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・ Kurando Baba (Tokyo University of Science)
Title: Backward mean curvature flow starting from equifocal submanifolds and building structures
Abstract: The notion of equifocal submanifolds in symmetric spaces of compact type was introduced by Terng-Thorbergsson (1995). This notion is a generalization of compact isoparametric submanifolds in Euclidean spaces and isoparametric hypersurfaces in spheres, and typical examples are provided by principal orbits of Hermann actions. In this talk, we give results on long-time solutions of backward mean curvature flow starting from equifocal submanifolds in symmetric spaces \(G/K\) of compact type, and explain the building structures on \(G/K\) associated with equifocal submanifolds. It is known that a Coxeter group is defined by using an equifocal submanifold \(M\) in \(G/K\). Then the closures of the Coxeter domains defined in the normal space of \(M\) give rise to a decomposition of \(G/K\) and a building structure via the normal exponential map of \(M\). We also explain the behavior of the long-time solutions of the backward mean curvature flow from the viewpoint of the Coxeter domains. This talk is based on joint work with Naoyuki Koike (Tokyo University of Science).
・ Mirei Itabashi (Ochanomizu University)
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・ Toru Kajigaya (Tokyo University of Science)
Title: Non-existence of stable discrete maps into homogeneous spaces of nonnegative curvature
Abstract: A piecewise smooth map from a finite graph into a smooth Riemannian manifold is called a discrete map.
For discrete maps, one can naturally define functionals such as length and energy, viewed as extensions of those for closed curves.
Maps that give critical points of such functionals are regarded as extensions of closed geodesics, and they are commonly referred to as stationary geodesic nets or discrete harmonic maps, which have been studied extensively.
It is known that when the target Riemannian manifold has negative curvature, critical points of the length or energy functional are always stable with respect to the second variation. In contrast to this situation, in this talk, we show the non-existence of nontrivial stable discrete maps into certain homogeneous Riemannian manifolds of nonnegative curvature.
・ ByeoRhi Kim (POSTECH)
Title: On the construction of foldings of branched covers along knots
Abstract: In this talk, we study braiding and folding of branched covers of the 3-sphere along knots, focusing on constructions derived from quandle colorings of knots and quipu diagrams for finite groups. These techniques were developed in earlier work. We present a detailed folding of the dihedral cover of \(S^3\) branched along the torus knot \(T(2,5)\), and describe a related example for the trefoil knot using the alternating group \(A_4\) . These constructions provide explicit geometric models for branched covers and suggest potential extensions to surface-knot branch sets in \(S^4\).
・ Yuka Kotorii (Hiroshima University)
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・ Sam Nelson (Claremont McKenna College)
Title: Biquandle Virtual Bracket Quivers
Abstract: Biquandle virtual brackets are quantum enhancements of the biquandle homset invariant using a three-part skein relation. In this talk (joint work with students Kazuma Okada and Rion Ostuka) we categorify this invariant via a quiver construction and obtain three new infinite families of polynomial invariants via decategorification.
・ Etsuo Segawa (Yokohama National University)
Title: A comfortable graph embedding on surfaces for quantum walks
Abstract: We propose a quantum walk model appearing in its time evolution operator for graph embeddings into closed surfaces. We describe how its stationary state reflects the embedding. Furthermore, we define a "comforability" in the stationary state that quantifies the accumulation of energy in the internal graph and discuss its dependence on the closed surface embedding.
・ Ayu Suzuki (Japan Women’s University)
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・ Satoshi Yamaguchi (The University of Osaka)
Title: On the Index of the Dirac Operator on the Lattice
Abstract: Lattice field theory provides one of the most reliable nonperturbative formulations of quantum field theory. On the other hand, the Atiyah-Singer index of the Dirac operator plays an important role, for example, in the description of anomalies in quantum field theory. In this talk, we investigate how the index of the Dirac operator can be formulated within lattice field theory. In our work, we define the index of the Dirac operator on the lattice and show that it reproduces the usual index of the continuum Dirac operator in the continuum limit. We also obtain a lattice counterpart of the Atiyah-Patodi-Singer index for the Dirac operator on manifolds with boundaries.
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