Abstracts are added as they become available.

See the Schedule page for the schedule.

**Limit laws for random matrices from traffic probability**

*Benson Au (University of California at Berkeley)*

We consider a random matrix ensemble that interpolates between the Wigner matrices and the random Markov matrices studied by Bryc, Dembo, and Jiang. For a real Wigner matrix $W_n$, let $D_n$ be the diagonal matrix whose entries correspond to the row sums of $W_n$, i.e., $D_n(i,i) = \sum_{j=1}^n W_n(i,j)$. For $p,q\in\mathbb{R}$, we study the asymptotic behavior of the matrices $M_{n,p,q}= pW_n + qD_n$ as $n\to\infty$. By realizing $M_{n,p,q}$ as a traffic of $W_n$ in the sense of Male, we show that when the entries of $W_n$ have finite moments of all orders, independent $M_{n,p,q}$ are asymptotically traffic independent with stable universal limiting traffic distribution. We obtain the limiting spectral distributions of the ensemble $(M_{n,p,q})_{p,q\in\mathbb{R}}$ as a consequence via the traffic central limit theorem.

**Efficient coupling for Brownian motion with redistribution**

*Iddo Ben Ari (University of Connecticut)*

We consider a model of Brownian motion on a bounded interval which upon exiting the interval is being redistributed back into the interval according to a probability measure depending on the exit point, then starting afresh, repeating the above mechanism indefinitely. It is not hard to show that the process is exponentially ergodic, although characterizing the rate of convergence is non-trivial. In this talk, after providing a general overview of the probabilistic method of coupling and its applications, I'll show how to study the ergodicity for the model through coupling, how it leads to an intuitive and geometric explanation for the rates of convergence previously obtained analytically, other insights, and more questions.

**The charming first nontrivial eigenpair**

*Mu-Fa Chen (Beijing Normal University)*

The talk surveys our study on the topic in a backward way. First, we present some new unexpected results on computing the maximal eigenpair for matrices. The key to this is the efficient initials for the known algorithms. The initials come from our previous study on the estimation of the leading eigenpair. Thus, in the second part of the talk, we survey shortly the unified basic estimates for the leading eigenvalues in various situation, mainly for one-dimensional elleptic operator (order 2). In the last part of the talk, we mention our original goal for the study is looking for mathematical tools to study the phase transitions in statistical mechanics.

**Consistency of modularity clustering on geometric random graphs**

*Erik Davis (University of Arizona)*

We consider random geometric graphs constructed from samples $\mathcal{X}_n = \{X_1,X_2,\ldots,X_n\}$ of independent, identically distributed observations of an underlying probability measure on a bounded domain $D\subset \mathbb{R}^d$. The popular "modularity" clustering method specifies a partition $\mathcal{U}_n$ of the set $\mathcal{X}_n$. In this talk we analyze the limiting behavior of modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is constrained, the discrete optimal partitions $\mathcal{U}_n$ converge in a certain sense to a continuum partition $\mathcal{U}$ of the underlying domain, characterized as the solution of a type of Kelvin's shape optimization problem.

**Exact sampling of combinatorial structures using probabilistic divide-and-conquer**

*Stephen DeSalvo (University of California at Los Angeles)*

We demonstrate a recent technique for exact sampling of combinatorial structures called probabilistic divide-and-conquer. By exploiting certain independence properties of the random variables which describe random component sizes, we are able to obtain efficient exact samplers, provably more efficient than exact Boltzmann samplers. The technique also generalizes to constrained structures with continuous-valued components. Applications include integer partitions, contingency tables, combinatorial polytopes.

**Tracy-Widom fluctuations for an inhomogeneous corner growth model**

*Elnur Emrah (University of Wisconsin at Madison)*

We consider a certain inhomogeneous generalization of the corner growth model with geometric weights. We show that the quenched limit fluctuations of the last-passage times in the strictly concave region of the shape function is governed by the Tracy-Widom GUE distribution.

