# Geometric Analysis Seminar

The geometric analysis seminar is held on Tuesdays at 11am in 210 Deady Hall.

### Spring Quarter, 2019

- April 2,
**Peter Gilkey** (UO)

An introduction to locally homogeneous affine surfaces with torsion
- April 30,
**Bradley Burdick** (UO)

Metrics of Positive Ricci Curvature on Connected Sums: Projective Spaces, Products, and Plumbings
- May 13,
**Luda Korobenko** (Reed College)

Regularity of degenerate elliptic PDEs and properties of associated metric measure spaces
**Abstract**: The central topic of this talk is regularity of weak solutions to second order infinitely degenerate elliptic equations. It is known that regularity of weak solutions can be studied by studying properties of certain metric spaces associated to the operator, namely, subunit metric spaces. When the degeneracy of the operator is of infinite type, the measures of subunit balls are non doubling. As a consequence, the classical Sobolev inequality – one of the main ingredients of the regularity theory – is also unavailable. However, weaker versions, such as log-Sobolev (or, more generally, Orlicz-Sobolev) inequalities can be used in an adaptation of the classical Moser or DeGiorgi iteration scheme to show improved regularity of weak solutions. In this talk I will discuss the connection between these weaker types of Sobolev inequalities and the doubling condition, and their connection to infinitely degenerate elliptic PDEs.

- May 21,
**Peng Lu** (UO)

Ancient solutions for Andrews hypersurface flow

### Winter Quarter, 2019

- February 12,
**Pablo Pisani** (Instituto de Física La Plata, Argentina)

Geodesics on homogeneous surfaces
**Abstract**: A connection on the tangent bundle of a smooth manifold defines an affine manifold and introduces the notions of parallel transport and geodesics, which describe the geometry of straight lines within the manifold. In this seminar we will show some advances in the determination of geodesic completeness of locally homogeneous affine surfaces.

- March 5,
**Bradley Burdick** (UO)

Metrics of positive Ricci curvature on connected sums and exotic smooth structures
**Abstract**: Given a set of Riemannian manifolds with positive Ricci curvatures, when does the connected sum of these manifolds admit a metric of positive Ricci curvature? Weakening the question to positive scalar curvature, it is always possible (in dimension 3 or more), but strengthening the question to positive sectional curvature, one can show it is not possible in general. Using the work of Perelman we can describe a class of a Ricci-positive Riemannian metrics that will be closed under connected sum. By proving a generalization of Perelman`s Ricci-positive gluing theorem to Riemannian manifolds with corners, we are able to prove that these class of metrics are closed under spherical fibrations as well. Combining this idea with the work of Wraith on Ricci-positive metrics on exotic spheres and some computations in surgery theory (due to many authors), we are able to give new examples of metrics of positive Ricci curvature on exotic smooth structures.

- March 12,
**Weiyong He** (UO)

Gursky-Streets equations and the uniqueness of sigma_2 problem

### Fall Quarter, 2018

- October 2,
**Peter Gilkey** (UO)

Affine Killing Vector Fields
- October 9,
**Boris Botvinnik** (UO)

Positive scalar curvature metrics on manifolds with fibered singularities
- October 16,
**Micah Warren** (UO)
- October 23,
**Guangbo Xu** (Simons Center)

Bershadsky-Cecotti-Ooguri-Vafa torsion of Landau-Ginzburg Models
**Abstract**: In their seminal work in 1994, Bershadsky-Cecotti-Ooguri-Vafa introduced a particular Ray-Singer analytic torsion of Calabi-Yau manifolds which coincides with the genus one topological string partition function. They also proved a holomorphic anomaly formula for this torsion which is related to the variation of Hodge structure and the Weil-Petersson geometry of deformation spaces. In this joint work with Shu Shen and Jianqing Yu, we consider the similar object in Landau-Ginzburg models. We prove an index theorem for the associated Dirac operator and rigorously define the BCOV torsion. We also obtain a partial result towards proving a holomorphic anomaly formula.

- October 30,
**Xiangwen Zhang**
- November 6,
**Bradley Burdick** (UO)
- November 13,
**Bo GUAN**

**Previous Schedule:** 2017 Fall – 2018 Spring, 2016 Fall – 2017 Spring, 2015 Fall – 2016 Spring, 2015 Spring, 2015 Winter, 2014 Fall, 2014 Spring, 2014 Winter, 2012 Winter, 2011 Fall, 2010 Spring, 2010 Winter