A reaction network is a graphical configuration that describes an interaction between species (molecules). If the abundances of the network system is small, then the randomness inherent in the molecular interactions is important to the system dynamics, and the abundances are modeled stochastically as a jump by jump fashion continuous-time Markov chain. One of challenging issues facing researchers who study biological systems is the often extraordinarily complicated structure of their interaction networks. Thus, how to characterize network structures that induce characteristic behaviors of the system dynamics is one of the major open questions in this literature. In this talk, I will provide an analytic approach to find a class of reaction networks whose associated Markov process has a stationary distribution. Moreover I will talk about the convergence rate for the process to its stationary distribution with the mixing time.

The contact process describes an elementary epidemic model by a continuous-time Markov chain on a given graph. In the process, each vertex is either infected or healthy, and an infected vertex gets healed at rate $1$ while it passes its disease to each of its neighbors at rate $\lambda$, where all the recoveries and infections are independent. In this talk, we discuss the phase diagram of the contact process on Galton-Watson trees and random graphs. To be specific, we show that the infection spread is subcritical for small enough $\lambda$ if $D$, the offspring distribution (resp. degree distribution) of the Galton-Watson tree (resp. random graph), has an exponential tail. On the other hand, when $D$ is subexponential, we prove that the contact process on the random graph is always supercritical. Our result on Galton-Watson trees, in particular, establishes a stronger version of the conjecture by [Huang-Durrett ‘18]. Joint work with Shankar Bhamidi, Oanh Nguyen and Allan Sly.

We consider a system of $N$-Bosons with a two-body interaction potential in three-dimensional space. It is known that the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding Hartree dynamics at time $t$ is less or equal to $Ce^{Kt}/N$ for Coulomb potential, and it is also known that the $N$-dependency of the bound is optimal. We prove that similar result holds for all interaction potential $V \in L^2(\mathbb{R}^3)+L^\infty (\mathbb{R}^3)$ which covers more singular potentials than Coulmomb potential. Moreover, we investigate the time dependency of the difference. To have sub-exponential bound, we use the results of time decay estimate for small initial data. For time dependency, we consider the interaction potential $V(x)$ of type $\lambda \exp(−\mu |x|)|x|^{-\gamma}$ for $\lambda\in\mathbb{R}$, $\mu \geq 0$, and 0 < $\gamma$ < 3/2, which covers Coulomb and Yukawa potential.

The jellium is a model, introduced by Wigner, for a gas of electrons moving in a uniform neutralizing background of positive charge. In two dimensions, the model is closely related to random matrices while in one dimension the model is used to study dimerization and crystallization. For the quantum jellium, Brascamp and Lieb (1975) proved crystallization (non-ergodicity of the Gibbs measures) at low densities of electrons. Using tools from probability theory we prove crystallization for the quantum one-dimensional jellium at all densities as well as similar behavior for quasi-1d systems. This talk is based on joint work with M. Aizenman and S. Jansen.

I will discuss joint work with Federico Camia and Jianping Jiang (arXiv:1707.02668) which proves exponential decay of correlations for the generalized random field that is the scaling limit of the near-critical (i.e., with small magnetic field at the critical temperature) two-dimensional Ising model. The proof involves both lattice and continuum FK (Fortuin-Kasteleyn) representations of the Ising model and the use of coupled conformal loop and measure ensembles.

We first review Krawtchouk polynomial and separability and entanglement of positive definite matrices in the product spaces. We discuss a characterization of integral zeros of binary Krawtchouk polynomials in terms of the range criterion for entangled states in quantum information theory.

I would like to discuss a triangle of ideas in one-dimensional manifold diffeomorphism groups, which consist of analysis, dynamics and group theory. I will first state and prove the Slow Progress Lemma, which justifies a meta-statement "smoother diffeomorphisms are slower". This lemma connects analysis to dynamics. Then I will illustrate how to justify another statement "slower representations have larger kernels", connecting dynamics to group theory. Joint work with Thomas Koberda.

