UCSB Differential Geometry Seminar 2020-2021

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

Spring 2021

Friday, April 2 at 9 am, Shu Shen (Sorbonne University)
Title: Coherent sheaves, superconnection, and the Riemann-Roch-Grothendieck formula

Abstract : In this talk, I will explain a construction of Chern character for coherent sheaves on a closed complex manifold with values in Bott-Chern cohomology. I will also show a corresponding Riemann-Roch-Grothendieck formula, which holds for general holomorphic maps between closed non-Kahler manifolds. Our proof is based on two fundamental objects : the superconnection and the hypoelliptic deformations. This is a joint work with J.-M. Bismut and Z. Wei arXiv:2102.08129.

Friday, April 9, Demetre Kazaras (Stony Brook University)
Title: The spacetime Laplace equation on initial data sets for Einstein’s equations

This talk will consider a ‘spacetime Laplace operator’ on an initial data set for the Einstein equations. This operator reflects the geometry of the underlying manifold in a manner similar to the Dirac operator appearing in Witten’s proof of the Positive Mass Theorem. By analyzing linear-growth spacetime harmonic functions and their level sets, we obtain a rather approachable refinement of the Positive Mass Theorem for asymptotically flat 3-dimensional initial data sets. Applications to asymptotically hyperbolic initial data sets are also considered. The work I will discuss includes collaborations with Hugh Bray, Sven Hirsch, Marcus Khuri, Daniel Stern, and Yiyue Zhang.

Friday, April 14, Aleksander Doan (Columbia University and the University of Cambridge)
Title: The Gopakumar-Vafa finiteness conjecture

Abstract: The Gopakumar-Vafa conjecture concerns the Gromov-Witten invariants of symplectic manifolds of dimension six. The first part of the conjecture, the integrality conjecture, which was proved recently by Ionel and Parker, asserts that the Gromov-Witten invariants can be expressed in terms of simpler, integer invariants called the BPS numbers. The second part of the conjecture, the finiteness conjecture, predicts that only finitely many of the BPS numbers are nonzero for every homology class. In this talk, based on joint work with E. Ionel and T. Walpuski, I will discuss a proof of the second part of conjecture. The proof combines ideas from the theory of pseudo-holomorphic curves, including Ionel and Parker’s proof of the integrality conjecture, and methods of geometric measure theory, especially Allard’s regularity theorem for currents with bounded mean curvature.

Friday, April 23, Jikang Wang (Rutgers University, New Brunswick)
Title: Ricci limit spaces are semi-locally simply connected

Abstract: In this talk, we will discuss local topology of a Ricci limit space $(X,p)$, which is the pointed Gromov-Hausdorff limit of a sequence of complete $n$-manifolds with a uniform Ricci curvature lower bound. I will show that $(X,p)$ is semi-locally simply connected, that is, for any point $x \in X$, we can find a small ball $B_r(x)$ such that any loop in $B_r(x)$ is contractible in $X$. We will also discuss a slice theorem for pseudo-group actions on the Ricci limit space and how to use this slice theorem to construct a homotopy map on the limit space. Partial of this material is joint work with Jiayin Pan.

Friday, April 30, Ruobing Zhang (Princeton University)
Title: Collapsing geometry of Ricci-flat Kaehler manifolds in dimension four

Abstract: We will present recent studies on the Ricci-flat Kaehler 4-manifolds in the collapsed setting. We will particularly introduce, from metric geometric point of view, some structure theorems on their Gromov-Hausdorff limits and the possible rescaling limits.

Friday, May 7, Ragini Singhal (University of Waterloo)
Title: Deformation theory of nearly $G_2$ manifolds

Abstract: In this talk we will talk about the deformations of nearly $G_2$ structures. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly $G_2$ structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly $G_2$ structure on the Aloff–Wallach space are all obstructed to second order. We also completely describe the cohomology of nearly $G_2$ manifolds. This is a joint work with Shubham Dwivedi.

Friday, May 14, Jintian Zhu (Peking University)
Title: Positive mass theorem for ALF and ALG manifolds

Abstract: We will show that positive mass theorem holds for ALF and ALG manifolds with necessary incompressible condition on fundamental group. After that we will make some discussions on fill-in problems as an application. This talk is based on my recent joint work arXiv:2103.11289 with Liu and Shi.

Friday, May 21, Charles Ouyang (University of Massachusetts)
Title: Compactification of the SL(3,R)-Hitchin component

Abstract: Hitchin components are natural generalizations of the classical Teichmüller space. In the setting of SL(3,R), the Hitchin component parameterizes the holonomies of convex real projective structures. By studying Blaschke metrics, which are Riemannian metrics associated to such structures, along with their limits, we obtain a compactification of the SL(3,R)-Hitchin component. We show the boundary objects are hybrid structures, which are in part flat metric and in part laminar. These hybrid objects are natural generalizations of measured laminations, which are the boundary objects in Thurston’s compactification of Teichmüller space. (joint work with Andrea Tamburelli)

Friday, May 28, Steve Rayan (University of Saskatchewan)
Title: Topology of ordinary, twisted, and wild Hitchin systems

Abstract: In this talk, I will survey the primary algebro-geometric and differential-geometric techniques for understanding the topology of Hitchin systems. The former utilizes the C*-action on the moduli space of stable Higgs bundles while the latter takes stock of the critical points of a Morse-Bott function defined on the space of solutions to the 2D self-dual Yang-Mills equations, otherwise known as the Hitchin equations. I will discuss early accomplishments for ordinary Hitchin systems that were made possible by these tools, but will also frame the discussion in terms of quiver varieties. Doing so sets the stage for more recent results in low genus and in the so-called “twisted” and “wild” cases that I have obtained independently and also in collaboration with each of Laura Fredrickson; Sheldon Miller and Jenna Rajchgot; and Evan Sundbo.

