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 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.

The random distribution Holomorphic multiplicative chaos (HMC) with Gaussian inputs is recently introduced independently by Najnudel, Paquette, and Simm as a limiting object on the unit complex circle of characteristic polynomial of circular beta ensembles, and by Soundararajan and Zaman as an analogue of random multiplicative functions. In this talk, we will explore this rich connection between HMC and random matrix theory, number theory, and Gaussian multiplicative chaos. We will also discuss the regularity of this distribution, alongside the fractional moments and tightness of its Fourier coefficients (also referred to as secular coefficients). Furthermore, we introduce non-Gaussian HMC, and discuss the Gaussian universality and two phase transitions phenomenon in the fractional moments of its secular coefficients. A transition from global to local effect is observed, alongside an analysis of the critical local-global case. As a result, we unveil the regularity of some non-Gaussian HMC and tightness of their secular coefficients. Based on joint work with Zhenyuan Zhang.

I will describe a central limit theorem for the rate of energy dissipation in a random network of resistors. In the continuum setting the model is an elliptic PDE with random conductivity coefficient. In the large network limit, homogenization occurs and the random dissipation rate can be approximated well by a normal random variable having the same mean and variance. I'll give error estimates for this approximation in total variation norm which have optimal scaling. The analysis is based on Stein's method and a recent result of Sourav Chatterjee.

Jigsaw percolation is a nonlocal process that iteratively merges elements of a partition of the vertices in a deterministic puzzle graph according to the connectivity properties of a random collaboration graph. We assume the collaboration graph is an Erdos-Renyi graph with edge probability p, and investigate the probability that the puzzle graph is solved, that is, that the process eventually produces the partition {V}. In some generality, for puzzle graphs with N vertices of degrees about D, this probability is close to 1 or 0 depending on whether pD(log N) is large or small. We give more detailed results for the one dimensional cycle and two dimensional torus puzzle graphs, where in many instances we can prove sharp phase transitions.

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.

This talk will be about the convergence problem in mean field control (MFC), i.e. the challenge of rigorously justifying the convergence of certain "symmetric" $N$-particle stochastic control problems towards their mean field counterparts. On the one hand, this convergence problem is already well-understood from a qualitative perspective, thanks to powerful probabilistic techniques based on compactness. On the other hand, quantitative results (i.e. rates of convergence) are more difficult to obtain, in large part because the value function of the mean field problem (which is also the solution to a certain Hamilton-Jacobi equation on the Wasserstein space) may fail to be $C^1$, even if all the data is smooth. After giving an overview of the convergence problem, I will discuss the results of two recent joint works with Cardaliaguet, Daudin, Delarue, and Souganidis, in which we use some ideas from the theory of viscosity solutions to overcome this lack of regularity and obtain rates of convergence of the $N$-particle value functions towards the value function of the corresponding MFC problem.

The nature of dynamical processes occuring on a directed network can depend qualitatively on the logic implemented at each node. In large random networks where nodes act as Boolean gates, there is a phase transition from quiescent to chaotic behavior as the average degree and/or the probabilities of assigning the different logic functions are varied. To understand the behavior at the transition, we are led to a special type of percolation problem in which the relevant question is not whether a cascade spans the system, but whether it covers the entire system. I will introduce the problem of exhaustive percolation, outline a method for solving it, and describe its application to random Boolean networks.

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.

New postdocs David Herzog, Kevn McGoff and Jessica Zuniga at Duke, and Sergio Almada, Chia Lee, and Xiaoming Song at UNC will give short talks to introduce themselves.

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.

We consider a process in which a finite set of n agents continually receive a 1 dollar allowance and gamble their fortunes, all in, with one another at a constant rate. This is a variation on the existing compulsive gambler process; in that process, initial fortunes are prescribed and no further allowances are given out. For our process, we find that after some time the distribution of wealth settles into a pattern in which most people have only a few dollars, a few are very wealthy, and a single person possesses most of the cash currently present in the population. In addition, eventually the only way to attain first rank is by winning a bet against the current champion. Moreover, if agents play a fair game, i.e., the probability of winning a bet is proportional to the players' fortunes, the title of champion is assumed by every player infinitely often, although it changes less and less frequently as time goes on. Finally, by examining the process from both the perspective of typical fortune, and that of large fortune, we can go one step further and obtain two distinct limiting processes as n --> infty, with each one admitting a detailed description of its dynamics.

