In this talk, we will discuss the LUE, focusing on the scaling limits. On the hard-edge side, we construct the $\alpha$-Bessel line ensemble for all $\alpha \in \mathbb{N}_0$. This novel Gibbsian line ensemble enjoys the $\alpha$-squared Bessel Gibbs property. Moreover, all $\alpha$-Bessel line ensembles can be naturally coupled together in a Bessel field, which enjoys rich integrable structures. We will also talk about work in progress on the soft-edge side, where we expect to have the Airy field as the scaling limit. This talk is based on joint works with Lucas Benigni, Pei-Ken Hung, and Greg Lawler.

In the last 10-15 years, Busemann functions have been a key tool for studying semi-infinite geodesics in planar first and last-passage percolation. We study Busemann functions in the semi-discrete Brownian last-passage percolation (BLPP) model and use these to derive geometric properties of the full collection of semi-infinite geodesics in BLPP. This includes a characterization of uniqueness and coalescence of semi-infinite geodesics across all asymptotic directions. To deal with the uncountable set of points in BLPP, we develop new methods of proof and uncover new phenomena, compared to discrete models. For example, for each asymptotic direction, there exists a random countable set of initial points out of which there exist two semi-infinite geodesics in that direction. Further, there exists a random set of points, of Hausdorff dimension 1/2, out of which, for some random direction, there are two semi-infinite geodesics that split from the initial point and never come back together. We derive these results by studying variational problems for Brownian motion with drift. Based on joint work with Timo Seppalainen.

In the recent twenty years, there has been a huge development in understanding the universal law behind a family of 2d random growth models, the so-called Kardar-Parisi-Zhang (KPZ) universality class. Especially, limiting distributions of the height functions are identified for a number of models in this class. Different from the height functions, the geodesics of these models are much less understood. There were studies on the qualitative properties of the geodesics in the KPZ universality class very recently, but the precise limiting distributions of the geodesic locations remained unknown. In this talk, we will discuss our recent results on the one-point distribution of the geodesic of a representative model in the KPZ universality class, the directed last passage percolation with iid exponential weights. We will give the explicit formula of the one-point distribution of the geodesic location joint with the last passage times, and its limit when the parameters go to infinity under the KPZ scaling. The limiting distribution is believed to be universal for all the models in the KPZ universality class. We will further give some applications of our formulas.

Given edge-lengths (t_e) assigned to the edges of Z^d for d \geq 2, each vertex draws a directed edge to its closest neighbor to form the ``nearest neighbor'' graph. Nanda-Newman studied these graphs when the t_e's are i.i.d. and continuously distributed and showed that the undirected version has only finite components, with size distribution whose tail decays like 1/n!. I will discuss recent work with B. Bock and J. Hanson in which the t_e's are only assumed to be translation-invariant and distinct. Here, we can show that there are no doubly-infinite paths and completely characterize the set of possible graphs. In particular, for d=2, the number of infinite components is either 0,1, or 2, and for d \geq 3, it can be any nonnegative integer. I will also mention relations to both geodesic graphs from first-passage percolation and the coalescing walk model of Chaika-Krishnan.

The diffusion coefficient of a Lévy process — the variance of its Gaussian part — can be viewed as an elementary case of a quantity known as the integrated volatility, whose estimation in the high-frequency sampling regime has been a topic of active research for the past 2+ decades. However, even in the fundamental case of a univariate Lévy process, there remain significant practical challenges to efficient estimation of this quantity when sample paths exhibit extreme jump behavior.

In this talk, I will discuss some recent work concerning optimal estimation of the diffusion coefficient of a Lévy process in the high-frequency sampling regime under extreme jump activity. This is joint with Yuchen Han and José E. Figueroa-López (Wash. Univ. in St. Louis).

Pareto peeling describes a family of algorithms for multidimensional sorting. I will discuss joint work with Peter Morfe in which we show that Pareto peeling of large random point clouds approximates the solution of a Hamilton-Jacobi equation. The limiting equations are closely related to the longest chain problem.