In this talk we study a finite horizon optimal contract problem with limited commitment in continuous time. We use the dual method and study the dual problem which is similar to an incremental irreversible investment problem. We transform the dual problem into an infinite series of optimal stopping problems, which essentially becomes a single optimal stopping problem. The optimal stopping problem has the same characteristics as that of finding the optimal exercise time of an American option, which has an integral equation representation. We recover the value function by establishing a duality relationship and provide some numerical results for optimal consumption strategies. This talk is based on a joint work with Hyeng Kuen Koo.

Diffusion equations is one of most famous partial differential equations (PDEs). Lots of generalized diffusion equations have appeared on the basis of scientific meaning. For instance, degenerate diffusion equations appeared long time ago due to observation of stopping diffusion intermittently. In this talk, we mostly introduce the reason why we study generalized diffusion equations far beyond the heat equation and we are going to discuss the change of regularity of solutions depending on the degeneracy of diffusion if time is allowed.

The bilinear Bochner-Riesz operator is a bilinear multiplier operator related to convergence of the Fourier series. In this talk, we are concerned with boundedness of the bilinear Bochner-Riesz operator on Lebesgue spaces. Especially, we focus on our recent result for the bilinear operator, which is a joint work with Sanghyuk Lee and Ana Vargas. We make use of a decomposition which relates the estimate for the bilinear Bochner-Riesz operator to those of the square function for the classical Bochner-Riesz operator.

In this talk we are concerned with the square-function estimate associated with the Bochner-Riesz means (also known as Stein's square function). Originally, the square function was introduced by Stein to study pointwise convergence of the Bochner-Riesz means but it later turned out that the square function has important applications to other problems, such as $L^p$ boundedness for radial multipliers and smoothing properties for the Schrödinger and wave operators. We discuss the sharp $L^p$ estimate for the square function including the recent development based on multilinear (adjoint) restriction estimate due to Bennett, Carbery and Tao.

The scenario of Type I singularity is a natural generalization of the self-similar or discretely self-similar singularities. Studying the problem of Type I singularities would be eventually helpful for future understanding of the possible singularities in the Euler system. In this talk, after short review of the studies of the self-similar solutions we survey recent works on scenario of the Type I blow-up. By applying the local analysis methods, which have been useful in the elliptic or parabolic regularity theories, we could make some progresses in our study of the Type I blow-up in the Euler equations. The project is jointly done with J. Wolf.

n this talk, I'll prove metastability of the zero range process on a finite set without using capacity estimates. The proof is based on the existence of certain auxiliary functions. One such function is inspired by Evans and Tabrizian's article, "Asymptotics for the Kramers-Smouchowski equations". This function is the solution of a certain equation involving the infinitesimal generator of the zero range process. Another relevant auxiliary function is from a work of Beltran and Landim. We also use martingale problems to characterize Markov processes. Let $p$ be the jump rates of a random walk on a finite set $S$. Assume that the uniform measure on $S$ is an invariant measure of this random walk for simplicity (we expect that our method is applicable for an arbitrary invariant measure $m$). Consider the zero range process on $S$, where the rate the particle jumps from a site $x$ to $y$ with $k$ particles at the site $x$ is given by $g(k)p(x,y)$. Here $g(0)=0, g(1)=1$, and $g(k)=(k/(k-1))^\alpha$, $k>1$ for some $\alpha > 1$. As total number of particles $N \rightarrow \infty$, most of the particles concentrate on a single site. In the time scale $N^{1+\alpha}$, the site of concentration evolves as a Markov chain whose jump rates are proportional to the capacities of the underlying random walk. This talk based on the joint work with F. Rezakhanlou.

The Swendsen-Wang dynamics is an MCMC sampler of the Ising/Potts model, which recolors many vertices at once, as opposed to the classical single-site Glauber dynamics. Although widely used in practice due to efficiency, the mixing time of the Swendsen-Wang dynamics is far from being well-understood, mainly because of its non-local behavior. In this talk, we prove cutoff phenomenon for the Swendsen-Wang dynamics on the lattice at high enough temperatures, meaning that the Markov chain exhibits a sharp transition from “unmixed” to “well-mixed.” The proof combines two earlier methods of proving cutoff, the update support [Lubetzky-Sly ’13] and information percolation [Lubetzky-Sly ’16], to establish cutoff in a non-local dynamics. Joint work with Allan Sly.

We discuss recent advances in the regularity estimates of solutions to nonlinear elliptic and parabolic problems.

