The geometric analysis seminar is held on Tuesdays at 11am in the Zoom Meeting with Meeting ID 933 0554 5643 and Passcode 324863.

Spring Quarter, 2022

• March 29, Xiangwen Zhang (UC Irvine)
  A geometric flow for Type IIA superstrings
  Abstract: The equations of flux compactifications of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow on symplectic Calabi-Yau 6-manifolds. In this talk, we will discuss the recent progress on the study of this Type IIA flow. This is based on joint work with Fei, Phong and Picard.
• April 5, Haotian Wu (The University of Sydney)
  Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up
  Abstract: The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. In this talk, we present some results, jointly with Isenberg (Oregon) and Zhang (Sydney), concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. Time permitting, we also present a numerical analysis, in collaboration with Garfinkle (Oakland University), Isenberg (Oregon) and Knopf (UT Austin), on the stability of Type-IIa singularity in MCF of non-compact hypersurfaces.
• April 12, Emily Windes (U Rochester)
  Special Lagrangians from the perspective of Morse theory
  Abstract: In this talk, we consider a Lagrange multipliers problem where the constraint is a section of a bundle E->M. We relate the Morse homology of a function restricted to the zero set of the section to the Morse homology of the associated Lagrange function on the total space E^*. Then we discuss a similar, infinite-dimensional Lagrange multipliers problem appearing in Donaldson and Segal’s paper “Gauge Theory in Higher Dimensions II”. The long term goal is to apply Floer theory to a functional whose critical points are a generalization of three-dimensional, special Lagrangian submanifolds.
• April 19, Ákos Nagy (UCSB)
  The asymptotic geometry of G_2-monopoles
  Abstract: G_2-monopoles are special solutions to the Yang-Mills-Higgs equation on G_2-manifolds, similar to 3-dimensional BPS monopoles.

  Donaldson and Segal conjectured that these gauge theoretic objects have a close relationship to the geometry of the underlying G_2-structure. Intuitively, G_2-monopoles with “large mass” are predicted to detect coassociative submanifolds.

  One of the first steps in proving this claim is understanding the analytic behavior of G_2-monopoles. In this talk, I will first introduce the proper analytic setup for the problem. Then I present results about the asymptotic form of G_2-monopoles with structure group \mathrm{SU} (2) on Asymptotically Conical manifolds. These are joint results with Gonçalo Oliveira and Daniel Fadel.

  Finally, I will also talk about further plans in this project, in particular:

  1. Generalizations of these results to manifolds with fibered end and higher rank gauge groups.
  2. Glue-in construction of monopoles.
  3. A concentration result for G_2-monopoles with large mass.

  These are ongoing research directions with Gonçalo Oliveira, Daniel Fadel, and Saman Habibi Esfahani.

• April 26, Boris Botvinnik (UO)
  Spin^c manifolds, positive scalar curvature and manifolds with fibered singularities
  Abstract: I will discuss a problem of existence of positive scalar curvature on manifolds with fibered singularities. It turns out there are necessary and sufficient conditions for a psc-metric to exist on such objects. There is a particular case of manifolds with fibered singularities when the fiber is a circle. This case leads to psc-metrics spin^c manifolds with special conditions near the singular locus. In particular, I describe some results concerning metrics on spin^c manifolds with positive “twisted scalar curvature,” where the twisting comes from the curvature of the spin^c line bundle. This work is joint with Jonathan Rosenberg.
• May 3, Jackson Goodman (UC Berkeley)
  Area extremality and Nonnegative Curvature in Dimension 4
  Abstract: Following Gromov, a Riemannian manifold is called “area extremal” if any modification of which increases scalar curvature must decrease the area of a 2-plane. Previous work of Llarull and Goette-Semmelmann has established area extremality for certain metrics with nonnegative curvature operator, and Kahler metrics with positive Ricci curvature. We show that in dimension 4 a larger class of nonnegatively curved metrics are area extremal, including on manifolds which do not admit metrics with nonnegative curvature operator or Kahler metrics. Following Lott, we examine area extremality on 4-manifolds with boundary, proving that all positively curved metrics are “locally” area extremal. This is joint work with Renato Bettiol.
• May 10, Maxwell Stolarski (ASU)
  Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers
  Abstract: Shrinking Ricci solitons are Ricci flow solutions that self-similarly shrink under the flow. Their significance comes from the fact that finite-time Ricci flow singularities are typically modeled on gradient shrinking Ricci solitons. Here, we shall address a certain converse question, namely, “Given a complete, noncompact gradient shrinking Ricci soliton, does there exist a Ricci flow on a closed manifold that forms a finite-time singularity modeled on the given soliton?” We’ll discuss recent work that shows the answer is yes when the soliton is asymptotically conical. No symmetry or Kahler assumption is required, and so the proof involves an analysis of the Ricci flow as a nonlinear degenerate parabolic PDE system in its full complexity. We’ll also discuss applications to the (non-)uniqueness of weak Ricci flows through singularities.
• May 31, Yu Yuan (UW)
  Rigidity for general semiconvex entire solutions to the sigma-2 equation
  Abstract: We present a rigidity result for general semiconvex entire solutions to the sigma-2 equation. Two decades ago, this result was obtained in three dimensions, as a byproduct of the work on special Lagrangian equations. A decade ago in a joint work with Chang, this result was shown for almost convex entire solutions. Warren’s rare saddle entire solution confirms the necessity of the semiconvexity assumption. Recall the classical Liouville rigidity for the Laplace equation or the sigma-1 equation and Jörgens-Calabi-Pogorelov rigidity for the Monge-Ampère equation or the sigma-n equation: all convex entire solutions to those equations in general n dimensions must be quadratic. This is joint work with Ravi Shankar.

