# UCSB Differential Geometry Seminar 2020-2021

The talks are held on every Friday 5-6 PM (Pacific Time) on Zoom.

## 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
Title: Witten Deformation on Non-compact Manifolds: Heat Kernel Expansion and Local Index Theorem

Abstract: Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on noncompact spaces are rarely explored, and even for cases as simple as $\mathbb{C}^n$ with (quasi-homogeneous) polynomials potentials, it’s already very complicated. Motivated by path integral formulation of the heat kernel, we introduced a parabolic distance, which also appeared in Li-Yau’s famous work on parabolic Harnack estimate. With the help of the parabolic distance, we derive a pointwise asymptotic expansion of the heat kernel with a strong remainder estimate. In particular, we obtain an asymptotic expansion of the heat kernel of the Witten Laplacian for the Witten deformation. When the deformation parameter of Witten deformation and time parameter are coupled, we derive a small time asymptotic expansion for the trace of the heat kernel of the Witten Laplacian. We introduce a novel rescaling technique to compute the local index density explicitly. If time permits, I will also briefly explain how to define analytic torsion for Witten Laplacian on noncompact spaces. This is joint work with Xianzhe Dai.

Friday, November 20: Qin Deng
Title: Hölder continuity of tangent cones in $RCD(K,N)$ spaces and applications to non-branching

Abstract: It is known by a result of Colding-Naber that for any two points in a Ricci limit space, there exists a minimizing geodesic where the geometry of small balls centred along the interior of the geodesic change in at most a Hölder continuous manner. This was shown using an extrinsic argument and had several key applications for the structure theory of Ricci limits. In this talk, I will discuss how to generalize this result to the setting of metric measure spaces satisfying the synthetic lower Ricci curvature bound condition $RCD(K,N)$. As an application, I will show that all $RCD(K,N)$ spaces are non-branching, a result which was previously unknown for Ricci limit spaces.

Friday, December 4 at 9 AM (Pacific Time): Szilárd Szabó
Title: Asymptotic Hodge theory in the Painlevé cases

Abstract: We will state two conjectures for moduli spaces of flat (irregular singular) connections and Higgs bundles on curves, namely the P=W conjecture (due to de Cataldo, Hausel and Migliorini) and its geometric counterpart (due to Simpson et al). We will explain our proof of these conjectures in some classical 4-dimensional cases called Painlevé cases.