UC San Diego
Geometry Seminar 2022-2023
Unless otherwise noted, all seminars are
on **Thursday 11:00 - 12:00 in Zoom setting or in person at APM 7321. Zoom ID:
953 0943 3365 and (after 11/03) Zoom ID 910 6959 2533** .
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** Fall 2022 Schedule**

October 6, 2022

Novack, Michael, UT Austin

**Title:**A mesoscale flatness criterion and its application to exterior isoperimetry

**Abstract:**
We introduce a "mesoscale flatness criterion" for hypersurfaces with bounded mean curvature, discussing its relation to and differences with classical blow-up and blow-down theorems, and then we exploit this tool for a complete resolution of relative isoperimetric sets with large volume in the exterior of a compact obstacle. This is joint work with Francesco Maggi (UT Austin).

October 13 , 2022, Zoom ID:
953 0943 3365

Xiaolong Li, Wichita, via zoom

**Title:**The Curvature Operator of the Second Kind

**Abstract:**I will first give an introduction to the notion of the curvature operator of the second kind and review some known results, including the proof of Nishikawa's conjecture stating that a closed Riemannian manifold with positive (resp. nonnegative) curvature operator of the second is diffeomorphic to a spherical space form (resp. a Riemannian locally symmetric space). Then I will talk about my recent works on the curvature operator of the second kind on Kahler manifolds and product manifolds. Along the way, I will mention some interesting questions and conjectures.

October 13 , 2022

Colloquium at 4:00pm APM 6402, Jeff Viaclovskiy, UCI

**Title:** Gravitational instantons and algebraic surfaces

**Abstract:**
ABSTRACT: Geometers are interested in the problem of finding a "best"
metric on a manifold. In dimension 2, the best metric is usually one
which possesses the most symmetries, such as the round metric on a
sphere, or a flat metric on a torus. In higher dimensions, there are
many more classes of geometrically interesting metrics. I will give a
general overview of a certain class of Einstein metrics in dimension 4
which have special holonomy, and which are known as "gravitational
instantons." I will then discuss certain aspects of their classification
and connections with algebraic surfaces.

October 20, 2022, Special time: 4--5pm (speaker local time 7:00-8:00 am), Zoom ID:
953 0943 3365

Xiaohua Zhu, Peking U

**Title: **Kaehler-Ricci flow on Fano G-manifolds

**Abstract:**
I will talk about a recent work jointly with Tian on Kaehler-Ricci flow on Fano G-manifolds. We prove that on a Fano G-manifold, the Gromov-Hausdorff limit of Kaehler-Ricci flow with initial metric in $2\pi c_1(M)$ must be a Q-Fano horosymmetric variety which admits a singular Keahler-Ricci soliton. Moreover, we show that the complex structure of limit variety can be induced by $C^*$-degeneration via the soliton vector field. A similar result can be also proved for Kaehler-Ricci flows on any Fano horosymmetric manifolds.

October 21-22, 5:00--9:00pm California time; Beijing Time: October 22-23, 8:00am-12:00noon, 2022

2022 Chongqing Geometric Analysis Workshop

Organizers: Xinyue Cheng, Fei He, Lei Ni, Bo Yang, Xiaokui Yang, Chunna Zeng, Yingying Zhang, Fangyang Zheng

Zoom link: https://ucsd.zoom.us/j/92856276102

**Program:**

Oct 22 (Sat) 8:am-12noon Beijing time:

8:00am-8:50am, V. Tosatti (NYU), (speaker local time Friday 8:00pm), host: X. Yang

9:00am-9:50am, J. Wang (U Minn), (speaker local time Friday 8:00pm), host: G. Liu

10:00am-10:50am, X. Li (Wichita State U), (speaker local time Friday 8:00pm), host: C. Zeng

11:00am-11:50am, G. Liu (East China Normal U), host: X. Cheng

Oct 23 (Sun) 8:am-12noon Beijing time:

8:00am-8:50am, J. Streets (UC Irvine), (speaker local time Sat 5:00pm), host: F. Zheng

9:00am-9:50am, Y. Ustinovskiy (Lehigh U), (speaker local time Sat 9:00pm), host: Y. Zhang

10:00am-10:50am, X. Zhang (UC Irvine), (speaker local time Sat 7:00pm), host: B. Yang

11:00am-11:50am, L. Ni (UC San Diego), (speaker local time Sat 8:00pm), host: F. He