I will talk about some models coming from Physics and Queueing Theory that give rise to singular reflected processes in their diffusion limit. Such diffusions are characterized by non-elliptic generators (which are not even hypoelliptic) in the interior, and ergodicity arises from non-trivial interactions between the diffusion, drift and reflection. I will introduce a regenerative process approach which identifies renewal times in diffusion paths and analyzes excursions between successive renewal times. This provides a detailed description of the stationary distribution even when closed form expressions are unavailable. Based on joint works with Chris Burdzy, Brendan Brown, Mauricio Duarte and Debankur Mukherjee.

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.

Axelrod's model is a voter model in which individuals have multiple opinions and neighbors interact at a rate proprtional to the fraction of opinions they share. I will describe physuicists predictions about the behavior of this model, recent results of Lanchier in one dimension, and a new result I have proved about the two dimensional case.

Social networks are often represented by directed graphs where the nodes are individuals and the edges indicate a form of social relationship. A simple way to model the diffusion of ideas, innovative behavior, or word-of-mouth effects on such a graph is to consider a stochastic process of ``infection'': each node becomes infected once an activation function of the set of its infected neighbors crosses a random threshold value. I will prove a conjecture of Kempe, Kleinberg, and Tardos which roughly states that if such a process is ``locally'' submodular then it must be ``globally'' submodular on average. The significance of this result is that it leads to a good algorithmic solution to the problem of maximizing the spread of influence in the network--a problem known in data mining as "viral marketing"'. This is joint work with Elchanan Mossel.

We consider first-passage percolation on the cylinder graph of length $n$ and width $h_n$ in the d-dimensional square lattice where each edge has an i.i.d.~nonnegative weight. The passage time for a path is defined as the sum of weights of all the edges in that path and the first-passage time between two vertices is defined as the minimum passage time over all paths joining the two vertices. We show that the first-passage time $T_n$ between the origin and the vertex $(n,0,\ldots,0)$ satisfies a Gaussian CLT as long as $h_n=o(n^{1/(d+1)})$. The proof is based on moment estimates, a decomposition of $T_n$ as an approximate sum of independent random variables and a renormalization argument. We conjecture that the CLT holds upto $h_n=o(n^{2/3})$ for $d=2$ and provide support for that. Based on joint work with Sourav Chatterjee.

The superprocess or Dawson-Watanabe process is one of the most intensively studied stochastic processes of the last quarter century. It arises as a limit of population processes, and includes information about the physical location of individuals. Usually the superprocess is measure valued, but In one dimension it has a density that satisfies a parabolic stochastic PDE. For a long time uniqueness for this equation was unknown. In joint work with Barlow, Mytnik, and Perkins, we show that nonuniquess holds for the superprocess equation and several related equations.

The Airy line ensemble arises in scaling limits of growth models, directed polymers, random matrix theory, tiling problems and non-intersecting line ensembles. This talk will mainly focus on the "non-intersecting Brownian Gibbs property" for this infinite ensemble of lines. Roughly speaking, the measure on lines is invariant under resampling a given curve on an interval according to a Brownian Bridge conditioned to not intersect the above of below labeled curves. This property leads to the proof of a number of previously conjectured results about the top line of this ensemble. We will also briefly touch on the KPZ line ensemble, which arises as the scaling limit of a diffusion defined by the Doob-h transform of the quantum Toda lattice Hamiltonian. The top labeled curve of this KPZ ensemble is the fixed time solution to the famous Kardar-Parisi-Zhang stochastic PDE. This line ensemble has a "softer" Brownian Gibbs property in which resampled Brownian Bridges may cross the lines above and below, but at exponential energetic cost. This is based on joint work with Alan Hammond.

Consider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

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.

