[
Department of Mathematics] [Shinshu University] Matsumoto, 390-8621 |

Editor of this web site; Takahiro Matsushita ---matsushita(at)shinshu-u.ac.jp |

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Title: | A non-commutative Reidemeister-Turaev torsion of homology cylinders |
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Speaker: | Yuta Nozaki (Yokohama National University) |

Room: | A-401, Faculty of Science, Shinshu University |

Abstract: | A homology cylinder is a 3-manifold that is homologically the product of a surface and an interval. The monoid of homology cylinders contains the Torelli group of a surface as a subgroup. In this talk, we introduce the Reidemeister-Turaev torsion of homology cylinders which takes values in the K_1-group of some completion of the group ring of the fundamental group of a surface over the rationals, and prove that a certain reduction of this torsion is a finite-type invariant. We also show that the 1-loop part of the LMO homomorphism and the Enomoto-Satoh trace can be recovered from the leading term of our torsion. This is joint work with Masatoshi Sato and Masaaki Suzuki. |

Title: | Index theory for quarter-plane Toeplitz operators via extended symbols |
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Speaker: | Shin Hayashi (Aoyama Gakuin University) |

Room: | A-401, Faculty of Science, Shinshu University |

Abstract: | Index theory for Toeplitz operators on a discrete quarter-plane has been investigated by Simonenko, Douglas-Howe, Park, and index formulas are obtained by Coburn-Douglas-Singer, Duducava. In this talk, we consider such operators of two-variable rational matrix function symbols and revisit Duducavafs idea to use Gohberg-Kreinfs theory for factorizations of matrix-valued functions from a geometric viewpoint. We see that, through matrix factorizations and analytic continuations, the symbols of Fredholm quarter-plane Toeplitz operators defined originally on a two-dimensional torus can canonically be extended to some three-sphere. By using K-theory, we show that their Fredholm indices coincide with the three-dimensional winding number of extended symbols. |

Title: | Torsion in classifying spaces of gauge groups |
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Speaker | Masaki Kameko (Shibaura Institute of Technology) |

Room: | A-401, Faculty of Science, Shinshu University |

Abstract: | Tsukuda showed that the integral homology of the classifying space of the gauge group of the nontrivial SO(3)-bundle over the 2-dimensional sphere has no torsion. SO(3) is isomorphic to the projective unitary group PU(2). I will generalize Tsukuda's result on the SO(3)-bundle to PU(n)-bundles. This talk is based on my paper with the same title, published online on April 1, 2024, in Proceedings of the Royal Society of Edinburgh, Section A: Mathematics. |

Title: | On hidden face contributions of configuration space integrals for long embeddings |
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Speaker | Leo Yoshioka (The University of Tokyo) |

Room: | A-401, Faculty of Science, Shinshu University |

Abstract: | In this talk, we study the space of long embeddings of Euclidean spaces with arbitrary codimension greater than one. During the 2010s, Sakai and Watanabe discovered that a geometric approach, called configuration space integrals, provides a formal relationship between the de Rham complex of this space of long embeddings and a cochain complex generated by graphs. However, these integrals may fail to give a cochain map in the general case, due to potential obstructions called hidden faces. In this talk, we first review which hidden faces need to be addressed. Then, we develop an approach to cancel some of these obstructions, by incorporating the bar construction of a certain dg algebra into the original graph complex. We show that this approach actually gives new non-trivial cocycles of the space of long embeddings. |

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