The variational principle, or the least action principle, offers a framework for the study of the Large Deviation Principle (LDP) for a stochastic system. The KPZ equation is a stochastic PDE that is central to a class of random growth phenomena. In this talk, we will study the Freidlin--Wentzell LDP for the KPZ equation through the lens of the variational principle. Such an approach goes under the name of the weak noise theory in physics. We will explain how to extract various limits of the most probable shape of the KPZ equation in the setting of the Freidlin--Wentzell LDP. We will also review the recently discovered connection of the weak noise theory to integrable PDEs. This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Yier Lin.

This talk presents the convergence of the KPZ equation to the directed landscape, which is the central object in the KPZ universality class. This convergence result is the first to the directed landscape among the positive temperature models.

The study of products of random matrices has been proposed many decades ago by Bellman and by Furstenberg and Kesten with the motivation to understand the properties of the Lyapunov exponents in this toy model for chaotic dynamical systems. In this talk, I will discuss the real eigenvalues of products of random matrices with i.i.d. Gaussian entries. In the critical regime where the size of matrices and the number of products are proportional in the large system, I will present the mean and variance of the number of real eigenvalues. Furthermore, in the Lyapunov scaling, I will introduce the densities of real eigenvalues, which interpolates Ginibre's circular law with Newman's triangular law. This is based on joint work with Gernot Akemann.

We show that an irrelevance result reminiscent of Modigliani and Miller holds in a non-public setting of a startup owned by two entities (an entrepreneur and VC) with divergent roles, interests, and bargaining power. The startup always maximizes innovation efficiency and firm value.

Keywords: Startup, Venture Capital, Local time, Reflected Brownian motion

The KPZ class is a very large set of 1+1 models that are meant to describe random growth interfaces. It is believed that upon scaling, the long time behavior of members in this class is universal and is described by a limiting random object, a Markov process called the KPZ fixed-point. The (one-type) stationary measures for the KPZ fixed-point as well as many models in the KPZ class are known - it is a family of distributions parametrized by some set I_ind that depends on the model. For k\in \mathbb{N} the k-type stationary distribution with intensities \rho_1,...,\rho_k \in I_ind is a coupling of one-type stationary measures of indices \rho_1,...,\rho_k that is stationary with respect to the model dynamics. In this talk we will present recent progress in our understanding of the multi-type stationary measures of the KPZ fixed-point as well as the scaling limit of multi-type stationary measures of two families of models in the KPZ class: metric-like models (e.g. last passage percolation) and particle systems (e.g. exclusion process). Based on joint work with Timo Seppalainen and Evan Sorensen.

No seminar

First passage percolation may be viewed as a way of creating a random metric on the integer lattice. Random passage times (iid) are assigned to the edges of the lattice, and the distance from $x$ to $y$ is defined to be the smallest sum of passage times among all paths from $x$ to $y$. We consider geodesics (shortest paths) in this context; specifically, what is the probability that two close--by nearly--parallel geodesics are disjoint? More precisely, if the geodesics have end-to-end distance $L$ and the endpoints are separated by distance $a$ at one end and $b$ at the other, how does the disjointness probability vary with $L,a,b$? We answer this question, at least at the level of the proper exponents for $L,a,$ and $b$.

In this talk, I will cover a study of the Loewner Differential Equation using Rough Path techniques, and beyond. The Loewner Differential Equation describes the evolution of a family of conformal maps. We rephrase this in terms of singular Rough Differential Equations (RDE). In this context, it is natural to study questions on the stability, and approximations of solutions of this equation. I will present a result on the continuity of the dynamics and related objects in the parameter kappa. The first approach will be based on Rough Path Theory, and the second approach will be based on a constructive method of independent interest: the square-root interpolation of the Brownian driver of the Loewner Differential Equation. I will also touch on the second question, which is the approximation of solutions of this singular RDE. I will present another approximation method with some further applications to the simulations of the SLE traces. In the final part, I will touch on some recent results on the Multiple SLE with Dyson Brownian motion driver, as well as on future potential investigations in this direction.

The t-PNG model is a one-parameter deformation of the polynuclear growth model that was recently introduced by Aggarwal, Borodin, and Wheeler, who studied its fluctuations using integrable probability methods. In this talk, we will discuss how to use techniques from interacting particle systems to prove a strong law of large numbers for this model. To do so, we will introduce a new colored version of the model that allows us to apply Liggett's subadditive ergodic theorem to obtain the hydrodynamic limit. The t-PNG model also has a close connection to the stochastic six vertex model. We will discuss this connection and explain how we can prove similar results for the stochastic six vertex model. This talk is based on joint work with Yier Lin.

This talk mainly concerns about a K sample high dimensional CLT over a class of hyper-polygons. This result finds an extremely useful application in the context of high dimensional MANOVA problems based on supremum type test statistics for the difference in means among the K groups. Here we can allow the number of groups (K) to diverge to infinity. The test procedure considered is free from any distribution and correlational assumptions which broadens its scope towards practical applications. The problem of generalized linear hypothesis testing has also been tackled using the previously mentioned CLT result. The asymptotic analysis of these tests in terms of controlling size and power has been theoretically validated. A detailed simulation study has been conducted which corroborates the findings done in the theoretical sections.

In this presentation we will talk about a class of random walks that build their own domain. I.e., at each step of the random walk, the walker may add new vertices to the underlying graph according to some distribution. We will see that under the right conditions the walker might be either transient or recurrent. Additionally, in the transient regiment, the walker may exhibit ballistic behavior.

We will also discuss some results from the perspective of the environment. That is, we will answer some questions regarding structural properties of the sequence of random graphs generated by the walker.

Although lattice Yang-Mills theory is easy to rigorously define, the construction of a satisfactory continuum theory is a major open problem in dimension d ≥ 3. Such a theory should assign a Wilson loop expectation to each suitable collection L of loops in d-dimensional space. One classical approach is to try to represent this expectation as a sum over surfaces having L as their boundaries. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities.

I will introduce the subject and show how to make sense of Yang-Mills integrals as surface sums for d=2, where the continuum theory is already understood. This perspective leads to alternative proofs of the Makeenko-Migdal equation and the Gross-Taylor expansion. The presentation is based on a joint work with Minjae Park (Chicago), Joshua Pfeffer (Columbia) and Pu Yu (MIT).