**Genealogies for a biased voter model**

*Wai Fan (University of Wisconsin at Madison)*

I will present rigorous results about the genealogies of a biased voter model introduced by Hallatschek and Nelson (2007). To investigate the lineage dynamics, we derived a system of stochastic partial differential equations (SPDE) from the tracer dynamics in which the particles have different colors. Brunet et al. (2006) have conjectured that genealogies in models of this type are described by the Bolthausen-Sznitman coalescent. However, there are no simultaneous coalescences in our model, since the dual branching coalescing random walk converges to a branching Brownian motion in which particles coalesce after an exponentially distributed amount of intersection local time. A new duality equation is established to show uniqueness of the SDPE. By generalizing results of Mueller and Tribe (1995), we also identified different scalings for which our biased voter model converges to either the Wright-Fisher SPDE or the deterministic FKPP. Joint work with Rick Durrett.

**Isoperimetry in the infinite cluster**

*Julian Gold (University of California at Los Angeles)*

We study the isoperimetric subgraphs of the infinite cluster $\textbf{C}_\infty$ for supercritical bond percolation on $\mathbb{Z}^d$ with $d \geq 3$. Specifically, we consider the subgraphs of $\textbf{C}_\infty \cap [-n,n]^d$ which have minimal open edge boundary to volume ratio. We prove a shape theorem for these subgraphs, obtaining that when suitably rescaled, these subgraphs converge almost surely to a translate of a deterministic shape. This deterministic shape is itself an isoperimetric set for a norm we construct. As a corollary, we obtain sharp asymptotics on a natural modification of the Cheeger constant for $\textbf{C}_\infty \cap [-n,n]^d$. This settles a conjecture of Benjamini for the version of the Cheeger constant defined here.

**The balanced random environment**

*Xiaoqin Guo (Purdue University)*

A random walk in a balanced environment is a Markov chain with a non-divergence form difference operator. In the talk we consider random balanced environment on the integer lattice, in both the elliptic and non-elliptic cases. We will discuss three related topics: invariant principles in balanced random environments, balanced directed percolation and harmonic analysis of balanced Laplace operators. The talk is based on joint works with Noam Berger, Jean-Dominique Deuschel and Alejandro Ramirez.

**Self-avoiding polygons and walks: counting, joining and closing**

*Alan Hammond (University of California at Berkeley)*

Self-avoiding walk of length $n$ on the integer lattice $Z^d$ is the uniform measure on nearest-neighbour walks in $Z^d$ that begin at the origin and are of length $n$. If such a walk closes, which is to say that the walk's endpoint neighbours the origin, it is natural to complete the missing edge connecting this endpoint and the origin. The result of doing so is a self-avoiding polygon. We investigate the numbers of self-avoiding walks, polygons, and in particular the "closing" probability that a length n self-avoiding walk is closing. Developing a method (the "snake method") employed in joint work with Hugo Duminil-Copin, Alexander Glazman and Ioan Manolescu that provides closing probability upper bounds by constructing sequences of laws on self-avoiding walks conditioned on increasing severe avoidance constraints, we show that the closing probability is at most $n^{-1/2 + o(1)}$ in any dimension at least two. Developing a quite different method of polygon joining employed by Madras in 1995 to show a lower bound on the deviation exponent for polygon number, we also provide new bounds on this exponent. We further make use of the snake method and polygon joining technique at once to prove upper bounds on the closing probability below $n^{-1/2}$ in the two-dimensional setting.

**Large time asymptotic for the parabolic Anderson model driven by spatially correlated noise**

*Jingyu Huang (University of Utah)*

We consider the parabolic Anderson model in multidimension driven by a Gaussian noise which is white in time and it has a correlated spatial covariance. Examples of such covariance include the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter $H>1/4$ in dimension one. We will discus existence and uniqueness of solution, Feynman-Kac formula for its moments, Lyapunov exponents and exponential growth indices. I'll discuss further related problems if time permits. This talk is based on joint works with Yaozhong Hu, Khoa Lê and David Nualart.

**Large deviations for some corner growth models with inhomogeneity**

*Christopher Janjigian (University of Wisconsin at Madison)*

The corner growth model is a classical model of growth in the plane. It is connected to other familiar models such as directed last passage percolation and the TASEP through various geometric maps. In the case that the waiting times are i.i.d. with exponential or geometric marginals, the model is well understood: the shape function can be computed exactly, the fluctuations around the shape function are known to be given by the Tracy-Widom GUE distribution, and large deviation principles corresponding to this limit have been derived. This talk considers the large deviation properties of a generalization of the classical model in which the rates of the exponential are drawn randomly in an appropriate way. We will discuss some exact computations of rate functions in the quenched and annealed versions of the model, along with some interesting properties of large deviations in these models. (joint work with Elnur Emrah)

