UC San Diego
Geometry Seminar 2018-2019
Unless otherwise noted, all seminars are
on **Wed 2:00 - 2:50 pm in Room APM 5829** .
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** Fall 2019 Schedule**

Michael Freedman, Station Q, Microsoft Research

October 4, 2019 (APM 6402, 11:00-11:50, special time and place)

**Title:The 2-width of 3-manifolds embedded in $\mathbb{R}^4$**

**Abstract:
** Morse theory can be generalized to the study of maps to Rk. I will discuss where this leads, focusing on
ambient Morse functions of embedded 3-manifolds. The relevance to the question of finite generation of the
Goeritz groups Gg will be explained

October 9, 2019

Luca Spolaor, UCSD

**Title: Epsilon-regularity for minimal surfaces near quadratica cones **

**Abstract:** Every area-minimizing hypercone having only an isolated singularity fits into a foliation by smooth, area-minimizing hypersurfaces asymptotic to the cone itself. In this talk I will present the following epsilon-regularity result: every minimal surfaces lying sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), is a perturbation of either the cone itself, or some leaf of its associated foliation. This result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces asymptotic
to a quadratic cone as either the cone itself, or some leaf of the foliation, and it also allows to study convergence to singular minimal hyper surfaces. This is a joint result with N. Edelen.

October 17 ( 1:00pm at 7321) , 2019, special date, time and place due to that this is joint seminar

Antonio De Rosa, NYU

**Title:Elliptic integrands in analysis **

**Abstract:** I will present the recent tools I have developed to prove existence and
regularity properties of the critical points of anisotropic functionals.
In particular, I will provide the anisotropic extension of Allard's
celebrated rectifiability theorem and its applications to the
anisotropic Plateau problem. Three corollaries are the solutions to the
formulations of the Plateau problem introduced by Reifenberg, by
Harrison-Pugh and by Almgren-David. Furthermore, I will present the
anisotropic counterpart of Allard's compactness theorem for integral
varifolds. To conclude, I will focus on the anisotropic isoperimetric
problem: I will provide the anisotropic counterpart of Alexandrov's
characterization of volume-constrained critical points among finite
perimeter sets. Moreover I will derive stability inequalities associated
to this rigidity theorem.
Some of the presented results are joint works with De Lellis, De
Philippis, Ghiraldin, Gioffré, Kolasinski and Santilli.

October 23, 2019

**Title: **

**Abstract:**
** **

October 31, 2019 (Re-scheduled)

Jianfeng Lin, UCSD

**Title:Comparing gauge theoretic invariants of homology S1 cross S3 **

**Abstract:**Since the ground breaking work of Donaldson in the 1980s, topologists has achieved huge success in using gauge theory to study smooth 4-manifolds with nonzero second homology. The case of 4-manifolds with trivial second homology is relatively less known.
In particular, when the 4-manifold have the same homology as S1 cross S3, there are several gauge theoretic invariants. The first one is the Casson-Seiberg-Witten invariant LSW(X) defined by Mrowka-Ruberman-Saveliev; the second one is the Fruta-Ohta invariant LFO(X). It is conjecture that these two invariants are equal to each other (This is an analogue of Witten’s conjecture relating Donaldson and Seiberg-Witten invariants.)
In this talk, I will recall the definition of these two invariants, give some applications of them (including a new obstruction for metric with positive scalar curvature), and sketch a proof of this conjecture for finite-order mapping tori. This is based on a joint work with Danny Ruberman and Nikolai Saveliev.
** **

November 6, 2019

Xiaolong Li, UCI

**Title:Sharp lower bound for the first eigenvalue of the weighted $p$-Laplacian on Bakry-Emery manifolds
**

**Abstract:**In this talk, we prove sharp lower bound of the first nonzero eigenvalue of the weighted $p$-Laplacian on compact Bakry-Emery manifolds, without boundary or with convex boundary and Neuman boundary condition. This is joint work with Kui Wang.

November 13, 2019

Benjamin Hoffman, Cornell

**Title: String domains for coadjoint orbits**

**Abstract:** For each regular coadjoint orbit of a compact group, we construct an exhaustion by symplectic embeddings of toric domains. As a by-product we arrive at a conjectured formula for the Gromov width of coadjoint orbits. Our method combines ideas from Poisson-Lie groups and from the geometric crystals of Berenstein-Kazhdan. We also prove similar results for multiplicity-free spaces. This is joint work with A. Alekseev, J. Lane, and Y. Li.

** **

November 20, 2019

Jianfeng Lin, UCSD

**Title: Comparing gauge theoretic invariants of homology S1 cross S3 **

**Abstract:** Since the ground breaking work of Donaldson in the 1980s, topologists has achieved huge success in using gauge theory to study smooth 4-manifolds with nonzero second homology. The case of 4-manifolds with trivial second homology is relatively less known.
In particular, when the 4-manifold have the same homology as S1 cross S3, there are several gauge theoretic invariants. The first one is the Casson-Seiberg-Witten invariant LSW(X) defined by Mrowka-Ruberman-Saveliev; the second one is the Fruta-Ohta invariant LFO(X). It is conjecture that these two invariants are equal to each other (This is an analogue of Witten’s conjecture relating Donaldson and Seiberg-Witten invariants.)
In this talk, I will recall the definition of these two invariants, give some applications of them (including a new obstruction for metric with positive scalar curvature), and sketch a proof of this conjecture for finite-order mapping tori. This is based on a joint work with Danny Ruberman and Nikolai Saveliev.

** **

December 4, 2019

Mattew Stoffregen , MIT

**Title: An infinite-rank summand of the homology cobordism group**

**Abstract: **The homology cobordism group of integer homology three-spheres is a natural invariant of interest to four-dimensional topologists. In this talk, we recall its definition and give a short introduction to involutive Floer homology, As an application, we see that there is an infinite-rank summand of the homology cobordism group. This includes joint work with Irving Dai, Jen Hom, and Linh Truong.

**Winter 2018 Schedule**

**Questions:** lni@math.ucsd.edu