Upcoming Seminars:
Wednesday, July 23, 2025, 16:30--18:00
Venue: A-401, Faculty of Science, Shinshu University
Speaker: Takumi Maegawa (The University of Tokyo)
TItle: A six-functor formalism and the Bauer-Furuta invarint
Past Seminars:
Wenesday, April 30, 2025, 16:30--18:00
Venue: A-401, Faculty of Science, Shinshu University
Speaker: Luigi Caputi (Bologna University)
Title: Bridging between überhomology and double homology
Abstract: Überhomology is a recently defined triply-graded homology theory of simplicial complexes, which yields both topological and combinatorial information. When restricted to (simple) graphs, a certain specialization of überhomology gives a categorification of the connected domination polynomial at -1; which shows that it is related to combinatorial quantities. On the topological side, überhomology detects the fundamental class of homology manifolds, showing that this invariant is a mixture of both. From a more conceptual viewpoint, we will show that a specification of überhomology of simplicial complexes can be identified with the second page of the Mayer-Vietoris spectral sequence, with respect to the anti-star covers. As a corollary, this allows us to connect überhomology to the double homology of moment angle complexes as defined by Limonchenko-Panov-Song-Stanley. This is joint work with D. Celoria and C. Collari.
Friday, May 23, 2025, 16:30--18:00
Venue: A-401, Faculty of Science, Shinshu University
Speaker: Norhiko Minami (Yamato University)
TItle: 純粋にトポロジーだけの範疇で定義される代数幾何的不変量
Friday, June 20, 2025, 16:30--18:00
Venue: A-401, Faculty of Science, Shinshu University
Speaker: Ryu Ueno (Hokkaido University)
TItle: Statistical Biharmonicity of Identity maps
Abstract: A statistical manifold $(M,g,\nabla)$ is a Riemannian manifold $(M,g)$ equipped with an affine connection $\nabla$, not necessarily the Levi-Civita connection. On a statistical manifold, there exists an important vector field $T$ known as the Tchebychev vector field, which characterizes the equiaffine structure of the statistical manifold. A statistical manifold is said to satisfy the equiaffine condition if $T=0$ holds.
Let $\nabla^g$ denote the Levi-Civita connection of $g$. Considering the identity map between statistical manifolds $(M,g,\nabla)$ and $(M,g,\nabla^g)$, our study finds that the tension field of this identity map is equal to $-T$. In Riemannian geometry, maps with vanishing tension fields are called harmonic maps. However, in the geometry of statistical manifolds, the tension field alone does not define classes of maps. Instead, statistical biharmonic maps between statistical manifolds are derived from a variational principle.
In this talk, we present results on the statistical biharmonicity of the identity map. The content is based on arXiv:2411.14156.
Organizer:
Takahiro Matsushita (Shinshu University)