# 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 $G_2$-monopoles

Abstract: $G_2$-monopoles are special Yang-Mills-Higgs configurations on $G_2$-manifolds. Donaldson and Segal conjecture that one can construct invariants of noncompact $G_2$-manifolds by counting $G_2$-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 $G_2$-monopoles, with structure group being $SU(2)$, on Asymptotically Conical $G_2$-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 $M$ be an open $n$-manifold of nonnegative Ricci curvature and let $p\in M$. We show that if $(M,p)$ has escape rate less than some positive constant $\epsilon(n)$, that is, minimal representing geodesic loops of $\pi_1(M,p)$ escape from any bounded balls at a small linear rate with respect to their lengths, then $\pi_1(M,p)$ 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 $N$ 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 $\sqrt{N \log N}$ 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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