Differential Geometry Seminar

Differential Geometry Seminar

Academic Year 2019-2020

Organizers: S. Picard, S.-T. Yau

The seminar meets on Tuesdays from 3pm to 4pm on Zoom. Everyone is welcome. If you would like to attend the seminar, email spicard@math.harvard.edu and we will send you the zoom link.

Spring Schedule

February 11
Yoosik Kim (Brandeis) - T-equivariant disc potentials of toric manifolds
Abstract

In this talk, we discuss how to derive the equivariant SYZ mirror of toric manifolds by counting holomorphic discs. In the case of (semi-)Fano toric manifolds, those mirrors recover Givental's equivariant mirrors, which compute the equivariant quantum cohomology. Also, we formulate and compute open Gromov-Witten invariants of singular SYZ fiber, which are closely related to the open Gromov-Witten invariants of Aganagic-Vafa branes. This talk is based on joint work with Hansol Hong, Siu-Cheong Lau, and Xiao Zheng.

February 18
Martin Lesourd (CMSA) - Dynamical Black Hole Formation
Abstract

In 2008, Christodoulou achieved a major breakthrough in the context of mathematical general relativity in being able to form trapped surfaces dynamically from initial data for the Einstein vacuum system. The results and methods which he lays out in his 600+ page manuscript has led to a flurry of activity in the last decade. I will give a rough overview of the basic ideas, describe how far theorems have come, and describe some recent progress - joint with Nikos Athanasiou - in this direction.

February 25
Jiewon Park (MIT) - Canonical identification between scales on Ricci-flat manifolds
Abstract

Let $M$ be a complete Ricci-flat manifold with Euclidean volume growth. A theorem of Colding-Minicozzi states that if a tangent cone at infinity of $M$ is smooth, then it is the unique tangent cone. The key component in their proof is an infinite dimensional Łojasiewicz-Simon inequality, which implies rapid decay of the $L^2$-norm of the trace-free Hessian of the Green function. In this talk we discuss how this inequality can be exploited to identify two arbitrarily far apart scales in $M$ in a natural manner through a diffeomorphism. We also prove a pointwise Hessian estimate for the Green function when there is an additional condition on sectional curvature, which is an analogue of various matrix Harnack inequalities obtained by Hamilton and Li-Cao in different time-dependent settings.

March 3
Man-Chun Lee (Northwestern) - Complete Kahler Ricci flow with unbounded curvature and applications
Abstract

In this talk, we will discuss the construction of Kahler Ricci flow on complete Kahler manifolds with unbounded curvature. As an corollary, we will discuss the application related to Yau's unformization problem and the regulartity of Gromov-Hausdorff's limit. This is joint work with L.F. Tam.

March 10
Christos Mantoulidis (MIT) - Ancient gradient flows of elliptic functionals and Morse index
Abstract

(Joint with Kyeongsu Choi.) We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, focusing on mean curvature flow for the talk. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or canonically shrinking equators. In the mean curvature flow case in S^3, we classify ancient flows with more relaxed area bounds: they are steady or canonically shrinking equators or Clifford tori. In the embedded curve shortening case in S^2, we completely classify ancient flows of bounded length: they are steady or canonically shrinking equators.

March 17
No seminar - Spring Break

March 24
Xiangwen Zhang (UC Irvine) - Canceled
Abstract

March 31
Chris Gerig (Harvard) - Canceled
Abstract

Most 4-manifolds do not admit symplectic forms, but most admit 2-forms that are "nearly" symplectic. Just like the Seiberg-Witten (SW) invariants, there are Gromov invariants that are compatible with the near-symplectic form. Although (potentially exotic) 4-spheres don't admit them, there is still a way to bring in near-symplectic techniques and I will describe my ongoing pseudo-holomorphic attempt(s) at analyzing them.

April 7
Valentino Tosatti (Northwestern) - Collapsing Calabi-Yau Manifolds
Abstract

I will report on some recent progress on the problem of understanding the collapsing behavior of Ricci-flat Kahler metrics on Calabi-Yau manifolds that admit a fibration structure, when the volume of the fibers shrinks to zero. Based on joint works with Gross-Zhang and with Hein.

April 14
Niky Kamran (McGill) - Non-uniqueness results for the Calderon inverse problem with local or disjoint data
Abstract

The anisotropic Calderon inverse problem consists in recovering the metric of a compact connected Riemannian manifold with boundary from the knowledge of the Dirichlet-to-Neumann map at fixed energy. A fundamental result due to Lee and Uhlmann states that there is uniqueness in the analytic case. We shall present counterexamples to uniqueness in cases when: 1) The metric smooth in the interior of the manifold, but only Holder continuous on one connected component of the boundary, with the Dirichlet and Neumann data being measured on the same proper subset of the boundary. 2) The metric is smooth everywhere and Dirichlet and Neumann data are measured on disjoint subsets of the boundary. This is joint work with Thierry Daude (Cergy-Pontoise) and Francois Nicoleau (Nantes).

