Geometry-Topology Archives for Academic Year 2019


Coupled Canonical Metrics in Kähler Geometry

When: Mon, September 9, 2019 - 3:00pm
Where: Kirwan Hall 3206

Speaker: Jakob Hultgren (University of Maryland) -

Abstract: A central theme in complex geometry is to study various types of canonical metrics, for example Kaehler-Einstein metrics and cscK metrics. Much of the interest in these come from connections to notions of algebro-geometric stability. I will talk about a new type of canonical objects introduced by Witt Nystroem and myself. These are k-tuples of Kaehler metrics that satisfy certain coupled Kaehler-Einstein equations. I will explain some basic existence and uniqueness results and indicate relations to algebraic geometry.

Non commutative cluster coordinates for Higher Teichmüller Spaces

When: Mon, October 7, 2019 - 3:00pm
Where: Kirwan Hall 3206

Speaker: Daniele Alessandrini (Columbia University) -
Abstract: Fock-Goncharov found a beautiful structure of cluster
variety on the decorated Hitchin components of punctured surfaces,
generalizing Penner's decorated Teichmueller Theory. This is an
algebraic theory based on the notion of positivity.
Hitchin components are an example of Higher Teichmueller Spaces, and
the spaces of Maximal Representations are another example. In this
latter case, we found new coordinates on these Higher Teichmueller
Spaces that give them a structure of non-commutative cluster
varieties, in the sense defined by Berenstein-Rethak. This is joint
work with Guichard, Rogozinnikov and Wienhard.

Singularities of Hermitian-Yang-Mills connections

When: Mon, October 21, 2019 - 3:00pm
Where: Kirwan Hall 3206
Speaker: Xuemiao Chen (University of Maryland) -
Abstract: The Donaldson-Uhlenbeck-Yau theorem confirms the existence of a Hermitian-Yang-Mills connection on a slope stable holomorphic vector bundle over a Kahler manifold. Later, it is generalized by Bando and Siu to the case of stable reflexive sheaves by using singular HYM connections with natural curvature bound. In this talk, I will explain how to understand the infinitesimal behavior of such a singular Hermitian-Yang-Mills connection near its singularities. The talk is based on joint work with Song Sun.

A D'Ambra Theorem in conformal Lorentzian geometry

When: Mon, October 28, 2019 - 3:00pm
Where: Kirwan Hall 3206

Speaker: Karin Melnick (University of Maryland) -

Abstract: D'Ambra proved in 1988 that the isometry group of a compact, simply connected, real-analytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture.

Curvature of the Weil-Petersson metric for the moduli space of Riemann surfaces

When: Mon, November 11, 2019 - 3:00pm
Where: Kirwan Hall 3206
Speaker: Scott Wolpert (University of Maryland) -

Abstract: The Weil-Petersson metric for the moduli space or Riemann surfaces is Kaehler, incomplete with negative sectional curvature. The metric is a tool for understanding the geometry of the moduli space. Applications include a proof that the compactified moduli space is projective, a solution of the Nielsen Realization Problem and a CAT(0) geometry. The behavior of the sectional curvature is an ingredient in the Burns-Masur-Wilkinson proof that the geodesic flow is ergodic. A non trivial upper bound is required to study the mixing rate of the geodesic flow. I will present the first non trivial upper bound for the sectional curvature and discuss the optimal expected bound.

TBA

When: Mon, December 9, 2019 - 3:00pm
Where: Kirwan Hall 3206
Speaker: Xuwen Zhu (University of California, Berkeley) -

TBA

When: Fri, December 13, 2019 - 11:00am
Where: Kirwan Hall 3206
Speaker: Carolyn Abbott (Columbia University) -