# 27th Southern California Geometric Analysis Seminar

## Speaker:

## Institution:

## Time:

**Conference website: **https://www.math.uci.edu/scgas

Please register at the conference website if planning to attend.

SCGAS

Conference

Saturday, February 8, 2020 - 11:00am to Sunday, February 9, 2020 - 12:30pm

**Conference website: **https://www.math.uci.edu/scgas

Please register at the conference website if planning to attend.

Yelin Ou

Texas A&M Commence

Tuesday, March 10, 2020 - 4:00pm to 5:00pm

RH 306

Biharmonic maps are maps between Riemannian manifolds which are critical points of the bi-energy. They are solutions of a system of 4thorder PDEs and they include harmonic maps and biharmonic functions as special cases. Biharmonic submanifolds (which include minimal submanifolds as special cases) are the images of biharmonic isometric immersions. The talk will review some problems, including classification of biharmonic submanifolds in space forms, biharmonic maps into spheres, biharmonic conformal maps, and unique continuation theorems, studied in this field and their progress since 2000. The talk also presents some recent work on equivariant biharmonic maps and the stability and index of biharmonic hypersurfaces in space forms.

Xiang Tang

Washington University in St Louis

Tuesday, January 7, 2020 - 4:00pm

RH 306

In this talk, we will present some recent study about the index

problem of longitudinal elliptic operators on open foliated manifolds. As

the operators under consideration are not elliptic on the whole (not

necessarily closed) manifold, they in general fail to be Fredholm. We will

introduce some operator algebra tools to study the index of such operators.

As an application, we will present a Lichnerowicz type vanishing result for

foliations on open manifolds.

Siyi Zhang

University of Notre Dame

Tuesday, October 22, 2019 - 3:00pm to 4:00pm

RH 440R

Joint with Analysis seminar.

Sean Paul

University of Wisconsin, Madison

Tuesday, November 5, 2019 - 4:00pm to 5:00pm

RH 306

Let (X,L) be a polarized manifold. Assume that the automorphism group is finite. If the height discrepancy of (X,L) is O(d^2) then (X,L) admits a csck metric in the first chern class of L if and only if (X,L) is asymptotically stable.

Spiro Karigiannis

University of Waterloo

Tuesday, November 10, 2020 - 4:00pm to 5:00pm

Remotely

I will discuss joint work with Jason Lotay from arXiv:2002.06444. We show how the three Bryant-Salamon G2 manifolds can be viewed as coassociative fibrations. In all cases the coassociative fibres are invariant under a 3-dimensional group and are thus of cohomogeneity one, In general there are both generic smooth fibres and degenerate singular fibres. The induced Riemannian geometry on the fibres turns out to exhibit asymptotically conical and conically singular behaviour. In some cases we also explicitly determine the induced hypersymplectic structure. In all three cases we show that the "flat limits" of these coassociative fibrations are well-known calibrated fibrations of Euclidean space. Finally, we establish connections with the multimoment maps of Madsen-Swann, the new compact construction of G2 manifolds of Joyce-Karigiannis, and recent work of Donaldson involving vanishing cycles and "thimbles".

Shoo Seto

UC Irvine

Tuesday, October 8, 2019 - 4:00pm

RH 306

In this talk, I will discuss some known results and recent improvements to lower bound on the first eigenvalue of the p-Laplacian as well as introduce a new p-Laplace type operator acting on differential forms.

Xianzhe Dai

UC Santa Barbara

Monday, October 28, 2019 - 4:00pm

RH 340P

Motivated by considerations from the mirror symmetry and

Landau-Ginzburg model, we consider Witten deformation on noncompact

manifolds.

Witten deformation is a deformation of the de Rham complex introduced by

Witten in an influential paper and has had many important applications,

mostly on compact manifolds. We will discuss some recent work with my

student Junrong Yan on the spectral theory of Witten Laplacian, the

cohomology of the deformation as well as its index theory.

A joint seminar with the Geometry & Topology Seminar series.

Jianfeng Lin

UC San Diego

Tuesday, October 1, 2019 - 4:00pm

RH 306

While classical gauge theoretic invariants for 4-manifolds are usually

defined in the setting that the intersection form has nontrivial positive

part, there are several invariants for a 4-manifold X with the homology S1

cross S3. 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 conjectured 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 prove of this conjecture for

finite-order mapping tori. This is based on a joint work with Danny Ruberman

and Nikolai Saveliev.

A joint seminar with the Geometry & Topology Seminar series.

Paula Burkhardt-Guim

UC Berkeley

Tuesday, October 15, 2019 - 4:00pm

RH306

We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C^0 metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.