UCR Geometry-Topology seminar 

Friday 11am-12 


 2024 Spring quarter

Title: The moduli space of solutions to the extended Bogomolny equations on ∑xR_+

Abstract: The extended Bogomolny equations (EBE) on the product of a Riemann surface ∑_g of genus g and a half line R_+ appear naturally in Witten's approach to Khovanov homology as dimensional reductions of the Kapustin-Witten (KW) equations. The physically interesting solutions satisfy a singular boundary condition at y = 0 specified by an effective divisor D in ∑  and a classical asymptotic condition as y goes to infinity. In this talk, fixing the effective divisor D, I will show that the moduli spaces of solutions to the EBE with the above properties are diffeomorphic to certain holomorphic Lagrangian submanifolds of the moduli space of Higgs bundles. Time permitting I will discuss how one can use this result to construct non-trivial knot solutions to the KW equations. 

Title: Minkowski inequality on static manifolds (on Zoom) TAKE 2

Abstract:  The classical Minkowski inequality provides a lower bound for the integral of mean curvature of a closed convex surface in Euclidean space. We describe how geometric flow method (Guan-Li, Brendle-Hung-Wang, etc) allows generalizations of the Minkowski inequality to non-convex surfaces in more general ambient spaces. The talk is based on joint work with Brian Harvie.

Title: Title: Dwyer-Kan theorem for model categories

Abstract: The homotopy theory of homotopy theories proved to be exceptionally useful tool in homotopy theory. However, the applications to model categories depend on the assumption that they are turned into small categories by changing the universe. It does not suit all purposes. In order to extend the techniques of homotopy theories of homotopy theory we suggest an analog for model categories of the results of Dwyer and Kan (87'), stating (in the modern terms) that a map of relative categories (A,U) -> (B,V) induces a Quillen equivalence of the categories of homotopy functors into simplicial sets iff the induced map between the respective simplicial localisations is an r-equivalence. Motivating example: the Quillen equivalence between the simplicial categories sSet and Top is well studied, but what about the categories of homotopy functors from these model categories to simplicial sets? We will present a framework that will make sense of this question and will provide an affirmative answer.

Title: Relative Calabi-Yau structures for microlocal sheaves/Fukaya categories

Abstract: Consider an exact symplectic manifold with a codimension 2 exact symplectic submanifold at infinity. One can associate a partially wrapped Fukaya category to the pair of symplectic manifolds, and a wrapped Fukaya category to the symplectic submanifold. The result of Ganatra-Pardon-Shende shows that the Fukaya categories are equivalent to certain categories coming from microlocal sheaf theory. Assume that the ambient symplectic manifold is a cotangent bundle. We will study duality and exact sequences arising from the pair of categories. This is a non-commutative analogue of the orientation class which induces the Poincare-Lefschetz duality on manifolds with boundary. This is joint work in preparation with Chris Kuo. 

Title:  Morse Theory on a point and a stack of broken lines

Abstract:  After introducing Morse theory, I will talk about what it means to study Morse theory on a point. I will then try to explain a surprising result at the interface of geometry and algebra: The stack encoding Morse theory on a point exactly encodes associative algebras. This is joint work with Jacob Lurie.

Title:  A Morse theory for the mapping cone of distinguished differential forms.

Abstract: Differential forms are basic objects of manifolds and encode invariants. Morse theory can describe how topological invariants are also encoded in a generic function of a manifold. For manifolds such as special holonomy and symplectic manifolds that have a distinguished closed form, we will develop new notions of Morse theory for the distinguished form, using the concept of the mapping cone from homological algebra. This talk is based on joint work with Xiang Tang and Li-Sheng Tseng.

Title: How rare are simple Steklov eigenvalues

Abstract: Steklov eigenvalues are eigenvalues of the Dirichlet-to-Neumann operator which are introduced by Steklov in 1902 motivated by physics. And there is a deep connection between the extremal Steklov eigenvalue problems and the free boundary minimal surface theory in the unit Euclidean ball as revealed by Fraser and Schoen in 2016. In the talk, we will discuss the question of how rare simple Steklov eigenvalues are on manifolds and its applications in nodal sets and critical points of eigenfunctions.

Title: Anosov representations of cubulated hyperbolic groups

Abstract: An Anosov representation of a hyperbolic group Γ is a representation which quasi-isometrically embeds Γ into a semisimple Lie group, in a way which mimics and generalizes the behavior of a convex cocompact representation into a rank one Lie group. It is unknown whether every linear hyperbolic group admits an Anosov representation. In this talk, I will discuss joint work with Sami Douba, Balthazar Flechelles, and Feng Zhu, which shows that every hyperbolic group that acts geometrically on a CAT(0) cube complex admits a 1-Anosov representation into SL(d, R) for some d. Mainly, the proof exploits the relationship between the combinatioral (cube complex) structure of right-angled Coxeter groups, and the projective geometry of a convex domain in real projective space on which a Coxeter group acts by reflections.