Fully nonlinear elliptic equations on real and complex manifolds

Speaker: 

Bo Guan

Institution: 

Ohio State University

Time: 

Monday, June 3, 2019 - 4:00pm to 5:00pm

Location: 

RH 340P

Joint with Analysis Seminar.

 

Abstract: Fully nonlinear elliptic and parabolic equations on manifolds play central roles in some important problems in real and complex geometry. A key ingredient in solving such equations is to establish apriori  estimates up to second order. For general Riemannian manifolds, or Kaehler/Hermitian manifolds in the  complex case, one encounters difficulties caused by the curvature (as well as torsion in the Hermian case) of the manifolds. 

In this talk we report some results in our effort to overcome these obstacles over the past  few years. We shall emphasize on understanding the roles of subsolutions and concavity of the equation based on which our techniques were developed. We are interested both in equations on closed manifolds, and the Dirichlet problem for equations on manifolds with boundary of arbitrary geometry. 

For the Dirichlet problem on manifolds with boundary, we prove that under some fundamental structure conditions which were first proposed by Caffarelli-Nirenberg-Spruck and are now standard in the literature, there exist a smooth solution provided that there is a C2 subsolution. 

For equations on closed manifolds, there have appeared two different notations of weak subsolutions, the C-subsolution introduced by Gabor Szekelyshidi (JDG, 2018) and "tangent cone at infinity" condition by myself (Duke J Math, 2014). We show for type I cones the two notations coincide. We also construct examples showing for the Dirichlet problem that the subsolution condition can not be replaced by the weaker versions.

Joint LA Topology Seminar

Institution: 

UCLA

Time: 

Monday, April 15, 2019 - 4:00pm to 6:00pm

Location: 

MS 6221

Talks at UCLA.  Please contact Li-Sheng Tseng if you plan to attend and would like to carpool.

 

Peter Lambert-Cole (Georgia Tech): Bridge trisections and the Thom conjecture
The classical degree-genus formula computes the genus of a nonsingular algebraic curve in the complex projective plane. The well-known Thom conjecture posits that this is a lower bound on the genus of smoothly embedded, oriented and connected surface in CP2. The conjecture was first proved twenty-five years ago by Kronheimer and Mrowka, using Seiberg-Witten invariants. In this talk, we will describe a new proof of the conjecture that combines contact geometry with the novel theory of bridge trisections of knotted surfaces. Notably, the proof completely avoids any gauge theory or pseudoholomorphic curve techniques.

James Conway (UC Berkeley): Classifying contact structures on hyperbolic 3-manifolds
Two of the most basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. In dimension 3, these questions have been answered for large classes of manifolds, but with a notable absence of hyperbolic manifolds. In this talk, we will see a new classification of contact structures on an family of hyperbolic 3-manifolds arising from Dehn surgery on the figure-eight knot, and see how it suggests some structural results about tight contact structures. This is joint work with Hyunki Min.

The \'etale descent problem in algebraic K-theory

Speaker: 

Akhil Mathew

Institution: 

University of Chicago

Time: 

Monday, May 13, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

Algebraic K-theory is an invariant of rings (or algebraic varieties) that sees deep geometric and arithmetic information (ranging from Chow rings to special values of L-functions), but is generally difficult to compute. One reason for the complexity of algebraic K-theory is that it fails to satisfy \'etale descent. A general principle in algebraic K-theory (going to Lichtenbaum-Quillen, and proved in the work of Voevodsky-Rost on the Bloch-Kato conjecture) is that it is not too far off from doing so. I will explain this principle and some new extensions of this (joint with Dustin Clausen) in p-adic settings.

Projective Geometry, Complex Hyperbolic Space, and Geometric Transitions.

Speaker: 

Steve Trettel

Institution: 

UCSB

Time: 

Monday, May 6, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

The natural analog of Teichmuller theory for hyperbolic manifolds in dimension 3 or greater is trivialized by Mostow Rigidity, so mathematicians have worked to understand more general deformations.  Two well studied examples, convex real projective structures and complex hyperbolic structures, have been investigated extensively and provide independently developed deformation theories.  Here we will discuss a surprising connection between the these, and construct a one parameter family of geometries deforming complex hyperbolic space into a new geometry built out of real projective space and its dual.

 

Sasaki-Einstein manifolds and AdS/CFT correspondence

Speaker: 

Dan Xie

Institution: 

Tsinghua University

Time: 

Monday, February 11, 2019 - 4:00pm

Location: 

RH 340P

Sasakian manifolds are odd dimensional analog of Kahler manifolds,
and it is an interesting question to determine when
a Sasakian manifold admits an Einstein metric. Five dimensional
Sasaki-Einstein (SE)  manifolds play an important role in AdS/CFT
correspondence, which relates a string theory and a quantum field theory. I
will discuss the existence of SE manifolds and its geometric properties
which will be of great interest to AdS/CFT correspondence.

Cohomology of the space of polynomial morphisms on A^1 with prescribed ramifications

Speaker: 

Oishee Banerjee

Institution: 

University of Chicago

Time: 

Monday, April 8, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

In this talk we will discuss the moduli spaces Simp^m_n of degree n+1 morphisms  \A^1_K\to \A^1_K  with "ramification length <m" over an algebraically closed field K. For each m, the moduli space Simp^m_n is a Zariski open subset of the space of degree n+1 polynomials over K up to Aut(\A^1_K). It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes- here we are prescribing, instead, the ramification data. We will also see why and how our results align, in spirit, with the long standing open problem of understanding the topology of the Hurwitz space.

Pages

Subscribe to RSS - Geometry and Topology