Mathematics

Liouville quantum gravity (LQG) is a natural model for a random fractal surface with origins in conformal field theory and string theory. A powerful tool in the study of LQG is conformal
welding, where multiple LQG surfaces are combined into a single LQG surface. The interfaces between the original LQG surfaces are typically described by variants of the random fractal curves known as Schramm-Loewner evolutions (SLE). We will present a few recent conformal welding results for LQG surfaces and applications to the integrability of SLE and LQG partition functions. Based on joint work with Ang and Sun and with Lehmkuehler.

Modeling "social" interactions within a large population has proven to be a rich subject of study for a variety of scientific communities during the past few decades. Specifically, with the goal of predicting the macroscopic effects resulting from microscopic-scale endogenous as well as exogenous interactions, many emblematic models for the emergence of collective behaviors have been proposed. In this talk we present a dynamical model for generic crowds in which individual agents are aware of their local environment, i.e., neighboring agents and domain boundary features, and may seek static targets. Our model incorporates features common to many other "active matter'' models like collision avoidance, alignment among agents, and homing toward targets. However, it is novel in key respects: the model combines topological and metrical features in a natural manner based upon the local environment of the agent's Voronoi diagram. With only two parameters, it is shown to capture a wide range of collective behaviors that go beyond the more classical velocity consensus and group cohesion. The work presented here is joint with R. Choksi and J.C. Nave at McGill

The box-ball system (BBS) was introduced by Takahashi and Satsuma as a simple discrete
model in which solitons (i.e. solitary waves) could be observed, and it has since been shown that it can be derived from the classical Korteweg-de-Vries equation, which describes waves in shallow water by a proper discretization. The last few years have seen a growing interest in the study of the BBS and related discrete integrable systems started from random initial conditions. Particular aims include characterizing measures that are invariant for the dynamics, exploring the soliton decompositions of random configurations, and establishing (generalized) hydrodynamic limits. In this talk, I will explain some recent progress in this direction.

Random field Ising model is a canonical example to study the effect of disorder on long
range order. In 70’s, Imry-Ma predicted that in the presence of weak disorder, the long-range
order persists at low temperatures in three dimensions and above but disappears in two
dimensions. In this talk, I will review mathematical development surrounding this prediction,
and I will focus on recent progress on exponential decay (joint with Jiaming Xia) and on
correlation length in two dimensions (joint with Mateo Wirth). In addition, I will describe a
recent general inequality for the Ising model (joint with Jian Song and Rongfeng Sun) which
has implications for the random field Ising model.

In this talk, we study an algebraic approach to fusions of synchronization models. The Lohe tensor model is a generalized synchronization model which contains three synchronization models; the Kuramoto model(on the circle), the swarm sphere model(on the sphere), and the Lohe matrix model(on the unitary group). Since the Lohe tensor model contains any synchronization models defined on any rank and size of tensors, we use this model to study fusions of synchronization models. The final goal of the study is to present a fusion of multiple Lohe tensor models for different rank tensors and sizes. For this, we identify an admissible Cauchy problem to the Lohe tensor model with a characteristic symbol consisting of a size vector, a natural frequency tensor, a coupling strength tensor, and an initial admissible configuration. In this way, the collection of all admissible Cauchy problems for the Lohe tensor models is equivalent to the space of characteristic symbols. On the other hand, we introduce a binary operation which we call “fusion operation," as a binary operation between characteristic symbols. It turns out that the fusion operation satisfies associativity and admits an identity element in the space of characteristic symbols that naturally form a monoid. By the fusion operation, the weakly coupled system of multi tensor models can be obtained by applying the fusion operation of multiple characteristic symbols corresponding to the Lohe tensor models. As a concrete example, we consider a weak coupling of the swarm sphere model and the Lohe matrix model and provide a sufficient framework leading to emergent dynamics to this coupled model.

Speaker Biography

Hansol Park was born and raised in the Republic of Korea(South Korea). He got a Ph. D. in mathematics in 2021 from Seoul National University(Advisor: Prof. Seung-Yeal Ha). During his doctoral period, he tried to integrate various types of synchronization models. Currently, he is a PIMS Postdoc at Simon Fraser University under Prof. Razvan C. Fetecau. So far, most of his researches are related to particle systems with interactions. Recently, he is interested in variation methods (minimization problem) and information geometry.