UCSB Differential Geometry Seminar 2020-2021

The talks are held on every Friday 5-6 PM on Zoom. Meeting ID TBA.

Fall 2020

Friday, October 9: Ákos Nagy
Title: The asymptotic geometry of G2G_2-monopoles

Abstract: G2G_2-monopoles are special Yang-Mills-Higgs configurations on G2G_2-manifolds. Donaldson and Segal conjecture that one can construct invariants of noncompact G2G_2-manifolds by counting G2G_2-monopoles. The first steps in achieving this goal is understanding the analytic properties of these monopoles, such as asymptotic and decay properties. In this talk, I introduce the proper analytic setup for the problem, and present our results about the asymptotic forms of G2G_2-monopoles, with structure group being SU(2)SU(2), on Asymptotically Conical G2G_2-manifolds. This is a joint project with Daniel Fadel and Gonçalo Oliveira.

Friday, October 16: Jiayin Pan
Title: Nonnegative Ricci curvature and escape rate gap

Abstract: Let MM be an open nn-manifold of nonnegative Ricci curvature and let pMp\in M. We show that if (M,p)(M,p) has escape rate less than some positive constant ϵ(n)\epsilon(n), that is, minimal representing geodesic loops of π1(M,p)\pi_1(M,p) escape from any bounded balls at a small linear rate with respect to their lengths, then π1(M,p)\pi_1(M,p) is virtually abelian.

Friday, October 23: Fedya Manin
Title: Filling random cycles

Abstract: I will explain an average-case isoperimetric inequality for certain combinatorial models of random cycles in cubes and spheres. For example, take a knot built by connecting a sequence of NN uniformly random points in the unit cube (a model introduced by Ken Millett). The minimal area of a Seifert surface for this knot is on the order of NlogN\sqrt{N \log N} with high probability. For all the models analyzed, this growth rate is the same, depending only on the codimension of the cycles; I suspect that Fourier analysis can be used to explain this apparent coincidence.

Friday, October 30: Yeping Zhang
Title: Quillen metric, BCOV invariant and motivic integration

Abstract: Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called the BCOV invariant. The BCOV invariant is conjecturally related to the Gromov-Witten theory via mirror symmetry. In this talk, we prove the conjecture that birational Calabi-Yau manifolds have the same BCOV invariant. We also build an analogue between the BCOV invariant and the motivic integration. The result presented in this talk is a joint work with Lie Fu.

Friday, November 6: Siqi He
Title: The Behavior of Solutions to the Hitchin-Simpson Equations

Abstract: The Hitchin-Simpson equations defined over a Kähler manifold are first order, non-linear equations for a pair of connection on a Hermitian vector bundle and a 1-form with values in the endomorphism bundle. We will describe the behavior of solutions to the Hitchin-Simpson equations with norms of these 1-forms unbounded. In addition, we will discuss the relationship between this behavior with the Taubes’ Z2 harmonic spinor and Hitchin’s WKB problem.

Friday, November 13: Junrong Yan
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Friday, November 20: Qin Deng
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