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Language: English
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Title: The Obstruction of Orientability of Real Monopole Floer Homology
Abstract: The Real Seiberg-Witten theory is one of a stong tool in knot theory and surface knot theory. For example, that can detect exotic P^2-knot and Kang-Park-Taniguchi proved that any non trivial cable of figure eight knot is not slice using Real Seiberg-Witten theory. This theory is also provide interesting phenomena in index theory, since Real Seiberg-Witten theory can be seen as a "non-linear version" of Atiya's Real K-theory. For example, there is an obstruction to orient moduli space of Real Seiberg-Witten eqations on closed 4-manifolds. Recently, Bonciocat obtained a obstruction to lift the real Heegaard Floer homology from mod 2 coefficient to Z-coefficient. In this talk, we give a canonical way to obtain Z- coefficient real monopole Floer homology of double branced cover of a link in homology S^3 in the case when the obstruction vanishes. We also proved that the Bonciocat's obstruction is also obstract Z-coefficient in real monopole, however the proof is different. This is joint work with Jiakai Li.
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Seminar by JSPS Fellow: Asa Conover
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Title: Strong positivity property for subdiffusion equations and applications to related inverse problems
Speaker: Prof. Yikan Liu (Kyoto University)
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Title: From image generation to three-dimensional turbulence: a physically interpretable diffusion-model from the perspective of deterministic PDEs
Speaker: Prof. Tsuyoshi Yoneda (Hitotsubashi University)
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Title: Mathematical analysis of generalized fractional viscoelastic models
Speaker: Prof. Hiromichi Itou (Chuo University)
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Seminar by Kwabena Boahen, Stanford University
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Title: Designing Complex Fluids: Leveraging Rheology for Advanced Manufacturing
Speaker: Dr. Chaimongkol Saengow, Lecturer, the Sirindhorn International Institute of Technology, Thammasat University
This seminar is co-hosted by the Mechanics and Materials Unit and the Micro/Bio/Nanofluidics Unit
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Title: Signs in real monopole Floer homology
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The computation of the 3-dimensional framed real Seiberg–Witten invariant of the branched double cover of a knot K is instrumental in Miyazawa's construction of exotic P^2-knots. This invariant, known as Miyazawa's degree invariant deg(K), is the Euler characteristic χ(HMR) of framed real monopole Floer homology (HMR). A priori deg(K) is defined only up to an overall sign, since the real Dirac operator has no complex orientation. Worse, signs associated to distinct real spin^c structures are not comparable. Resolving the latter issue is essential for developing computational techniques for real Seiberg--WItten invariants.
I will explain how to fix the sign of the 3-dimensional count, in other words, define an absolute mod-two grading, via a mod-two index-theoretic correction term. This yields a total Euler characteristic \tilde{χ} that sums over all real spin^c structures, the analogue of \hat{χ} in real Heegaard Floer homology (HFR). Recent work of Srivastava provides the motivating results: an absolute mod-two grading in \hat{HFR} and the evaluation of \hat{χ} in terms of the Alexander polynomial at \sqrt{-1}. This is joint work with Ciprian Manolescu.