Winter Quarter, 2022

• January 11, Weiyong He (UO)
  Existence of Polyharmonic maps in the critical dimensions
  Abstract: We discuss the existence of biharmonic maps and Polyharmonic maps in the critical dimensions.

  The regularity of Polyharmonic maps was completely solved in 2009. Recently we made a step further to prove that given a null homotopy condition, there is always a Polyharmonic map in the critical dimensions for each free homotopy class [M, N].

  We will also discuss some possible progress without a null homotopy condition.

• January 20, Harish Seshardi (Indian Institute of Science, Bangalore)
  Special Time: 8:00am
  Limits of Positive Einstein metrics on the 4-sphere
  Abstract: I will talk about a compactness theorem for positive Einstein metrics on S4. For general 4-manifolds, it is known that a limit of such metrics is an orbifold Einstein metric. In the case of S4, it turns out that the limit is always a smooth Einstein metric.

  This is joint work with Kazuo Akutagawa and Hisaaki Endo.

• January 25, Rafe Mazzeo (Stanford University)
  The flow of networks by curvature
  Abstract: This work constitutes a very preliminary step in the study of geometric heat flows on stratified spaces. It concerns the one-dimensional case of singular networks. Work on this goes back several decades (motivated by metallurgy!), and I will review some of the main points of progress, and then discuss a new formulation of the problem which clarifies short-time existence around “irregular vertices”, and is expected to be particularly useful for the higher dimensional version. This is joint work with Jorge Lira, Alessandra Pluda and Mariel Saez.
• February 3, Harish Seshardi (Indian Institute of Science, Bangalore)
  Special Time: 8:00am
  Limits of Positive Einstein metrics on the 4-sphere
  Abstract: I will talk about a compactness theorem for positive Einstein metrics on S4. For general 4-manifolds, it is known that a limit of such metrics is an orbifold Einstein metric. In the case of S4, it turns out that the limit is always a smooth Einstein metric.

  This is joint work with Kazuo Akutagawa and Hisaaki Endo.

• February 8, Dan Knopf (University of Texas at Austin)
  Ricci solitons, conical singularities, and nonuniqueness
  Abstract: In dimension n=3, there is a complete theory of weak solutions of Ricci flow — the singular Ricci flows introduced by Kleiner and Lott. These are unique across singularities, as was proved by Bamler and Kleiner. In joint work with Angenent, we show that uniqueness should not be expected to hold for Ricci flow weak solutions in dimensions n>4, not even topologically. Specifically, we exhibit a discrete family of asymptotically conical gradient shrinking soliton singularity models, each of which admits non-unique forward continuations by gradient expanding solitons.
• February 24, Vamsi Pingali (Indian Institute of Science, Bengaluru)
  Special Time: 8:00am
  The differential geometry of Hartshorne-ample bundles
  Abstract: I shall discuss a conjecture of Griffiths and give a survey of results related to it. I shall also state a formulation of a parabolic version. Some of the results are part of a joint work with Indranil Biswas.
• March 8, Paul Allen (Lewis & Clark)
  Yamabe problem for asymptotically hyperbolic geometries
  Abstract: Given a Riemannian manifold, the Yamabe problem seeks to find a metric of constant scalar curvature conformally equivalent to the original metric. For compact manifolds, the problem presents an interesting combination of geometric and analysis challenges. In the asymptotically hyperbolic setting, there are additional challenges associated to the regularity of the conformal boundary. In this talk I explain the nature of these challenges, review existing results, and discuss work that is presently underway.

Fall Quarter, 2021

• September 28, Laura Fredrickson (UO)
  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.
• October 5, Peng Lu (UO)
  Conformal Bach flow
  Abstract: We introduce conformal Bach flow and establish its well-posedness on closed manifolds. We also obtain its backward uniqueness. To give an attempt to study the long-time behavior of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi’s type -estimate of derivatives of curvatures are derived.

  This is a joint work with Jiaqi Chen and Jie Qing.

• October 12, Otis Chodosh (Stanford University)
  Special Time : 10am
  Stable minimal hypersurfaces in R^4
  Abstract: I will explain why a stable minimal hypersurface in R^4 is flat. This is joint work with Chao Li.
• October 19, Demetre Kazaras (Duke)
  If Ricci is bounded below, then mass is in control!
  