April 21
Alexander Mramor (Johns Hopkins) - Unknottedness of noncompact self shrinkers (3-4pm)
Abstract

In this talk I’ll discuss work in preparation on the unknottedness of asymptotically conical self shrinkers in R^3.

April 28
Yang Li (IAS) - Weak SYZ conjecture for hypersurfaces in the Fermat family (3-4pm)
Abstract

The SYZ conjecture predicts that for polarised Calabi-Yau manifolds undergoing the large complex structure limit, there should be a special Lagrangian torus fibration. A weak version asks if this fibration can be found in the generic region. I will discuss my recent work proving this weak SYZ conjecture for the degenerating hypersurfaces in the Fermat family. Although these examples are quite special, this is the first construction of generic SYZ fibrations that works uniformly in all complex dimensions.

May 7
Mario Garcia-Fernandez (ICMAT) - Gauge theory for string algebroids (special time and date: 4-5pm on Thursday May 7)
Abstract

We introduce a moment map picture for holomorphic string algebroids, a special class of holomorphic Courant algebroids introduced in arXiv:1807.10329. An interesting feature of our construction is that the Hamiltonian gauge action is described by means of Morita equivalences, as suggested by higher gauge theory. The zero locus of the moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical Calabi problem and the Hull-Strominger system. Our main results are concerned with the geometry of the moduli space of solutions, and assume a technical condition which is fulfilled in examples. We prove that the moduli space carries a pseudo-Kähler metric with Kähler potential given by the dilaton functional, a topological formula for the metric, and an infinitesimal Donaldson-Uhlenbeck-Yau type theorem. Finally, we relate our topological formula to a physical prediction for the gravitino mass in order to obtain a new conjectural obstruction for the Hull-Strominger system. This is joint work with Roberto Rubio and Carl Tipler.

May 12
Mu-Tao Wang (Columbia) - Angular momentum in general relativity (3-4pm)
Abstract

In the theory of general relativity, defining a valid notion of angular momentum is proven to be an even more challenging task than the definition of energy/mass. In this talk, I shall discuss this fundamental notion from the quasilocal level to null infinity.

Fall Schedule

September 10
Ahil Alaee (CMSA) - Geometric inequalities for quasi-local masses
Abstract

In this talk, we establish lower bounds for quasi-local masses in terms of charge, angular momentum, and horizon area. In particular, we treat three quasi-local masses based on a Hamiltonian approach, namely the Brown-York, Liu-Yau, and Wang-Yau masses. The geometric inequalities are motivated by analogous results for the ADM mass. They may be interpreted as localized versions of these inequalities, and are also closely tied to the conjectured Bekenstein bounds for entropy of macroscopic bodies. In addition, we give a new proof of the positivity property for the Wang-Yau mass which can be used to remove the spin condition in higher dimensions. Furthermore, we generalize a recent result of Lu and Miao to obtain a localized version of the Penrose inequality for the static Liu-Yau mass. This is joint work with Marcus Khuri and Shing-Tung Yau.

September 17
Xin Zhou (UCSB) - Existence of hypersurfaces with prescribed mean curvature
Abstract

I will present some results on the construction of closed hypersurfaces with prescribed mean curvature (PMC) in closed manifolds. In particular, I will start by reporting a joint work with Jonathan Zhu, in which we proved the existence of PMC hypersurfaces for a large class of prescribing functions by min-max method. I will also talk about Morse index bounds for those min-max PMC hypersurfaces. This topic is related to Problem 59 in Professor Yau's famous 1982 problem section.

September 24
Carl Lian (Columbia) - Enumerating pencils with moving ramification on curves
Abstract

We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E->P^1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.

October 1
Julian Scheuer (Freiburg) - The Minkowski inequality in de Sitter space
Abstract

The classical Minkowski inequality in the Euclidean space provides a lower bound on the total mean curvature of a hypersurface in terms of the surface area, which is optimal on round spheres. In this talk we present a curvature flow approach to prove a properly defined analogue in the Lorentzian de Sitter space.

October 8
Daren Cheng (Chicago) - Bubble tree convergence of conformally cross product preserving maps
Abstract

We study a class of weakly conformal maps, called Smith maps, which parametrize associative 3-folds in 7-manifolds equipped with G2-structures. These maps satisfy a system of first-order PDEs generalizing the Cauchy-Riemann equation for J-holomorphic curves, and we are interested in their bubbling phenomena. Specifically, we first prove an epsilon-regularity theorem for Smith maps in W^{1, 3}, and then explain how that combines with conformal invariance to yield bubble trees of Smith maps from sequences of such maps with uniformly bounded 3-energy. When the G2-structure is closed, we show that both the 3-energy and the homotopy are preserved in the bubble tree limit. The result can be viewed as an associative analogue of the bubble tree convergence theorem for J-holomorphic curves. This is joint work with Spiro Karigiannis and Jesse Madnick.

