Geometry-Topology Archives for Academic Year 2016

Mass in Kahler geometry

When: Mon, September 14, 2015 - 3:00pm
Where: Math 1313
Speaker: Hans-Joachim Hein (UMD) -
Abstract: In General Relativity, isolated gravitating systems are modeled by complete Riemannian 3-manifolds asymptotic to flat R^3 at infinity. The "mass" of such a manifold is a real number defined in terms of the higher order asymptotics of the metric. As its name suggests, in the examples coming from physics, it represents the total mass of the gravitating system. From a mathematical point of view, it makes sense as an invariant of asymptotically Euclidean manifolds in all dimensions, and has been widely studied as such. We prove an explicit formula for the mass in the Kahler case, which implies the Positive Mass Theorem for Kahler manifolds. Joint work with Claude LeBrun.

Deforming Convex Projective Manifolds

When: Mon, September 21, 2015 - 3:00pm
Where: Math 1313
Speaker: Daryl Cooper (UCSB) -
Abstract: This talk concerns a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many generalized cusps. We extend a theorem of Koszul which asserts that for a compact manifold without boundary the holonomies of properly convex structures is an open subset of the representation variety.

Families of projective domains and neck-separating degenerations of convex RP^2 surfaces

When: Mon, September 28, 2015 - 3:00pm
Where: Math 1313
Speaker: John Loftin (Rutgers) -
Abstract: Given a bounded convex domain Omega in RP^n, Cheng-Yau provide a unique solution to the Monge-Ampere equation det v_{ij} = (-1/v)^{n+2}, for convex v and Dirichlet boundary value v = 0. This solution v then determines a unique hypersurface, a hyperbolic affine sphere, asymptotic to the boundary of the cone over Omega in R^{n+1}. The hyperbolic affine sphere carries tensors, the Blaschke metric and cubic form, which are invariant under special linear automorphisms of the cone. These tensors descend to Omega to be projectively invariant. Benoist-Hulin show that if Omega_i \to Omega in the Hausdorff topology, then these tensors converge in the local C^\infty topology. We discuss examples of convergence of projective domains, especially in the case of dimension 2.

A 2-dimensional hyperbolic affine sphere carries a conformal structure induced by the metric, and the cubic tensor is equivalent to a holomorphic cubic differential. A quotient of such a domain Omega by a subgroup of PGL(3,R) acting discretely and properly discontinuously then is called a real projective surface, and the conformal structure and cubic differential descend to the quotient. We will discuss a recent result relating degenerations of convex real projective surfaces along necks in terms of the geometry of the bundle of regular cubic differentials over the Deligne-Mumford compactification of the moduli space of Riemann surfaces. In addition to the Benoist-Hulin convergence results mentioned above, the proof also uses analytic techniques of Dumas-Wolf and Wolpert.

Geodesics of positive Lagrangians and almost Calabi-Yau reduced spaces

When: Mon, October 12, 2015 - 3:00pm
Where: Math 1313
Speaker: Amitai Yuval (Hebrew University) -
Abstract: The space of positive Lagrangians in an almost Calabi-Yau manifold is an open set in the space of all Lagrangian submanifolds. A Hamiltonian isotopy class of positive Lagrangians admits a natural Riemannian metric, which gives rise to a notion of geodesics. The geodesic equation is a fully non-linear degenerate elliptic PDE, and it is not known yet whether the initial value problem and boundary problem have solutions in general.

We will talk about Hamiltonian classes of positive Lagrangians which are invariant under a Lie group Hamiltonian action. Such a Hamiltonian class is isometric to the corresponding class in the symplectic reduced space, which has a natural almost Calabi-Yau structure. We will show that when the symplectic reduced space is of real dimension 2, both the initial value problem and boundary problem have unique solutions. As examples, we will discuss Hamiltonian classes of symmetric positive Lagrangians in toric Calabi-Yau manifolds and Milnor fibers. As time permits, we will show as an application that in these cases, the Riemannian metric induces a metric space structure on every Hamiltonian isotopy class, and that the obtained metric spaces can be embedded isometrically in L^2 spaces.

All the funny words will be explained in the talk. Joint work with Jake Solomon.

New developments in strongly positive curvature

When: Mon, October 26, 2015 - 3:00pm
Where: Math 1313
Speaker: Renato Bettiol (UPenn) -
Abstract: Strongly positive curvature is an intermediate condition between positive-definiteness of the curvature operator and positive sectional curvature (sec > 0), defined in terms of modifying the curvature operator with a 4-form to make it positive-definite. It stems from the work of Thorpe in the 1970s, but has also been implicitly studied by others. In this talk, I will report on a recent classification result for homogeneous manifolds with strongly positive curvature, constructions of manifolds with strongly nonnegative curvature, and work in progress using the Bochner technique to find topological obstructions. This is joint work with R. Mendes (WWU Munster).

