with y and picks a new neighbor at random (i) from the graph, or (ii) from those with the same opinion as x. One model has a discontinuous transition, the other a continuous one.

We propose a framework for modelling stochastic systems which satisfy detailed balance (or in other terminology, time-reversibility). Rather than specifying the dynamics through a state-dependent drift and diffusion coefficients, we specify an equilibrium probability density and a state-dependent diffusion coefficient. We argue that our framework is more natural from the modelling point of view and has a distinct advantage in situations where either the equilibrium probability density or the diffusion coefficient is discontinuous. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and elegantly handles discontinuities in the coefficients. This is joint work with Xin Yang.

The KPZ equation was introduced in 1986, and has become the default model in physics for random interface growth. It is a member of a large universality class with non-standard fluctuations, including directed random polymers. Even in one dimension, it turned out to be difficult to interpret and analyze mathematically, but at the same time to have a large degree of exact solvability. We will survey the history and recent progress.

The efficient delivery of proteins and other molecular products to their correct location within a cell (intracellular transport) is of fundamental importance to normal cellular function and development. Moreover, the breakdown of intracellular transport is a major contributing factor to many degenerative diseases. There are two major types of transport. (I) Passive diffusion within the cytosol or the surrounding plasma membrane of the cell. Since the aqueous environment (cytosol) of a cell is highly viscous at the length and velocity scales of macromolecules (low Reynolds number), a diffusing particle can be treated as an overdamped Brownian particle where inertial effects are ignored. (II) Active motor-driven transport along polymerized filaments such as microtubules and F-actin that comprise the cytoskeleton. At appropriate length and time scales, active transport can either be modeled as a velocity-jump process or as an advection-diffusion process. In this talk I present various PDE models of active and passive transport within cells. The bulk of the talk will focus on three examples: synaptic democracy and vesicular transport in axons and dendrites; stochastically gated diffusion in bounded domains; cytoneme-based transport of morphogens during embryogenesis. (A cytoneme is a thin actin-rich filament that forms direct contacts between cells and is thought to provide an alternative to diffusion-based morphogen gradient formation.) Other applications include cellular length control, cell polarization, and synaptogenesis in C. elegans.

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.

Over the past decade, the theory of tumor evolution has largely focused on the selective sweeps model. According to this theory, tumors evolve by a succession of clonal expansions that are initiated by driver mutations. In a 2015 paper, Sottoriva et al collected genetic data of various types from 349 individual tumor glands were sampled from the opposite sides of 15 colorectal tumors and large adenomas. Based on this the authors proposed an alternative theory of tumor evolution, the so-called {\bf Big Bang model}, in which one or more driver mutations are acquired by the founder gland, and the evolutionary dynamics within the expanding population are predominantly neutral. In this talk we will describe a simple mathematical model that reproduces the observed phenomena and makes quantitative predictions.

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 Ron Peled and Ofer Zeitouni.

A deterministic walk in a random environment can be understood as a general finite-range dependent random walk that starts repeating the loop once it reaches a site it has visited before. Such process lacks the Markov property. We will talk about the exponential decay of the probabilities that the walk will reach sites located far away from the origin.

Given a continuous time Markov Chain \( q(x,y)\) on a finite set S, the associated noisy voter model is the continuous time Markov chain on \(\{0,1\}^S\) which evolves by (i) for each two sites x and y in S, the state at site x changes to the value of the state at site y at rate \( q(x,y) \) and (ii) each site rerandomizes its state at rate 1. We show that if there is a uniform bound on the rates \(q(x,y)\) and the corresponding stationary distributions are ``almost'' uniform, then the mixing time has a sharp cutoff at time \(\log |S|/2\) with a window of order 1. Lubetzky and Sly proved cutoff with a window of order 1 for the stochastic Ising model on toroids: we obtain the special case of their result for the cycle as a consequence of our result.

Consider a standard branching Brownian motion whose particles have varying mass. At time t, if a total mass m of particles have distance less than one from a fixed particle x, then the mass of particle x decays at rate m. The total mass increases via branching events: on branching, a particle of mass m creates two identical mass-m particles. One may define the front of this system as the point beyond which there is a total mass less than one (or beyond which the expected mass is less than one). This model possesses much less independence than standard BBM. Nonetheless, it is possible to prove that (in a rather weak sense) the front is at distance ~ c t^{1/3} behind the typical BBM front.

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.

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!).

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.

We introduce the term `superconcentration' to describe the phenomenon when a function of a Gaussian random field exhibits a far stronger concentration than predicted by classical concentration of measure. We show that when superconcentration happens, the field becomes chaotic under small perturbations and a `multiple valley picture' emerges. Conversely, chaos implies superconcentration. While a few notable examples of superconcentrated functions already exist, e.g. the largest eigenvalue of a GUE matrix, we show that the phenomenon is widespread in physical models; for example, superconcentration is present in the Sherrington-Kirkpatrick model of spin glasses, directed polymers in random environment, the Gaussian free field and the Kauffman-Levin model of evolutionary biology. As a consequence we resolve the long-standing physics conjectures of disorder-chaos and multiple valleys in the Sherrington-Kirkpatrick model, which is one of the focal points of this talk.

We study a random walk on the integers Z which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, p and q. R consecutive right jumps from a site in the q-mode are required to switch it to the p-mode, and L consecutive left jumps from a site in the p-mode are required to switch it to the q-mode. From a site in the p-mode the walk jumps right with probability p and left with probability (1-p), while from a site in the q-mode these probabilities are q and (1-q). We prove a sharp cutoff for right/left transience of the random walk in terms of an explicit function of the parameters $\alpha = \alpha(p,q,R,L)$. For $\alpha > 1/2$ the walk is transient to $+\infty$ for any initial environment, whereas for $\alpha < 1/2$ the walk is transient to $-\infty$ for any initial environment. In the critical case, $\alpha = 1/2$, the situation is more complicated and the behavior of the walk depends on the initial environment. We are able to give a characterization of transience/recurrence in many instances, including when either R=1 or L=1 and when R=L=2. In the noncritical case, we also show that the walk has positive speed, and in some situations are able to give an explicit formula for this speed. This is joint work with my former post-doc, Nick Travers, now at Indiana University.

Branching random walk can be viewed as particles performing random walks while branching at integer time. We review some of the existing results on the maximal (or minimal) displacement, when each particle moves and branches independently according the same step distribution and the same branching law. Then we will compare them with similar but different models. Roughly speaking, in one variation, we will consider the asymptotic behavior of the particle at time n, whose ancestors’ location are consistently small. In another variation, we will consider the maximal displacement for the model, where the step distributions vary with respect to time.