**Poincare Duality and Bakry-Émery estimate on Hino index-1 spaces**

*Daniel Kelleher (Purdue University)*

In the last few years, there has been much work in defining curvature bounds in general metric measure spaces. In this work, we are interested in fractals, which are spaces for which good notions of curvature have been elusive so far. Our main result is that the Bakry-Émery estimate is satisfied on the Sierpinski Gasket with harmonic energy measures. To obtain this result, the space of differential forms on these spaces is classified using a measurable analogue of Poincaré duality for classical differential forms. We investigate then the consequences of this estimate on a large class of metric measure Dirichlet spaces including the Sierpinski gasket as a special case. We are in particular interested in boundedness of Riesz transforms and isoperimetric inequalities.

**Bose-Einstein condensation: from many quantum particles to a quantum "superparticle"**

*Kay Kirkpatrick (University of Illinois at Urbana-Champaign)*

Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss recent progress on a couple of quantum central limit theorems. (Joint work with Gerard Ben Arous, Michael Brannan, Benjamin Schlein, and Gigliola Staffilani.)

**A mathematical perspective on Data Science**

*Tom LaGatta (Splunk Inc.)*

As with all things, the process of analyzing data admits a mathematical description. As a mathematician-turned-data-scientist, I will describe my approach to problem solving, and attempt to loosely formalize "stakeholders", "use cases", "data" and "deliverables" in mathematical language for the enjoyment of this mostly academic audience. In particular, I will describe how query languages are inherently functional, acting as functional transformations of Data into Data, which obey the usual functional composition law. The process of analyzing data results in an iterative sequence of queries, converging to a final query which is satisfactory to the use case. These queries are then organized into deliverables, which can be "dashboards" (web pages with visualizations) or "data products" (with scheduled jobs & analyses running in the background). When this process is done right, it results in the extraction of "value" for stakeholders, which can be measured tangibly in terms of revenue, costs or risk metrics. Sometimes this has a fancy name like "data science", but more often than not, is just the normal operational work of a good data-savvy IT, Security, Tech or Business department in modern enterprises and governmental agencies. There will be no proofs, but I would be very interested to discuss rigorous approaches to social organization & problem solving after the talk.

**Diffusion in a randomly switching environment**

*Sean Lawly (University of Utah)*

Driven by diverse applications to neuroscience, biochemistry, and physiology, several recent models impose randomly switching boundary conditions on either a PDE or SDE. In this talk, I will describe the mathematical tools for analyzing these systems and highlight the interesting behavior that they can exhibit. Special attention will be given to establishing mathematical connections between these classes of stochastic processes.

**One point fluctuations of periodic TASEP with step and flat initial conditions**

*Zhipeng Liu (New York University, Courant Institute)*

We consider the periodic TASEP model on the space $\{(x_1,\cdots,x_N)\in Z^N; x_1 < \cdots < x_N < x_1+L\}$. This model can be described as the TASEP on the ring with length $L$, or directed last passage percolation with periodic entries, or directed last passage percolation on a cylinder. We are interested in the fluctuations of a fixed particle as $N$, $L$, and time $t$ all go to infinity. For the step initial condition, if the density of particles $N/L$ is fixed, we prove that the limiting distribution is given by either GUE Tracy-Widom distribution or the product of two GUE Tracy-Widom distributions when $t < N^{3/2}$. We also find the explicit formula for the limit of the one point distribution in when $t\sim N^{3/2}$. For flat initial conditions, we have a similar formula. This is a joint work with Jinho Baik.

**Distilling mixtures with a single characterized component**

*Manuel Lladser (University of Colorado at Boulder)*

This talk addresses probabilistic and statistical problems associated with the distilling of mixtures with possibly unknown components, specially in the context biomedical high-throughput datasets where the mixture and only a single of its components can be observed indirectly through data. We call this the "characterized component" of the mixture. Despite being a largely ill-posed problem, in this talk I will present necessary and sufficient conditions to estimate either exactly or approximately the unknown weight of the unique characterized component via a U-statistic, with a kernel that depends on the sample size, and determine the asymptotic distribution of this statistic. This work is in collaboration with Andrew Smith (USC) and Antony Pearson (CU-Boulder).

