UCSD Differential Geometry Seminar 2022

Unless otherwise noted, all seminars are on Thurs 11:00 - 12:00 am in Room 7321.


Winter 2023 Schedule

January 12

Tristan Collins, MIT, Special Colloquium 4-5 pm at APM 6402
Title: Complete Calabi-Yau metrics: recent progress and open problems
Abstract:Complete Calabi-Yau metrics are fundamental objects in Kaehler geometry arising as singularity models or "bubbles" ​in degenerations of compact Calabi-Yau manifolds. The existence of these metrics and their relationship with algebraic geometry are the subjects of several long standing conjectures due to Yau and Tian-Yau. I will describe some recent progress towards the question of existence, and explain some future directions, highlighting connections with notions of algebro-geometric stability.​

January 19

Jingze Zhu, MIT
Title: Spectral quantization for ancient asymptotically cylindrical flows
Abstract asymptotically cylindrical flows are ancient solutions to the mean curvature flow whose tangent flow at $-\infty$ are shrinking cylinders. In this talk, we study quantized behavior of asymptotically cylindrical flows. We show that the cylindrical profile function u of these flows have the asymptotics $u(y,\omega, \tau) = \frac{y^{T}Qy - 2tr Q}{|\tau|} + o(|\tau|^{-1})$ as $\tau\rightarrow -\infty$, where $Q$ is a constant symmetric $k\times k$ matrix whose eigenvalues are quantized to be either 0 or $-\frac{\sqrt{2(n-k)}}{4}$. Assuming non-collapsing, we can further draw two applications. In the zero rank case, we obtain the full classification. In the full rank case, we obtain the $SO(n-k+1)$ symmetry of the solution. This is joint work with Wenkui Du.

January 26

Juilan Chaidez, Princeton, Colloquium 4-5 at APM 6402
Title: Symplectic dynamics without Floer homology
Abstract Symplectic dynamics is the study of dynamical systems using tools from symplectic topology, like Floer homology. Many natural dynamical systems in physics and topology can be studied fruitfully through this lens, including many-body problems, billiards and surface diffeomorphisms. In this talk, I will give an overview of three tools in symplectic dynamics that are independent of Floer homology: the Ruelle invariant, piecewise-smooth contact geometry and min-max spectral invariants. I will discuss several recent applications to open problems, including the Viterbo conjecture and the (strong) closing lemma for higher dimensional Reeb flows.

February 2

Peter Petersen, UCLA
 Title:Lichnerowicz Laplacians
Abstract: I will explain the relevance of the Lichnerowicz Laplacian in several situations and how one can easily understand the zeroth order term in the expression. This leads to simple proofs of some classical results and also to new results with more general curvature conditions.

February 9

Antonio De Rosa, Univ. Maryland
Title: Min-max construction of anisotropic CMC surfaces
Abstract: We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard [Invent. Math.,1983] in dimension 3.

February 16


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February 23


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March 2


Zihan Wang, Princeton
Title: Translating mean curvature flow with prescribed end.
Abstract: Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on the joint work with Ao Sun.
Yang Li, MIT, 4-5 pm(Special time), 7321
Title: A variational program for the Thomas-Yau conjecture
Abstract: The Thomas-Yau conjecture is concerned with relating the existence question of special Lagrangians to stability conditions and Fukaya categories. After a very brief recap on Floer theory, we will sketch a variational framework to tackle the conjecture, and the main bulk of the talk is on the analytical aspects. It turns out that a very weak minimizer exists under a suitable Thomas-Yau semistability condition, and the main open problem is to prove the Euler Lagrange equation, which is the special Lagrangian condition.

March 16


Wilderich Tuschmann, KIT, Germany
Title: Moduli spaces of Riemannian metrics
Abstract: Consider a smooth manifold with a Riemannian metric satisfying some sort of curvature or other geometric constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasaki, etc. A natural question to ponder is then what the space of all such metrics does look like. Moreover, one can also pose this question for corresponding moduli spaces of metrics, i.e., quotients of the former by (suitable subgroups of) the diffeomorphism group of the manifold, acting by pulling back metrics. The study of spaces of metrics and their moduli has been a topic of interest for differential geometers, global and geometric analysts and topologists alike, and I will introduce to and survey in detail main results and open questions in the field with a focus on non-negative Ricci or sectional curvature as well as other lower curvature bounds on closed and open manifolds, and, in particular, also discuss broader non-traditional approaches from metric geometry and analysis to these objects and topics.

Spring 2022 Schedule

Fall 2022 Schedule

Questions: lni@math.ucsd.edu