CMC surfaces in Minkowski space

Speaker: 

Peter Smillie

Institution: 

Caltech

Time: 

Tuesday, January 14, 2020 - 4:00pm

Location: 

RH 306

In joint work with F. Bonsante and A. Seppi, we solve a
Dirichlet-type problem for entire constant mean curvature hypersurfaces in
Minkowski n+1-space, proving that such surfaces are essentially in bijection
with lower semicontinuous functions on the n-1-sphere. This builds off of
existence theorems by Treibergs and Choi-Treibergs, which themselves rely on
the foundational work of Cheng and Yau. I'll present their maximum principle
argument as well the extra tool that leads to our complete existence and
uniqueness theorem. Time permitting, I'll compare with the analogous problem
of constant Gaussian curvature and present a new result on their intrinsic
geometry.

Some study of biharmonic maps and submanifolds since 2000 (Cancelled)

Speaker: 

Yelin Ou

Institution: 

Texas A&M Commence

Time: 

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

Host: 

Zhiqin Lu

Location: 

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.

A longitudinal index theorem for open foliated manifolds

Speaker: 

Xiang Tang

Institution: 

Washington University in St Louis

Time: 

Tuesday, January 7, 2020 - 4:00pm

Location: 

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.

Faltings Heights, Igusa Local Zeta Functions, and the Stability Conjectures in Kahler Geometry

Speaker: 

Sean Paul

Institution: 

University of Wisconsin, Madison

Time: 

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

Host: 

Jeff Viaclovsky

Location: 

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.

Bryant-Salamon G2 manifolds and coassociative fibrations

Speaker: 

Spiro Karigiannis

Institution: 

University of Waterloo

Time: 

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

Location: 

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".
 

Witten deformation on noncompact manifolds

Speaker: 

Xianzhe Dai

Institution: 

UC Santa Barbara

Time: 

Monday, October 28, 2019 - 4:00pm

Location: 

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.

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