Japan-Netherlands Online Seminar on Index Theory, Geometry and Physics
1-2 February 2022

M. Yamashita - Twisted differential KO-theories via fermionic mass terms

In this talk, I explain my work with Kiyonori Gomi on differential KO theory. We construct a model for differential KO theory which can be regarded as classifying "fermionic mass terms" in physics. We generalize the construction to twisted differential KO theory, where twists come from bundles of graded central simple algebras. I also explain the differential pushforward (integration) in this model, which is expected to be related to partition functions for massive fermions in physics.

H. Konno - Floer K-theory for knots

This talk is based on joint work with Jin Miyazawa and Masaki Taniguchi where we established a version of Seiberg-Witten Floer K-theory for knots. This framework is used to prove a version of “10/8-inequality for knots”, which effectively extracts difference between topological and smooth categories in knot theory. I will explain concrete applications including stabilizing numbers and relative genus bounds, as well as how we constructed this framework.

H. Inoue - An index theorem in scattering theory with 2 Hilbert spaces

Levinson’s theorem theorem is a fundamental result in non-relativistic quantum mechanics. It relates the scattering matrix to the number of bound states of the underlying quantum system. Topological methods for this relation have been widely popularized for the last about ten years. In this talk, we provide index theoretic interpretation of Levinson’s theorem for a model consisting of conducting wires attached to a finite graph. In this model, the perturbed and unperturbed Hamiltonians act on different Hilbert spaces, and the already existing C*-algebraic framework can not be directly applied although quantities in both sides of Levinson’s theorem are well-defined. We also discuss a possible generalization of our framework to topological insulators in place of finite graphs. This talk is based on a joint work with Johannes Kellendonk and Hermann Schulz-Baldes.

M. Seri - A semiclassical approach to the dynamics of Bloch electrons

The Fermi surface is an important concept in solid state physics and in the theory of transport of electron in metals. While physically this has been thoroughly investigated in decades of experiments, the mathematics has somehow lagged behind. In this talk I will start from the physical definition of Fermi surface and a very brief review of recent results by Novikov and Maltsev to justify the interest for a new approach. I will then suggest a rigorous definition of the physical Fermi surface in presence of perturbations in the semiclassical regime, inspired by recent developments in the mathematics of topological insulators. The talk is based on a joint work with Max Lein and Giuseppe De Nittis.