Geometry-Topology Archives for Academic Year 2020


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.

Moduli spaces, uniformization and singularities

When: Mon, December 9, 2019 - 3:00pm
Where: Kirwan Hall 3206
Speaker: Xuwen Zhu (University of California, Berkeley) -
Abstract: The classical uniformization theorem transforms the study of moduli spaces of marked Riemann surfaces into the study of constant curvature metrics with singularities. I will give a survey on constant curvature metrics with cusp and conical singularities, including works joint with Richard Melrose, Rafe Mazzeo and Bin Xu, where new analytic tools have been developed to understand the uniformization problem. "Resolution of singularities" is the key idea in the analysis, which can be seen as an analogue of the Deligne--Mumford compactification of Riemann moduli spaces.

Ordering actions on metric spaces

When: Fri, December 13, 2019 - 11:00am
Where: Kirwan Hall 3206

Speaker: Carolyn Abbott (Columbia University) -


Abstract: Geometric group theory is the study of groups through their (isometric) actions on metric spaces. Given a finitely generated group, it admits many different actions on different metric spaces, and they are not all equally useful for studying the algebra of the group. For example, every group acts isometrically on a point, but this action provides no information about the group. In this talk, I will describe how to put a partial order on the set of isometric actions of a given group on metric spaces that encodes how much information about the group the action provides, turning this set of actions into a poset. I will focus on actions on metric spaces which are negatively curved, that is, metric spaces whose geometry is similar to that of the hyperbolic plane or a tree. After adding this restriction on the geometry of the space (as well as a reasonable restriction on the action), I will give several results about the structure of this poset for the class of acylindrically hyperbolic groups. This is a large and diverse collection of groups including infinite mapping class groups, non-elementary hyperbolic and relatively hyperbolic groups, most fundamental groups of 3--manifolds, and the Cremona group, among many others.

Coordinates for real projective structures using ideal triangulations.

When: Mon, February 3, 2020 - 3:00pm
Where: Kirwan Hall 3206
Speaker: Alex Casella (Florida State University) -
Abstract: Since the work of Thurston in the late 1970s, ideal triangulations are known to be a great tool in the study of (possibly incomplete) hyperbolic structures on 3–manifolds. In this talk we describe recent developments in generalising Thurston’s machinery to (possibly convex) real projective structures. This project is a joint work with Sam Ballas.

Notions of hyperbolicity for affine and projective structures

When: Mon, February 10, 2020 - 3:00pm
Where: Kirwan 3206
Speaker: Bill Goldman (University of Maryland) -
Abstract: Based on the holomorphic properties of bounded domains in
complex spaces, Caratheodory and Kobayashi developed a theory of intrinsic metrics
for complex manifolds. Later Kobayashi, Vey and Wu developed an analogous theory
of intrinsic pseudodistances for flat affine and projective structures.

One consequence of the work of Kobayashi and Vey is the following.
Say that an affine manifold is "completely incomplete" if it possesses no
complete geodesic (this is the opposite of geodesic completeness).

Then a closed affine manifold is completely incomplete if and only if
it is a. quotient of a sharp convex cone.

The heights theorem for infinite Riemann surfaces

When: Mon, February 17, 2020 - 3:00pm
Where: Kirwan 3206
Speaker: Dragomir Saric (City University of New York) -
Abstract: Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first kind. In fact, we establish a more general result: the horizontal map which assigns to each integrable holomorphic quadratic differential a measured lamination obtained by straightening the horizontal trajectories of the quadratic differential is injective for an arbitrary Riemann surface with a conformal hyperbolic metric. This was established by Strebel in the case of the unit disk.
When a hyperbolic surface has a bounded geodesic pants decomposition, the horizontal map assigns a bounded measured lamination to each integrable holomorphic quadratic differential. When surface has a sequence of closed geodesics whose lengths go to zero, then there exists an integrable holomorphic quadratic differential whose horizontal measured lamination is not bounded. We also give a sufficient condition for the non-integrable holomorphic quadratic differential to give rise to bounded measured laminations.




Positive scalar curvature on pseudomanifolds

When: Mon, February 24, 2020 - 3:00pm
Where: Kirwan 3206
Speaker: Jonathan Rosenberg (University of Maryland) -
Abstract:
Famous results by Gromov-Lawson and Stolz completely characterize what
simply connected spin manifolds (in dimensions 5 and up) admit Riemannian metrics of positive
scalar curvature. I describe joint work with Boris Botvinnik and Paolo Piazza in
which we solve a similar problem for certain simply connected spin pseudomanifolds
(of depth one) with Riemannian metrics well adapted to the singularity structure.

This problem involves a blend of several very different techniques:
index theory on singular spaces, use of O'Neill's formulas for curvature of Riemannian
submersions, and bordism and surgery theory.


SU(2) : A Mathematical Case Study

When: Fri, February 28, 2020 - 2:00pm
Where: Kirwan Hall 1310
Speaker: Ian Teixeira (University of Maryland) -