We briefly present issues on uniqueness of entropy solutions to the compressible Euler system, which are related to the long standing conjecture: The compressible Euler equations admit a unique entropy weak solution in a class of vanishing viscosity solutions as inviscid limits of solutions to the associated viscous system that is compressible Navier-Stokes system. As a study on the conjecture for an entropy shock, we prove the contraction property for any weak perturbations of viscous shocks of the barotropic Navier-Stokes systems with various viscosity coefficients, by constructing a weighted relative entropy and time-dependent shift. The contraction property of the shocks does not depend on the viscosity coefficient. Therefore, this provides a weak compactness for the inviscid limit problem, that is, entropy shocks for the isentropic Euler system are stable and unique in the class of weak inviscid limits of solutions to the Navier-Stokes system.

According to the results by Krylov and Röckner, unique strong solution exists to stochastic differential equations with non-degenerate noise when the drift vector field belongs to $L^q([0,T],L^p_x)$ for $p,q\in (1,\infty)$ satisfying $\frac{2}{q}+\frac{d}{p}$ < 1. In this talk, we discuss two different directions to generalize the previous results: the critical case $\frac{2}{q}+\frac{d}{p}=1$, and the degenerate noise. I will introduce a Lorentz space to deal with the critical case and hypoelliptic theory to consider the degenerate case. Brief theory of the analysis on Lie groups will be covered and I will explain how this theory can be applied to prove the probabilistic results.

The limiting eigenvalue statistics of a Wigner matrix is given by the semiclrcle law. For the semicircle law, the behavior of the spectral edge is characterized by the square-root decay and the location of the edge. The behavior of the spectral edge for other random matrices can be understood by complex analytic method based on the Stiejtes transform. In this talk, we consider examples such as deformed Wigner matrices and sparse random matrices and analyze the behavior of the spectral edges of those random matrices.

We will discuss about few types of multi-linear operators with singular multipliers and their wide range of $L^p$ estimates by applying time-frequency analysis technique.

We construct a local well-posedness class for the $3D$ incompressible axi-symmetric Euler equations, and show that within this class there exist compactly supported initial data which blows up in finite time. This local well-posedness class is critical in terms of the natural scaling transformation of the Euler equations and consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on $2D$ Euler and Boussinesq systems. An essential idea is the robustness of scale-invariant dynamics for inviscid transport systems. This is joint work with Tarek Elgindi.

For each real number $a=k+s\ge1,$ where $k=[a],$ we define $Diff(R;a)$ as the group of compactly supported $C^k$-diffeomorphisms of the real line whose $k$-th derivatives are $s$-Hölder-continuous. We prove that there exists a finitely generated group $G$ inside $Diff(R;a)$ such that $G$ admits no injective homomorphisms into the group $U \{ Diff(R;b) : b > a \}.$ The cases $a=0$ (Thurston; Calegari for $S^1$) and $a=1$ (Navas) are previously known. This is a joint work with Thomas Koberda.

We will begin by reviewing how a certain modification of the moment method was utilized to obtain limit theorems concerning fluctuations of extreme eigenvalues of various random matrix ensembles. Then we describe how to apply the method to analyze the asymptotic behavior of the first, second, and so on rows of stochastically decaying partitions. This is based on joint work with Sasha Sodin.

We study a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as adjacency matrices of Erdos-Renyi graphs with $p=1/N.$ Our matrices have real entries which are i.i.d. up to symmetry. The distribution of entries depends on $N,$ and we require sums of rows to converge in distribution; it is then well-known that the limit must be infinitely divisible.
We show that a limiting empirical spectral distribution (LSD) exists, and via local weak convergence of associated graphs, the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges. One example covered are matrices with i.i.d. entries having infinite second moments, but normalized to be in the Gaussian domain of attraction. In this case, the LSD is a semi-circle law.

We prove that $(F_2 X Z) * Z$ does not embed into Diff$^{\,2}(S^1).$ Then we classify RAAGs that admit faithful smooth actions on the circle, answering a question raised in a paper of M. Kapovich. (Joint with T. Koberda)

Which finitely presented groups arise as subgroups of PSL(2,R)? The general (indiscrete) case of this question is wide-open. We propose a class of torsion-free groups, called "flexible groups"; these groups admit uncountably many, independent, indiscrete faithful representations into PSL(2,R). We prove a combination theorem for this class of groups. A key underlying idea is a version of Klein--Maskit combination theorem. This is a joint work with Thomas Koberda and Mahan MJ.

