Differential Geometry and Geometric Analysis

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.

The work of Misha Gromov has revolutionized geometry in many respects, but at the same time introduced a geometric point of view in many questions. His impact is very broad and one can say without exaggeration that many fields are not the same after the introduction of Gromov's ideas.I will try and explain several avenues that Gromov has been pursuing, stressing the changes in points of view that he brought in non-technical terms.Here is a list of topics that the lecture will touch:

1. The h-Principle
2. Distance and Riemannian Geometry
3. Group Theory and Negative Curvature
4. Symplectic Geometry
5. A wealth of Geometric Invariants
6. Interface with other Sciences
7. Conceptualizing Concept Creation

After reviewing ordinary finite-dimensional Morse theory, I will explain how Morse generalized Morse theory to loop spaces, and how Floer generalized it to gauge theory on a three-manifold. Then I will describe an analog of Floer cohomology with the gauge group taken to be a complex Lie group (rather than a compact group as assumed by Floer), and how this is expected to be related to the Jones polynomial of knots and Khovanov homology.