The theory of large deviations has been used in the development of Monte Carlo methods for estimating quantities defined in terms of a specific rare event, such as ruin probabilities or buffer overflow probabilities. However, rare events also play an important role when estimating functionals of an invariant distribution, where straightforward simulation will converge very slowly when parts of the state space do not communicate well. Problems of this sort are common in statistical inference, engineering and the physical sciences. After reviewing some of the methods used to accelerate the convergence of Monte Carlo, we consider the use of the large deviation rate for the empirical measure as a performance measure and introduce a new class of algorithms (which we call infinite swapping schemes) that optimize this rate.

There has been a flurry of recent activity in studying the topology of point cloud data. However, there is a feeling that we are lacking rigorous null hypotheses to compare with the results. This is one motivation for the following: Take n points, independently and identically distributed in R^d, according to some distribution (for example, a standard normal distribution). Connect them if they are close (within distance epsilon, a function of n), and then build the Cech complex or Rips complex. What can one say about the homology of this complex as n approaches infinity? Or the persistent homology with respect to the radius? Using a variety of techniques, including Poissonization, Stein's method, and discrete Morse theory, we are able to identify phase transitions, and for certain ranges of epsilon prove central limit theorems for the Betti numbers. This is joint work with Gunnar Carlsson and Persi Diaconis.

Integrable stochastic particle systems in one space dimension, like the Totally Asymmetric Simple Exclusion Process (TASEP), have been studied for over 50 years (introduced simultaneously in biology and mathematics in 1969-70). They strike a balance between being simple enough to be mathematically tractable and complicated enough to describe many interesting phenomena. Many natural questions about these systems can be generalized by introducing multiple parameters. The interplay between these parameters is powered by the Yang-Baxter equation, which brings new intriguing results to the well-traveled territory. In particular, I will discuss new Lax-type equations for the Markov semigroups of the TASEP and its relatives. Based on a joint work with Axel Saenz.

Importance sampling is an accelerated Monte Carlo algorithm that can reduce variance when estimating small probabilities. The design of the algorithm involves the choice of a change of measure, and based on this choice the performance can range from substantially better than standard Monte Carlo to substantially worse. One approach to choosing a change of measure involves embedding the problem of interest in a sequence of processes that satisfies a large deviations principle, and then basing the change of measure on subsolutions to the Hamilton-Jacobi-Bellman equation associated the large deviations rate function. This approach has the benefit of guaranteeing a certain level of asymptotic performance based on the subsolution, but different embeddings can lead to different rate functions, subsolutions, and consequently different algorithms. I will contrast the strengths and weaknesses of two different embeddings, one using a scaling commonly referred to as the standard large deviations scaling and the other using a scaling referred to as moderate deviations.

Over the last few years a wide array of random graph models have been postulated to understand properties of empirically observed networks. Most of these models come with a parameter t (usually related to edge density) and a (model dependent) critical time t_c which specifies when a giant component emerges. There is evidence to support that for a wide class of models, under moment conditions, the nature of this emergence is universal and looks like the classical Erdos-Renyi random graph, in the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like n^{2/3}) and (b) the structure of components (rescaled by n^{-1/3}) converge to random fractals related to the continuum random tree. Till date, (a) has been proven for a number of models using different techniques while (b) has been proven for only two models, the classical \erdos random graph and the rank-1 inhomogeneous random graph. The aim of this paper is to develop a general program for proving such results. The program requires three main ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent (ii) scaling exponents of susceptibility functions in the barely subcritical regime are the same as the Erdos-Renyi random graph and (iii) macroscopic averaging of expected distances between random points in the same component in the barely subcritical regime. We show that these apply to a number of fundamental random graph models including the configuration model, inhomogeneous random graphs modulated via a finite kernel and bounded size rules. Thus these models all belong to the domain of attraction of the classical Erdos-Renyi random graph. As a by product we also get the first known results for component sizes at criticality for a general class of inhomogeneous random graphs. This is joint work with Xuan Wang, Sanchayan Sen and Nicolas Broutin.

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.

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.