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Place: N902
Title: 2 harmonic 1-forms and related problems
Abstract: Z2 harmonic 1-forms are a higher-dimensional generalization of quadratic differentials of Riemannian surface, with tight connections to gauge theory, low-dimensional topology, and calibrated geometry. According to Taubes’ work, Z2 harmonic 1-forms are the ideal boundary of the flat SL(2,C) connections. Previously, most known examples of Z2 harmonic 1-forms are coming from complex manifold. In the first half of the talk, we will discuss about the existence problem of Z2 harmonic 1-form. We will introduce a topological condition to construct examples of Z2 harmonic 1-forms, which could be satisfied for real Riemannian manifolds. In the second half of the talk, a geometry application of Z2 harmonic 1-forms will be discussed. We will explain how to construct branched deformations of special Lagrangian submanifolds using nondegenerate Z2 harmonic 1-forms.
Place: N902
Title: Extendabilities of periodic surfaces maps over n-spheres
Abstract: A self-homeomorphism f of a surface S is said to be extendable over a given manifold M, if S can be embedded into M such that f extends to a self-homeomorphism of M. We will discuss the extendabilities of periodic maps on closed orientable surfaces over 3-sphere and 4-sphere. For 3-sphere we give equivalent conditions for extendabilities and construct all the extendable maps. For 4-sphere we provide non-extendable examples for almost every genus.
Place: N902
Title: SO(3) representations and the four color theorem
Abstract: The four color theorem was proved by Appel and Haken in 1989 using a computer. At ICM 2018, Kronheimer and Mrowka proposed an alternative way based on gauge theory in low-dimensional topology that might give a new proof of this theorem. This talk is a survey of their approach. First, I’ll explain the relation between the SO(3) representation variety of a trivalent graph and the four color theorem. Then I’ll review some properties of singular instanton Floer homology J^# of a trivalent graph.
Place: N902
Title: Symplectic excision and noncompact Moser theorem
Abstract: A symplectic excision is a sympletomorphism from a noncompact symplectic manifold to the complement of a closed subset. We will discuss the existence of symplectic excisions by explicit constructions and examples. Then we introduce a Moser stability theorem on noncompact manifolds.
Place: N902
Title: Turaev–Viro TQFT and the Rank versus Genus Conjecture
Abstract: We introduce the idea of a topological quantum field theory (TQFT), and the construction of Turaev–Viro state sum TQFT. As an application we obtain a lower bound of the Heegaard genus of a 3-manifold, and provide some known counterexamples to the rank versus genus conjecture.
Place: N902
Title: Fractional structures on bundle gerbe modules and fractional classifying spaces
Abstract: We study the homotopy aspects of the twisted Chern classes of torsion bundle gerbe modules. Using Sullivan’s rational homotopy theory, we realize the twisted Chern classes at the level of classifying spaces. The construction suggests a notion, which we call fractional U-structure serving as a universal framework to study the twisted Chern classes of torsion bundle gerbe modules from the perspective of classifying spaces. Based on this, we introduce and study higher fractional structures on torsion bundle gerbe modules parallel to the higher structures on ordinary vector bundles.
This is a joint work with Fei Han and Varghese Mathai.
Place: N902
Title: The mapping class group of manifolds which are like projective planes
Abstract: A manifold which is like a projective plane is a simply-connected closed smooth manifold whose homology equals three copies of Z. In this talk I will discuss our computation of the mapping class group of these manifolds, as well as some applications in geometry.
This is a joint work with WANG Wei from Shanghai Ocean University.
Place: N902
Title: Endotrivial modules of Hopf algebras
Abstract:
Place: N902
Title: Infinite not contact isotopic embeddings into the standard sphere.
Abstract: I will explain that there exist infinitely many formally contact isotopic, but not contact isotopic embeddings of the standard cosphere bundle of a sphere into the standard contact sphere when the dimension is at least 7. This confirms a conjecture of Casals and Etnyre except for the dimension 5 case.