**The Breiman conjecture**

*David Mason (University of Delaware)*

Let $Y,Y_{1},Y_{2},\ldots $ be positive, nondegenerate, i.i.d. $G$ random variables, and independently let $X,X_{1},X_{2},\ldots $ be i.i.d. $F$ random variables. Breiman (1965) conjectured that if $X$ is nondegenerate, $% \mathbb{E}\left\vert X\right\vert <\infty $ and $\mathbb{T}_{n}=\sum X_{i}Y_{i}/\sum Y_{i}\rightarrow _{d}T$, where $T$ is nondegenerate, then necessarily $\overline{G}=1-G$ is regularly varying at infinity with index $% 0\leq \beta <1$, written $G\in D\left( \beta \right) $. In this talk we discuss the recent progress of Péter Kevei and David Mason towards resolving this conjecture. We have shown that whenever for some $F\in \mathcal{F}$, in a specified class of distributions $\mathcal{F}$, $\mathbb{T% }_{n}$ converges in distribution to a nondegenerate limit then necessarily $% G\in D\left( \beta \right) $ with $0\leq \beta <1.$ The class $\mathcal{F}$ contains the distributions of nondegenerate $X$ with a finite second moment, as well as those of $X$ in the domain of attraction of a stable law with index $1<\alpha <2 $. Our results will appear in the

*Proceedings of the AMS*. We shall also discuss the limiting distributional behavior of these self-normalized sums along subsequences and their Lévy process analogs.

**Abstract Wiener groups**

*Tai Melcher (University of Virginia)*

Gaussian measure has for decades been recognized as the appropriate measure to use in infinite-dimensional analysis, and calculus on such measure spaces has become a valuable tool in the analysis of stochastic processes and their applications. For infinite-dimensional curved spaces, the analogue of Gaussian measure is heat kernel measure. We'll discuss heat kernel measures in a special class of infinite-dimensional spaces and provide motivation for the construction. In particular, these spaces admit a natural hypoelliptic structure, and we're able to show smoothness results for heat kernel measures under both elliptic and hypoelliptic conditions. Parts of this talk are based on joint work with Fabrice Baudoin, Daniel Dobbs, Bruce Driver, Nate Eldredge, and Masha Gordina.

**Information, concentration, and transportation of exponentially concave functions**

*Soumik Pal (University of Washington at Seattle)*

Exponentially concave functions are functions whose exponentials are nonnegative concave functions. These functions have recently appeared independently at the intersection or probability, analysis, and geometry. We consider exponentially concave functions on the unit simplex. Its gradient map turns out to be the solution to a remarkable Monge-Kantorovich optimal transport problem. Exponential stochastic integrals of such gradient maps arise naturally in modern portfolio theory. We will show how concentration of measure of Dirichlet distributions in high-dimensional simplex determines a model-free behavior of such integrals. Finally, we show that exponentially concave functions on the unit simplex, seen as a statistical manifold, give rise to a new information geometry which is an exponential version of the celebrated classical information geometry of Bregman divergences such as relative entropy. To the best of our knowledge this is the only example of a geometry which is not dually-flat where a Pythagoras theorem holds.

**The six phases of a two-parameter scaled Brownian penalization**

*Hugo Panzo (University of Connecticut)*

In a series of papers, Roynette-Vallois-Yor initiated a study of weak limits of Wiener measure weighted by various path functionals, the so-called Brownian penalizations. A particular two-parameter penalization they considered exhibited three distinct phases corresponding to three regions of the parameter plane. We extend their results by considering a scaled version of this model and show the existence of three additional "critical" phases corresponding to the rays separating the three regions.

**Spectral properties of large non-Hermitian random matrices**

*David Renfrew (University of Colorado at Boulder)*

The study of the spectrum of non-Hermitian random matrices with independent, identically distributed entries was introduced by Ginibre and Girko. I will present a generalization of this model when the identically distribution assumptions are relaxed and discuss applications.

**From genomes to trees and beyond: Recent progress in Mathematical Phylogenomics**

*Sebastien Roch (University Wisconsin at Madison)*

The reconstruction of the Tree of Life is a classical problem in evolutionary biology that has benefited from many branches of mathematics, including probability, information theory, combinatorics, and geometry. Modern DNA sequencing technologies are producing a deluge of new genetic data - transforming how we view the Tree of Life and how it is reconstructed. I will survey recent progress on some mathematical questions that arise in this context. No biology background will be assumed.