Solutions to a large family of stochastic PDEs (SPDEs) exhibit tall peaks on some small regions and this phenomenon is called intermittency. In this talk, we consider the geometric structure of the tall peaks for two types of SPDEs: intermittent SPDEs (SPDEs with nonlinear noise) and non-intermittent SPDEs (SPDEs with additive noise). Using the Barlow-Taylor macroscopic Hausdorff dimension, we will show that there are infinitely many length scales in the tall peaks for both types of SPDEs. This is based on an on-going work with Davar Khoshnevisan and Yimin Xiao.

Gabor systems are structured function systems consisting of time-frequency translates of a window function. I will discuss some of the central problems in Gabor analysis and progress towards them. In particular, I will present recent results on stability of the spanning properties of such systems under deformations of the underlying set of time-frequency nodes. The deformations that we consider are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. The proofs involve a characterization of Gabor frames and Gabor Riesz sequences in the style of Beurling's characterization of sets of sampling for bandlimited functions ('without inequalities').

Consider a system of interacting Brownian motions. Because of the complex nature of interactions among particles, it is difficult to obtain the typical or atypical behavior of a single particle. Varadhan and his co-workers have extensively studied such a problem for interacting particle systems on lattice. In the first part of the presentation, we review the theory proposed by Varadhan et al., which is now called the hydrodynamic limit theory. In the second part or the presentation, we explain our recent generalization of this theory to a system of interacting Brownian motions and discuss the universality of the results.

In the inhomogeneous heat equation $$u_t(t,x) = \Delta u(t,x)+f(t,x),$$ the term $f$ models the interruptions to the heat diffusion along time and on space locations. Especially, the effect of $f$ in time direction is more troublesome and the regularity of $u$ is subject to the regularity of $f.$ In this talk we discuss a type of modeling of $f$ in the form $N_t(\omega)g(t,x),$ where $N$ is a random noise process which can be whiter than the white noise. We also discuss a regularity relation between $N, g$ and $u.$

We will show the asymptotics of the heat kernel $p(t,x,y)$ as $t$ goes to infinity for Riemannian manifolds of negative curvature. We will explain how to use dynamics of the geodesic flow and certain Gibbs measures. This is a joint work with F. Ledrappier.

Stochastic partial differential equations (SPDEs) are differential equations which include the effects of random forces and environments.
The theory of SPDE was started in 1970s, and $L_p$-theory of SPDE was first introduced by Krylov in 1994. Since then, $L_p$-theory has become one of main approaches to study the regularity of solutions to SPDEs.
In this talk, I will give a short description on SPDEs and introduce classical $L_p$-theory of second-order SPDEs. I will also introduce recent results on SPDEs having non-local operators.

In this talk we consider a large class of non-local operators on metric measure spaces. We first show that a uniform and scale invariant boundary Harnack principle holds. Then we study Martin boundaries corresponding to non-local operators. If infinity is accessible from an open set D, then there is only one Martin boundary point of D associated with it, and this point is minimal. Under suitable further assumptions there is only one Martin boundary point associated with infinity, and that this point is minimal if and only if infinity is accessible from D. This is based on joint works with Renming Song and Zoran Vondracek.

The local eigenvalue statistics of a large class of random matrices exhibits the same limiting behaviour, and such property is known as the universality. A typical strategy to prove the universality is to first compute the local statistics for the simplest case, usually the Gaussian case, and compare the given model with the simple model. In this talk, I will explain the comparison methods and how we can apply them to prove universality for complicated models such as deformed Wigner matrices or sample covariance matrices with general population.

One intriguing direction of research in surface theory is the analogy between mapping class groups and higher rank lattices. However, current understanding of finite index subgroups of mapping class groups are still rudimentary. We prove the result in the title, which was originally asked by Farb, and which is analogous to Ghys and Burger--Monod theorem about obstructions of higher rank lattice actions on the circle. (Joint work with Hyungryul Baik and Thomas Koberda)