Seminars in 2023

2023/06/30(Fri.) 16:00-17:00@H230（普段と場所が違いますのでご注意ください）

Haru Negami (根上春)氏 (Chiba University)

Title:Title: Construction of representations of braid groups by homological Euler transformation

Abstract: In this talk, we will introduce a method to obtain braid group representation. The Long-Moody construction is a technique to get representations of braid groups introduced by Long and Moody. Through this method, various braid group representations can be constructed, including Burau representations, the unreduced Gassner representation of pure braid group, and the Lawrence-Krammer-Bigelow representation. Based on the analytical background that the representation of the braid group is associated with monodromy representations of KZ-type equations, we generalized the Long-Moody construction and enabled to obtain infinitely many (unitary) representations of braid group. This talk includes joint work with Kazuki Hiroe.

Reference: Long-Moody construction of braid representations and Katz middle convolution https://arxiv.org/pdf/2303.05770.pdf

2023/06/13(Tue.) 16:00-17:00@H213 （火曜日です！ご注意ください）

Yulan Qing (Fudan University)

Title: Boundary of Groups

Abstract: Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it is an essential tool in the study of the coarse geometry of hyperbolic groups. In this study we introduce a generalization of the Gromov boundary for all finitely generated groups. We construct the sublinearly Morse boundaries and show that it is a QI-invariant topological space and it is metrizable. We show the geometric genericity of points in this boundary using Patterson Sullivan measure on the visual boundary of CAT(0) spaces. As an application we discuss the connection between the sublinearly Morse boundary and random walk on groups. We answer open problems regarding QI-invariant models of random walk on CAT(0) groups and on mapping class groups. If time permits, we also will also look at how this boundary behaves under sublinear bilipschitz equivalences.

2023/04/28(Fri.) 16:00-17:00@H213

Daniel Ruberman氏（Brandeis University）

Title: Diffeomorphisms of 4-manifolds and embedding spaces

https://zoom.us/j/95948063911?pwd=ZTRMWkZYSFhneWdrbXJwSWxpWWpNdz09

Abstract: A phenomenon that is unique to dimension 4 is the existence of infinite families of manifolds that are homeomorphic but not diffeomorphic. This is shown via a combination of gauge theory (Seiberg-Witten theory or Yang- Mills theory) with Freedman’s topological classification results. In a joint project with Dave Auckly, we find similar ‘exotic’ behavior comparing the topology of the groups of diffeomorphisms and homeomorphisms of a smooth 4-manifold. Our main theorem is that the kernel of the map on homotopy groups induced by the inclusion Diff(X) → Homeo(X) can be infinitely generated. I will describe similar results about spaces of embeddings of surfaces and 3-manifolds in 4-manifolds.

2023/04/21(Fri.) 16:00-17:00@西８号館W1101***場所が通常と異なりますのでご注意ください****

Serban Matei Mihalache氏（Tohoku University,  東北大学）

Title: Refined Dijkgraaf-Witten invariant of spin 3-manifold

Abstract:

We give a construction of a state sum invariant of oriented closed spin 3-manifold based on super 3-cocycle (α, ω) and a combinatorial representation of spin 3-manifold based on branche ideal triangulation, where ω is a Z2-valued cocycle and ̃α is a 3-cochain satisfying a 3-cocycle condition with a sign coming from the 2-cocycle ω. The definition of the invariant is similar to the state sum construction of the Dijkgraaf-Witten invariant, except it uses the spin structure to take care of the sign in the 3-cocycle condition. We also give an example of the invariant and see that it is sensitive to the spin structure.