**Bak-Sneppen backwards**

*Mackenzie Simper (University of Utah)*

The Bak-Sneppen model is a Markov chain which serves as a simplified model of evolution in a population of spatially interacting species. We study the backwards Markov chain for the Bak-Sneppen model and derive its corresponding reversibility equations. We show that, in contrast to the forwards Markov chain, the dynamics of the backwards chain explicitly involve the stationary distribution of the model, and from this we derive a functional equation that the stationary distribution must satisfy. We use this functional equation to derive differential equations for the stationary distribution of Bak-Sneppen models in which all but one or all but two of the fitnesses are replaced at each step.

**Small-particle limits in regularized Laplacian random growth models**

*Alan Sola (University of South Florida)*

We study global scaling limits as well as fluctuations for conformal mapping models of planar aggregation phenomena. The main interest as well as difficulty in these models is the presence of long-range dependecy of the aggregation process on its past. Joint work with F. Johansson Viklund and A. Turner.

**Long-lasting effects of small random perturbations on dynamical systems: theoretical and computational tools**

*Eric Vanden-Eijnden (Courant Institute, New York University)*

Small random perturbations may have a dramatic impact on the long time evolution of dynamical systems, and large deviation theory (LDT) is often the right theoretical framework to understand these effects. At the core of the theory lies the minimization of an action functional, which in many cases of interest has to be computed by numerical means. Here I will review the theoretical and computational aspects behind the calculation of the quasipotential of LDT whose role is key to understand the long time effect of the random perturbations on the system, including the mechanism of transitions events induced by these perturbations. As illustration, I will use various examples from material sciences, fluid dynamics, atmosphere/ocean sciences, and reaction kinetics. In terms of models, these examples involve stochastic (ordinary or partial) differential equations with multiplicative or degenerate noise, Markov jump processes, and systems with fast and slow degrees of freedom, which all violate detailed balance, so that simpler computational methods are not applicable. Time permitting, I will also discuss situations in which LDT fails to describe metastability, explain why that is the case, and indicate what one can try to do in these cases.

**The Sine$_\beta$ operator**

*Bálint Virág (University of Toronto)*

We show that Sine$_\beta$, the bulk limit of the Gaussian $\beta$-ensembles is the spectrum of a self-adjoint random differential operator \[ f\to 2 {R_t^{-1}} \left[ \begin{array}{cc} 0 &-\tfrac{d}{dt}\\ \tfrac{d}{dt} &0 \end{array} \right] f, \qquad f:[0,1)\to \mathbb R^2, \] where $R_t$ is the positive definite matrix representation of hyperbolic Brownian motion with variance $4/\beta$ in logarithmic time. The result connects the Montgomery-Dyson conjecture about the Sine$_2$ process and the non-trivial zeros of the Riemann zeta function, the Hilbert-Pólya conjecture and de Brange's attempt to prove the Riemann hypothesis. We identify the Brownian carousel as the Sturm-Liouville phase function of this operator. We provide similar operator representations for several other finite dimensional random ensembles and their limits: finite unitary or orthogonal ensembles, Hua-Pickrell ensembles and their limits, hard-edge $\beta$-ensembles, as well as the Schrödinger point process. In this more general setting, hyperbolic Brownian motion is replaced by a random walk or Brownian motion on the affine group. Our approach provides a unified framework to study $\beta$-ensembles that has so far been missing in the literature. In particular, we connect Itô's classification of affine Brownian motions with the classification of limits of random matrix ensembles.

**Estimates for some perturbations of Bessel processes**

*Lidan Wang (University of Washington at Seattle)*

For a reflecting Bessel process, the inverse local time at $0$ is an $\alpha$-stable subordinator, and the subordinate Brownian motion is a symmetric $\alpha$-stable process. We expect to add some perturbations to the generator of Bessel process, then get estimates of the subordinators and the corresponding subordinate Brownian motions.

**University of the stochastic Bessel operator**

*Patrick Waters (Temple University)*

It is known that the smallest several singular values of a Laguerre $\beta$-ensemble $n \times n$ random matrix converge in distribution to singular values of a Stochastic Bessel Operator as $n\rightarrow \infty$. We consider a perturbed Laguerre $\beta$-ensemble in which the potential function is changed from $V(\lambda)=\lambda$ to an arbitrary polynomial function such that $V(\lambda^2)$ is convex. We prove that the Stochastic Bessel operator limit is universal in the sense that it is achieved for all such $V$. Joint work with Brian Rider.