I will present exact probability formulas, obtained via the Bethe Ansatz, for interacting particle systems on a one-dimensional lattice. The Bethe Ansatz, originally introduced in 1931 by Hans Bethe, is a method to diagonalize some linear operators that have certain symmetries; I will make this more precise in the talk. In particular, I will focus on two models: the Heisenberg-Ising spin-1/2 chain (aka XXZ spin-1/2 chain) and the asymmetric simple exclusion process (ASEP). I will present some well-known results from the literature for the ASEP and some recent results based on joint work with C.A. Tracy for the XXZ.

The KPZ equation is a fundamental stochastic PDE that can be viewed as the log-partition function of continuum directed random polymer (CDRP). In this talk, we will first focus on the fractal properties of the tall peaks of the KPZ equation. This is based on separate joint works with Li-Cheng Tsai and Promit Ghosal. In the second part of the talk, we will study the KPZ equation through the lens of polymers. In particular, we will discuss localization aspects of CDRP that will shed light on certain properties of the KPZ equation such as ergodicity and limiting Bessel behaviors around the maximum. This is based on joint work with Weitao Zhu.

We show a limit shape result of the KPZ equations under weak noise scaling, when we condition the value of the KPZ equations at one point to be very large. We also show that under such conditioning, the shape of the noise in the KPZ equation is asymptotically given by the optimizer of the L^4 Gagliardo-Nirenberg-Sobolev inequality.

The class, ID, of infinitely divisible distributions appears as the class of limiting distributions of uniformly infinitesimal triangular arrays. It is intimately connected with Lévy stochastic processes. On the level of characteristic functions (Fourier transforms) infinitely divisible distributions are characterized by the Lévy-Khintchine formula. In this lecture, following Kazimierz Urbanik (1972 and 1973), we will present an increasing sequence of subclasses (convolution semigroups) that begins with the class of normal distributions and ends (after taking closure) with the whole class ID. Each subclass will be described, among others, via the random integrals and characteristic functions. [Also we may mention the general conjecture on random integral representations.]

We examine statistical properties of integers when they are sampled using the Riemann zeta distribution and compare to similar properties when they are sampled according to "uniform" or harmonic distributions. For example, as the variable in the Riemann zeta function approaches 1, a central limit theorem and large and moderate deviations for the distinct number of prime factors for the sampled integer can be readily derived. These results can then be deduced for the uniform distribution via a Tauberian Theorem.

After giving a general and self-contained introduction to the homogenization of Hamilton-Jacobi (HJ) equations, including the classical results in the cases where the Hamiltonian is periodic in the spatial variable $x$ or convex in the gradient variable $p$, I will focus on viscous HJ equations in one space dimension with separable Hamiltonians of the form $G(p) + V(x,\omega)$, where $G$ is a nonconvex function and $V$ is a stationary & ergodic random potential that satisfies a valley & hill condition (which holds in many natural examples, but fails in the periodic case). I will present several recent results on this class of HJ equations (by various combinations of A. Davini, E. Kosygina, O. Zeitouni and myself), where homogenization is established by showing that, outside of the intervals where the effective Hamiltonian turns out to be flat (due to the valley & hill condition), there is a unique sublinear corrector (which is a notion I will introduce) with certain properties. In the special case where $G$ is the minimum of two identical parabolas, these sublinear correctors have convenient representations involving Brownian motion in a random potential. More generally, the existence & uniqueness of these sublinear correctors can be proved using ODE methods that bypass the need for explicit representations, which I will demonstrate when $G$ is quasiconvex.

We introduce the diffusion K-means clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion K-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as similarities given by a kernel that captures the local geometry of manifolds. The diffusion K-means is a multi-scale clustering tool that is suitable for data with non-linear and non-Euclidean geometric features in mixed dimensions. Given the number of clusters, we propose a polynomial-time convex relaxation algorithm via the semidefinite programming (SDP) to solve the diffusion K-means. In addition, we also propose a nuclear norm regularized SDP that is adaptive to the number of clusters. In both cases, we show that exact recovery of the SDPs for diffusion K-means can be achieved under suitable between-cluster separability and within-cluster connectedness of the submanifolds, which together quantify the hardness of the manifold clustering problem. We further propose the localized diffusion K-means by using the local adaptive bandwidth estimated from the nearest neighbors. We show that exact recovery of the localized diffusion K-means is fully adaptive to the local probability density and geometric structures of the underlying submanifolds. Joint work with Yun Yang (UIUC).