Mathematics

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.

Molecular interactions lie at the core of biochemistry and biology, and their understanding is crucial to the advancement of biotechnology, therapeutics, and diagnostics. Most existing tools make “ensemble” measurements and report a single result, typically averaged over millions of molecules or more. These measurements can miss rare events, averaging out the natural variations or sub-populations within biological samples, and consequently obscure insights into multi-step and multi-state reactions. The ability to make and connect robust and quantitative measurements on multiple scales - single molecules, cellular complexes, cells, tissues - is a critical unmet need. In this talk, I will introduce a general method called “CLiC” imaging to image molecular interactions one molecule at time with precision and control, and under cell-like conditions. CLiC works by mechanically confining molecules to the field of view in an optical microscope, isolating them in nanofabricated features, and eliminates the complexity and potential biases inherent to tethering molecules. By imaging the trajectories of many single molecules simultaneously and in a dynamic manner, CLiC allows us to investigate and discover the design rules and mechanisms which govern how therapeutic molecules or molecular probes interact with target sites on nucleic acids; and how molecular cargo is released inside cells from lipid nanoparticles. In this talk, I will discuss applications of our imaging platform to better understand DNA, RNA, protein interactions, as well as emerging classes of genetic medicines, gene editing and drug delivery systems. I will highlight current and potential future applications to connect our observations from the level of single molecule to single cells, and opportunities for collaboration as we set up our labs at UBC.

I will introduce a skeleton obtained directly from monomial relations in a finite quiver without cycles, and relate the construction to some classical examples in mirror symmetry. This is work in progress with David Favero.