The 17th HU-SNU Symposium on Mathematics

 Abstract: The lace expansion is one of the few methods to rigorously prove critical behavior for various models in high dimensions. It was initiated by David Brydges and Thomas Spencer in 1985 to show degeneracy of the critical behavior for weakly self-avoiding walk in d>4 dimensions to that for random walk. This is one of the reasons Brydges was awarded the Poincaré Prize at ICMP this summer. Self-avoiding walk is a standard model for linear polymers in a good solvent. Other models to which the lace expansion has been successfully applied are percolation, lattice trees/animals, the contact process, the Ising and phi^4 models.

 In the colloquium talk, I will explain the lace expansion for self-avoiding walk (past) and how it has been extended to other models listed above (present).  If time permits, I will also explain the current status of my ongoing work on the lace expansion for self-avoiding walk on random conductors (future).

Abstract: We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF) and radial nonlinear heat equation (NLH). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices D≥3; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of $\dot{H}^1$-bounded radial solutions to energy-critical heat equations in dimensions N≥7, building upon soliton resolution for such solutions. This is a joint work with Frank Merle (IHES and CY Cergy-Paris University).

 Abstract: In this talk, we propose a stochastic approximation that converges the viscosity solution to the weighted $p$-Laplace equation. We consider stochastic two-player zero-sum game controled by a random walk, players choices and the gradient of the weight function. Using the generalized viscosity method, we obtain an approximation under the weight. These results are extend previous findings for non-weighted $p$-Laplace equation [Manfredi, Parviainen, Rossi, 2012].

 Abstract: In this talk, we derive the lower bounds for gradients of viscosity solutions to Hamilton-Jacobi equations, where the convex Hamiltonian depends on the unknown function. We obtain several gradient estimates using different methods. First, we utilize the equivalence between viscosity solutions and Barron-Jensen solutions and study the properties of the inf-convolution. Second, we examine the Lie equation to understand how initial gradients propagate along its solutions.

Abstract: In this talk, we will discuss some global regularity results for weak solutions to fractional Laplacian type equations. Specifically, the operator under consideration involves a weight function $A(·, ·)$. Under suitable assumptions on the boundary of the domain and the right hand side (source term), we obtain global Calderón–Zygmund type estimates for the function $u/d^s$ , where $d(x) = dist(x, ∂Ω)$ is the distance to the boundary function. More precisely, we will discuss some sharp regularity results for the function u/ds in the sense of Lebesgue, Sobolev and Hölder depending on various assumptions on the weight function $A(·, ·)$. This talk is based on a joint work with S.-S. Byun and K. Kim. 

 Abstract: In this talk, we study the local behaviors of weak solutions, with possible singularities, of nonlocal nonlinear equations. We first prove that sets of capacity zero are removable for harmonic functions under certain integrability conditions. By adopting this removability theorem, we characterize the behavior of singular solutions near an isolated singularity. Our approach relies on the substantial use of local estimates such as Caccioppoli estimates and the Harnack inequality, together with the nonlocal nonlinear potential theory.

 Abstract: We consider the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation by using a game-theoretic approximation, to which we extend that of [Kohn Serfaty 2006]. Kohn and Serfaty give a deterministic two-person zero-sum game whose value functions approximate the solution to the level set mean curvature flow equation without obstacle functions. We prove that moving curves governed by the mean curvature flow converge in time to the boundary of the convex hull of obstacles under some assumptions on the initial curves and obstacles. Convexity of the initial set, as well as smoothness of the initial curves and obstacles, are not needed. We pursue both open and closed evolutions (with open and closed obstacle respectively) by game-theoretic and topological arguments. Compared to last year's presentation in HU-SNU, we have especially extended the result of closed evolutions.

 Abstract: Clustering behavior is ubiquitous in natural phenomena, such as birds flocking and fish swarming. In 2007, Cucker and Smale introduced the well-known Cucker-Smale model to describe the collective behavior of bird flocks. The population of birds and fish in nature is very large, therefore how to characterize its clustering behavior when the system size goes to infinity is an important issue. The kinetic model provides such a tool to describe the clustering dynamics of the infinite particle system using the density function. In this talk, we will discuss the clustering dynamics of the kinetic Cucker-Smale model with non-compact support.

Abstract: Finding a ground state of a given Hamiltonian of an Ising model on a graph G = (V,E) is an important but hard problem. The standard approach for this kind of problem is the application of algorithms that rely on single-spin-flip Markov chain Monte Carlo methods, such as the simulated annealing based on Glauber or Metropolis dynamics. In this work, we investigate new algorithms, the so-called Digital Annealer's algorithm, and some particular kinds of stochastic cellular automata, the SCA and the ε-SCA. We prove that if we consider exponential cooling schedules, the algorithms converge to approximations of ground states. We also provide some simulations of these algorithms and show their superior performance compared to the conventional simulated annealing.

 Abstract: Free independence, which differs from (classical) independence, is a concept in noncommutative probability theory, and it emerges as the limit of the distribution of eigenvalues of certain independent random matrices. In this talk, we review the regularity properties of the sum and product of free independent random variables and present some extensions of previous results.

Abstract: In the last decades, Free probability theory has become a topic of great interest in the random matrix community, since it was observed by Voiculescu that conjugation by Haar unitary matrices generates free independence. As a consequence, understanding spectral properties of the sum of two large matrices conjugated by Haar unitary matrices comes to understanding their free additive convolution. In this talk, I will consider the free additive convolution of two probability measures that are supported on more than one interval and will discuss new results on the local behavior of its density close to the endpoints of its support. More specifically, I will show that for a large class of probability measures that verify a power law behavior with exponents strictly between -1 and 1, the free additive convolution exhibits a universal behavior: it only decays as a one-sided square root or as a two-sided cubic root at any endpoint of its support. I will then consider the free additive convolution semigroup and will determine the behavior of its density at any singular point of its support.

Abstract: A metastable state is a state which is not a real stable state, but can be observed for a long time. This is a concept that appears in several areas of natural science, like chemical kinetics, meteorology, neuroscience, etc. Some of these phenomena can be modeled as a dynamical system, but traditional dynamical systems theory was developed by analyzing behaviors of the system in the infinite time limit, so mathematical theory for understanding (non-trivial) dynamics on metastable time scales would be far from complete (although there are several important developments recently). In this talk, I try to concentrate on a (famous) toy model dynamics, which is a piecewise expanding interval map without statistical stability (i.e. its "physical" invariant measure does not vary continuously under perturbations), to make the presentation of our idea/formulation transparent. Our results include central limit theorems and large deviation principles on metastable time scales. This is based on a joint work in progress with J. Atnip, C. Gonzalez-Tokman, G. Froyland, and S. Vaienti.

Abstract: In this lecture, we review classic theory on the Kesten's stochastic recurrence equation which naturally appears in the analysis of various time series such as SGD or GARCH. We then explain our recent result on the escape time analysis on two major regimes which connects the Lyapunov exponent of the process and the escape time from a domain. Then, we try to connect our result to the empirically observed behavior of the time series quoted above.

Abstract: Probabilistic methods play significant roles across many fields, including abstract harmonic analysis and Banach algebra. Among the various applications, this talk aims to explain two applications of the non-commutative Khintchine inequality. The first addresses the random Fourier series and the associated Lp-lq boundedness problems. The second focuses a Banach algebra problem, specifically, the complete representability of the convolution algebra L^1(G) of general compact quantum groups.