  Abstract: The ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields. In a previous work, we showed how to compute this invariant (in 3 spatial dimensions) by studying harmonic functions. Now I’ll use this formula to consider the following question: How flat is an asymptotically flat manifold with very little total mass? We make progress on this problem and confirm special cases of conjectures made by Huisken-Ilmanen and Sormani. The main results asserts that in the class of reasonably-behaved asymptotically flat manifolds with non-negative scalar curvature satisfying a uniform lower bound on Ricci curvature, small mass implies Gromov-Hausdorff closeness to flat space.
• October 26, Aleksander Doan (Columbia University)
  The Gopakumar-Vafa finiteness conjecture
  
  Abstract: Gromov-Witten invariants, which encode information about pseudo-holomorphic curves contained in a given symplectic manifold, provide powerful tools for solving problems in symplectic geometry. However, their geometric meaning is not always clear. These invariants are rational numbers and therefore cannot be simply interpreted as a count of pseudo-holomorphic curves. Moreover, a single curve can contribute to infinitely many Gromov-Witten invariants. The Gopakumar-Vafa conjecture predicts that for symplectic manifolds of dimension six, the information encoded in the Gromov-Witten invariants can be repackaged into a collection of numbers, called the BPS invariants, which don’t suffer from these drawbacks: they are integers and only finitely many of them are nonzero in every homology class. The first part of the conjecture was proved in 2018 by Ionel and Parker. I will discuss a proof of the second part, which relies on combining Ionel and Parker’s cluster formalism with results from geometric measure theory. The talk is based on joint work with Eleny Ionel and Thomas Walpuski.
• November 2, Mark Walsh (National University of Ireland Maynooth)
  Homotopy Invariance of the Space of PSC-metrics on Manifolds with Singularities
  Abstract: We consider the space of metrics of positive scalar curvature (PSC-metrics) on certain manifolds with singularities. The main result we will discuss shows that the homotopy type of this space is an invariant of particular surgeries on both the non-singular and singular parts of the manifold.

  This is joint work with Boris Botvinnik.

• November 9, Zilu Ma (UC San Diego)
  Geometry at Infinity of Singularity Models of Ricci Flow
  Abstract: Based on Hamilton’s program, Perelman solved the Poincaré conjecture and Thurston’s geometrization conjecture using Ricci flow with surgeries. It is crucial to understand the singularity formation in order to perform surgeries. To capture the geometry in the large of the singularity models, Perelman developed a space-time comparison geometry, the L-geometry, to show the existence of the asymptotic shrinkers, which is a smooth blow-down limit as time goes to ∞. Despite the miraculous success in dimension 3, Perelman’s machinery is not suitable for higher dimensions partly due to the complexity of curvatures. Recently, Richard Bamler developed a compactness and partial regularity regularity theory for noncollapsed Ricci flows, which greatly advanced the study of Ricci flows in dimension greater or equal to 4. Among other important results, Bamler introduced a notion of tangent flow at infinity, which is a blow-down limit with respect to Bamler’s new F-distance, and its existence is canonical thanks to Bamler’s compactness theory. In a recent work with Chan and Zhang, roughly speaking, we proved that the two notions of blow-downs coincide. If time permits, we also mention the study of tangent flows at infinity of 4-dimensional steady Ricci solitons and how the geometry at infinity determines the geometry in the large, which is a recent work joint with Bamler, Chow, Deng and Zhang.
• November 16, Qi S. Zhang (UC Riverside)
  Log gradient estimates of the heat equation on manifolds
  Abstract: First, we will review Li-Yau and Hamilton’s log gradient estimates for positive solutions of the heat equation on manifolds and some applications. Second, we will report new results on a sharp Li-Yau gradient estimate on compact manifolds.
• November 30, Mark Walsh (National University of Ireland Maynooth)
  Homotopy Invariance of the Space of PSC-metrics on Manifolds with Singularities
  Abstract: We consider the space of metrics of positive scalar curvature (PSC-metrics) on certain manifolds with singularities. The main result we will discuss shows that the homotopy type of this space is an invariant of particular surgeries on both the non-singular and singular parts of the manifold.

  This is joint work with Boris Botvinnik.

• December 7, Saman Habini Esfahani (Stony Brook University)
  Gauge theory, Floer homology and manifolds with special holonomy
  Abstract: Gauge theory and Floer homology have proven to be useful in the study of low-dimensional manifolds. Donaldson and Thomas put forward the idea of extending these techniques to higher dimensions, especially to study manifolds with special holonomy groups. In this talk, after a brief introduction to the Donaldson-Thomas program — and Doan-Walpuski’s approach — we focus on the differential-geometric aspects and study some of the fundamental questions in this direction, including the singularities, compactness and gluing problems, and give solutions to some of them. Parts of this work is based on an on-going project, joint with Daniel Fadel, Àkos Nagy and Gonçalo Oliveira.

Previous Schedule: 2020, 2019, 2018, 2017, 2016, 2015 Spring, 2015 Winter, 2014 Fall, 2014 Spring, 2014 Winter, 2012 Winter, 2011 Fall, 2010 Spring, 2010 Winter