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.

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.

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.

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.

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.

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.

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.

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 3â€“manifolds. In this talk we describe recent developments in generalising Thurstonâ€™s machinery to (possibly convex) real projective structures. This project is a joint work with Sam Ballas.

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.

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.

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.

Where: Kirwan Hall 1310

Speaker: Ian Teixeira (University of Maryland) -