Friday, June 4 at 9 am, Steven Gindi (Binghamton University)
Title: Long Time Limits of Generalized Ricci Flow

Abstract: We derive rigidity and classification results for blowdown limits of invariant generalized Ricci flow solutions on principal bundles. We focus on the nonabelian case, where Perelman-type functional methods cannot be used. Our arguments are based on new maximum principles and renormalized flows.

Winter 2021

Friday, January 15: Detang Zhou (UFF-Universidade Federal Fluminense, Brazil)
Title: Volume growth of complete submanifolds in gradient Ricci Solitons

Abstract: It is well-known that the volume of geodesic balls has polynomial growth and at least linear growth on complete noncompact Riemannian manifolds of nonnegative Ricci curvature. We study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if the ambient manifold is of bounded geometry, we prove that such a submanifold must have at least linear volume growth. In particular, we show that a properly immersed complete noncompact hypersurface in the Euclidean space with bounded Gaussian weighted mean curvature must have polynomial volume growth and at least linear volume growth. This is a joint work with Xu Cheng and Matheus Vieira.

Friday, January 22 at 9 AM: Anna Siffert (Max Planck Institute for Mathematics)
Title: Construction of explicit $p$-harmonic functions

Abstract: The study of $p$-harmonic functions on Riemannian manifolds has invoked the interest of mathematicians and physicists for nearly two centuries. Applications within physics can for example be found in continuum mechanics, elasticity theory, as well as two-dimensional hydrodynamics problems involving Stokes flows of incompressible Newtonian fluids.
In my talk I will focus on the construction of explicit p-harmonic functions on rank- one Lie groups of Iwasawa type. This joint wok with Sigmundur Gudmundsson and Marko Sobak.

Friday, January 29: Laura Fredrickson (University of Oregon)
Title: ALG Gravitational Instantons and Hitchin Moduli Spaces

Abstract: Four-dimensional complete hyperkaehler manifolds can be classified into ALE, ALF, ALG, ALG*, ALH, ALH* families. It has been conjectured that every ALG or ALG* hyperkaehler metric can be realized as a 4d Hitchin moduli space. I will describe ongoing work with Rafe Mazzeo, Jan Swoboda, and Hartmut Weiss to prove a special case of the conjecture, and some consequences. The hyperkaehler metrics on Hitchin moduli spaces are of independent interest, as the physicists Gaiotto–Moore–Neitzke give an intricate conjectural description of their asymptotic geometry.

Friday, February 5: Darong Chen (University of Waterloo)
Title: Existence of constant mean curvature 2-spheres in Riemannian 3-spheres

Abstract: In this talk I’ll describe recent joint work with Xin Zhou, where we make progress on the question of finding closed constant mean curvature surfaces with controlled topology in 3-manifolds. We show that in a 3-sphere equipped with an arbitrary Riemannian metric, there exists a branched immersed 2-sphere with constant mean curvature H for almost every H. Moreover, the existence extends to all H when the target metric is positively curved. This latter result confirms, for the branched immersed case, a conjecture of Harold Rosenberg and Graham Smith.

Friday, February 12,: Paula Burkhardt-Guim (UC Berkeley)
Title: Pointwise lower scalar curvature bounds for $C^0$ metrics via regularizing Ricci flow

Abstract: We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from $C^0$ initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from $C^0$ initial data.

Friday, February 26, Matthias Wink (UCLA)
Title: Vanishing and estimation results for Betti numbers

Abstract: We prove that manifolds with $\lceil n/2 \rceil$-positive curvature operators are rational homology spheres. This is a consequence of a general vanishing and estimation theorem for the $p$-th Betti number for manifolds with a lower bound on the average of the lowest $(n-p)$ eigenvalues of the curvature operator. Our main tool is the Bochner Technique. We will also discuss similar results for the Hodge numbers of Kaehler manifolds. This talk is based on joint work with Peter Petersen.

Friday, March 5, Otis Chodosh (Stanford)
Title: Soap bubbles and topology of manifolds with positive scalar curvature

Abstract: I will describe recent work with Chao Li concerning new topological obstructions to positive scalar curvature.

Friday, March 12, Eric Chen (UCSB)
Title: The Yamabe flow on asymptotically flat manifolds

Abstract: Although the long-time existence and convergence of the Yamabe flow on compact manifolds is by now essentially settled, the situation on noncompact manifolds is less clear. In the setting of asymptotically flat manifolds, I will describe why long-time existence also holds in general, and why the flow converges if and only if the initial metric is Yamabe positive. This is joint work with Yi Wang.

Friday, March 19, Rima Chatterjee (Louisiana State University)
Title: Knots and links in overtwisted contact manifolds

Abstract: Knot theory associated to overtwisted manifolds is less explored. There are two types of knots/links in an overtwisted manifold namely loose and non-loose. These knots are different than the knots in tight manifolds in many ways. In this talk, I’ll start with an overview of these knots/links and discuss my recent results on classifying loose null-homologous links. Next, I’ll talk about an invariant named support genus of knots and links and show that this invariant vanishes for loose links. I’ll end with some interesting open questions and future work directions.

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.