October 15
Shubham Dwivedi (Waterloo) - A gradient flow of isometric $G_2$ structures
Abstract

We will talk about a flow of isometric $G_2$ structures. We consider the negative gradient flow of the energy functional restricted to the class of $G_2$ structures inducing a given Riemannian metric. We will discuss various analytic aspects of the flow including global and local derivative estimates, a compactness theorem and a local monotonicity formula for the solutions. We also study the evolution equation of the torsion and show that under a modification of the gauge and of the relevant connection, it satisfies a nice reaction-diffusion equation. After defining an entropy functional we will prove that low entropy initial data lead to solutions that exist for all time and converge smoothly to a $G_2$ structure with divergence free torsion. We will also discuss finite time singularities and show that at the singular time the flow converges to a smooth $G_2$ structure outside a closed set of finite 5- dimensional Hausdorff measure. Finally, we will prove that if the singularity is Type-I then a sequence of blow-ups of a solution has a subsequence which converges to a shrinking soliton of the flow. This is a joint work with Panagiotis Gianniotis (University of Athens) and Spiro Karigiannis (University of Waterloo).

October 22
No seminar - Ding Lecture

October 29
No seminar

November 5
Kyeongsu Choi (MIT) - The Gauss curvature flow - the shape of worn stones
Abstract

The Gauss curvature flow was introduced to model the shape of worn stones. It is a parabolic Monge-Ampere type flow for convex hypersurfaces, involving regularity issues for fully nonlinear equations and free boundary problems, and involving several entropy. In this talk, we will discuss the free boundary problem concerning worn stone with flat sides, and the optimal regularity of non-concave fully nonlinear equations. When time permits, we also talk about the singularity analysis, especially about the relations to the affine geometry at the critical case, to fully nonlinear equations at the super-critical case, and to convex geometry and spectral analysis at the sub-critical case.

November 12
Yu-Shen Lin (BU) - On the SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
Abstract

The Strominger-Yau-Zaslow conjecture predicts the existence of special Lagrangians fibration in Calabi-Yau manifolds near large complex structure limit. The SYZ conjecture has been an important guiding principle for mirror symmetry and many of the implications are verified. In this talk, I will report on the recent progress on the SYZ fibration on certain log Calabi-Yau surfaces using the Lagrangian mean curvature flow and the theory of J-holomorphic curves. As a bi-product, we produce many new special Lagrangian submanifolds. I will also explain the its applications in mirror symmetry, including the tropical/holomorphic correspondence for log Calabi-Yau surfaces and a mathematical realization of renormalization process of Hori-Vafa. Part of the talk is based on the joint work with T. Collins, A. Jacob and S-C. Lau, T-J. Lee.

November 19
John Loftin (Rutgers) - Limits of Hitchin Representations for Rank-2 Lie Groups
Abstract

It is well known that on a closed oriented surface S of genus at least 2, Teichmuller space can be viewed a component of the representation variety of the fundamental group of S into PSL(2,R). Hitchin used Higgs bundles to formulate a generalization of this component for higher rank split real Lie groups such as PSL(n,R). In the rank 2 cases PSL(3,R), PSp(4,R), and G_2, there is additional interesting structure: 1) Labourie showed that one may reformulate Hitchin's construction, which involves a fixed background conformal structure on S, invariantly under the action of the mapping class group, and 2) Baraglia showed that Hitchin's equations, which in general produce a metric on a vector bundle, reduce in this case to coupled equations which are amenable to study by the maximum principle. We present this theory in terms of limits of representations for which the conformal structure on S degenerates to a nodal curve, and we discuss new joint work in progress with Qiongling Li and Andrea Tamburelli in the case of PSp(4,R) representations.

November 26
Or Hershkovits (Stanford) - Ancient solutions to the Mean Curvature Flow and applications
Abstract

In the last 35 years, geometric flows have proven to be a powerful tool in geometry and topology. The Mean Curvature Flow is, in many ways, the most natural flow for hypersurfaces in Euclidean space. In this talk, which will assume no prior knowledge, I will present recent progress in classifying ancient solutions to the mean curvature flow (including joints work with Kyeongsu Choi, Robert Haslhofer and Brian White). I will also explain how this classification assists in answering fundamental questions regarding the singularity formation of the flow, and describe what are the remaining challenges in converting the mean curvature flow into the powerful tool we hope it can become.

December 3
(Canceled due to weather)
Niky Kamran (McGill) - Non-uniqueness results for the Calderon inverse problem with local or disjoint data
Abstract