The aim of this study group is to introduce both mathematicians and researchers of iTHEMS in other fields to various recent research topics and the variety of mathematical research areas. Another important aim is to enhance the mathematical interaction between researchers from various fields.
To achieve these aims, iTHEMS math seminar holds various events every semester. Usually, these are seminars or lecture series. Seminar talks are held around ten to fifteen times per semester. They consist of two parts. In the first part, the speaker gives an introductory talk on the subject, aimed at researchers in theoretical, mathematical, and computational sciences, but not necessarily familiar with the specific research field. The second part of the talk provides the opportunity of delving more into detail on the subject and is directed at a narrower audience. However, it is up to the speaker to choose content and targets for this second part. For instance, the second part can elaborate reasoning from the first part, provide rigorous mathematical proofs on statements from the first part, or create different perspectives, like indicating connections to different fields such as quantum mechanics.
Narutaka Ozawa (Professor, Research Institute for Mathematical Sciences (RIMS), Kyoto University)
Suppose you are given a large finite set G and want to estimate the size |G| or see how a typical element x in G looks like. In this talk, G will be a finite group generated by g_1,...,g_d. The "Product" Replacement Algorithm" is a popular algorithm for random sampling in the group G. The PRA shows outstanding performance in practice, but the theoretical explanation has remained mysterious. I will talk how an infinite-dimensional topological-algebraic analysis (operator algebra theory) connects this problem to a convex (semidefinite) optimization problem that can be rigorously solved by computer.
This talk is intended for a general audience.
Aug. 2, 16:00-17:00, on Zoom + Common Room
Hayato Imori (Ph.D. Student, Division of Mathematics and Mathematical Sciences, Graduate School of Science, Kyoto University)
Floer theory is an infinite-dimensional version of Morse theory and has provided powerful invariants in the study of low-dimensional topology. In the context of Yang-Mills gauge theory, some versions of Floer homology groups for knots have been developed. These knot invariants are called instanton knot homology groups and are strongly related to representations of the fundamental group of the knot complement.
In this talk, the speaker introduces basic constructions of instanton knot homology groups and recent developments related to the equivariant version of instanton knot homology theory.
Jul. 25, 16:00-18:00, on Zoom + Common Room
Hokuto Konno (Assistant Professor, Graduate School of Mathematical Sciences, The University of Tokyo)
I will survey a mathematical object called the Seiberg-Witten Floer homotopy type introduced by Manolescu. This is a machinery that extracts interesting aspects of 3- and 4-dimensional manifolds through the Seiberg-Witten equations. This framework assigns a 3-manifold to a "space" (more precisely, the stable homotopy type of a space), and this space contains rich information that is strong enough to recover the monopole Floer homology of the 3-manifold, which is known already as a strong invariant.
I shall sketch how this theory is constructed along Manolescu's original work, and introduce major applications. If time permits, I will also explain recent developments of Seiberg-Witten Floer homotopy theory.
Jul. 15, 14:00-16:30, on Zoom + Common Room