**Abstract:**
** Abstracts of Talks**

October 27, 2020

Alessandro Pigati, NYU

**Title: **Partial results on the anisotropic Michael-Simon inequality

**Abstract:**In geometric measure theory, the monotonicity formula for the area functional is a basic
tool upon which many other basic fundamental facts depend.
Some of them also follow from a weaker analytic tool, which is the Michael-Simon inequality.
For anisotropic integrands (which generalize the area), monotonicity does not hold, while the latter inequality
is conjectured to be true (under appropriate assumptions); actually, the latter is more essential to geometric measure theory,
in that it turns out to be equivalent to the compactness of the classes of rectifiable and integral varifolds.
In this talk we present some partial results, one of which is a slight improvement of a posthumous result of Almgren,
namely the validity of this inequality for convex integrands close enough to the area, for surfaces in R^3.
Our technique relies on a nonlinear inequality bounding
the L^1-norm of the determinant of a function, from the plane to 2x2
matrices, with the L^1-norms of the divergence of the rows, provided the
matrix obeys some pointwise nonlinear constraints.
This is joint work with Guido De Philippis (NYU).
** **

November 3, 2022, 3-4pm (speaker local time 8-9 am) via zoom 910 6959 2533

Ramiro Lafuente, Queensland

**Title:
** Non-compact Einstein manifolds with symmetry

**Abstract:**We will discuss Einstein manifolds which are invariant under an isometric Lie group action. Our main goal is to explain the proof of the 1975 Alekseevskii Conjecture on non-compact homogeneous Einstein spaces, recently obtained in collaboration with Christoph Boehm (Muenster). To that end, we will also present new structure results for Einstein metrics on principal bundles. The talk will conclude with open questions and future directions.

November 10, 2022

Antoine Song, UCB

**Title: **The spherical Plateau problem: existence and structure

**Abstract:**Consider a countable group G acting on the unit sphere S in the space of L^2 functions on G by the regular representation. Given a homology class h in the quotient space S/G, one defines the spherical Plateau solutions for h as the intrinsic flat limits of volume minimizing sequences of cycles representing h. In some special cases, for example when G is the fundamental group of a closed hyperbolic manifold of dimension at least 3, the spherical Plateau solutions are essentially unique and can be identified. However not much is known about the properties of general spherical Plateau solutions. I will discuss the questions of existence and structure of non-trivial spherical Plateau solutions.

** **

November 17, 2022

Yury Ustinovskiy, Lehigh

**Title: ** The generalized Kahler Calabi-Yau problem

**Abstract:**
In this talk we define the fundamental geometric constructions behind the generalized Kahler geometry introduced by Hitchin and Gualtier and set up an appropriate generalization of the Calabi problem. Similarly to Cao's approach to the solution of the classical Calabi problem, we study the existence and convergence of the generalized Kahler-Ricci flow (GKRF) on relevant backgrounds. In particular, we prove that on a Kahler Calabi-Yau background, the GKRF converges to the unique classical Ricci-Flat structure. This result has non-trivial applications to understanding the space of generalized Kahler structures, and as a special case yields the topological structure of natural classes of Hamiltonian symplectomorphisms on hyperKahler manifolds.
Based on a joint work with Apostolov, Fu and Streets.

** **

December 1, 2022

Li-Sheng Tseng, UCI

**Title: ** A Cone Story for Smooth Manifolds

**Abstract:**Differential forms are basic objects of manifolds and encode
invariants. This talk will motivate the usefulness of considering pairs
of differential forms together with a map linking them. We will show
how this can lead to novel functionals and geometric flows. As an
application, it leads to new notions of flat connections and Morse theory
on symplectic manifolds. This talk is based on joint works with Jiawei
Zhou, David Clausen and Xiang Tang.

Thomas Walpuski, Humbolt University, 1:00-2:00pm at APM 7321

**Title: ** Gopakumar–Vafa finiteness: an application of geometric measure theory to symplectic geometry

**Abstract:**The purpose of this talk is to illustrate an application of the powerful machinery of geometric measure theory to a conjecture in Gromov–Witten theory arising from physics. Very roughly speaking, the Gromov–Witten invariants of a symplectic manifold (X,ω) equipped with a tamed almost complex structure J are obtained by counting pseudo-holomorphic maps from mildly singular Riemann surfaces into (X,J). It turns out that Gromov–Witten invariants are quite complicated (or “have a rich internal structure”). This is true especially for if (X,ω) is a symplectic Calabi–Yau 3–fold (that is: dim X = 6, c_1(X,ω) = 0).
In 1998, using arguments from M–theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi–Yau 3–folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov–Witten invariants by a transformation of the generating series. The Gopakumar–Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition.
The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that Gromov’s compactness theorem can (and should!) be replaced with the work of Federer–Flemming, Allard, and De Lellis–Spadaro–Spolaor. This upgrade of Ionel and Parker’s cluster formalism proves both the integrality and finiteness conjecture.
This talk is based on joint work with Eleny Ionel and Aleksander Doan.

** **

**Spring 2022 Schedule**

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