Symmetric Spaces: Lie Triple Systems

When: Tue, October 27, 2015 - 2:00pm
Where: Math 0104
Speaker: Ryan Hunter (UMD) -

Kahler-Einstein metrics: from cones to cusps

When: Mon, November 2, 2015 - 3:00pm
Where: Math 1313
Speaker: Henri Guenancia (Stony Brook) -
Abstract: In this talk, I will explain the following result and outline its proof: Let X be a compact Kahler manifold and D a smooth divisor such that K_X + D is ample. Then the negatively curved Kahler-Einstein metric with cone angle beta along D converges to the cuspidal Kahler-Einstein metric of Tian-Yau when beta tends to zero.

Maximal SO(2,3) surface group representations and Labourie's conjecture

When: Mon, November 9, 2015 - 3:00pm
Where: Math 1313
Speaker: Brian Collier (UIUC) -
Abstract: The nonabelian Hodge correspondence provides a homeomorphism between the character variety of surface group representations into a real Lie group G and the moduli space of G-Higgs bundles. This homeomorphism however breaks the natural mapping class group action on the character variety. Generalizing techniques and conjectures of Labourie for Hitchin representations, we restore the mapping class group symmetry for all maximal SO(2, 3) = PSp(4, R) surface group representations. More precisely, we show that for each maximal SO(2, 3) representation there is a unique conformal structure in which the corresponding equivariant harmonic map to the symmetric space is a conformal immersion, or, equivalently, a minimal immersion. This is done by exploiting finite order fixed point properties of the associated maximal Higgs bundles.

Complex deformations of Anosov representations

When: Mon, November 23, 2015 - 3:00pm
Where: Math 1313
Speaker: Andy Sanders (UIC) -
Abstract: An Anosov representation of a hyperbolic surface group is a homomorphism from the surface group into a semi-simple Lie group which satisfies a certain dynamical property: from this property one deduces that Anosov representations are discrete, faithful and the set of all Anosov representations is an open subset of the space of all homomorphisms. In recent years, Guichard-Wienhard produced examples of co-compact domains of discontinuity for Anosov representations, which lie in various homogeneous spaces, thus giving an answer to the question of whether or not Anosov representations appear as monodromies of locally homogeneous geometric structures on manifolds. In this talk, which comprises joint work with David Dumas, I will discuss some of the complex analytic features of these locally homogeneous geometric manifolds in the case the relevant homogeneous space is a generalized flag variety. In particular, we will give sufficient conditions to compute the space of all infinitesimal deformations of the complex manifold underlying these manifolds. Time permitting, we will discuss the problem of deforming a pair (M,Z) where M is a holomorphic locally homogeneous manifold and Z is a complex sub-manifold and indicate an application to the study of Anosov representations.

Asymptotically Conical Calabi-Yau manifolds

When: Mon, November 30, 2015 - 3:00pm
Where: Math 1313
Speaker: Ronan Conlon (UQAM) -
Abstract: Asymptotically Conical (AC) Calabi-Yau manifolds are Ricci-flat Kahler manifolds that are modelled on a Ricci-flat Kahler cone at infinity. I will describe a method to determine every AC Calabi-Yau manifold modelled on some given Ricci-flat Kahler cone. I will then present some examples. This is joint work with Hans-Joachim Hein (University of Maryland).

Straightening out degeneracy in CR geometry: When can it be done?

When: Mon, December 7, 2015 - 3:00pm
Where: Math 1313
Speaker: Curtis Porter (Texas A&M) -
Abstract: CR geometry studies boundaries of domains in C^n and their generalizations. A central role is played by the Levi form L of a CR manifold M, which measures the failure of the CR bundle to be integrable, so that when L has a nontrivial kernel of constant rank, M is foliated by complex manifolds. If the local transverse structure to this foliation still determines a CR manifold N, then we say M is CR-straightenable, and the Tanaka-Chern-Moser classification of CR hypersurfaces with nondegenerate Levi form can be applied to N. It remains to classify those M for which L is degenerate and no such straightening exists. This was accomplished in dimension 5 by Ebenfelt, Isaev-Zaitzev, and Medori-Spiro. I will discuss their results as well as my recent progress on the problem in dimension 7 (

Character Varieties of Free Groups are Gorenstein but not always Factorial

When: Mon, December 14, 2015 - 3:00pm
Where: Math 1313
Speaker: Sean Lawton & Chris Manon (GMU) -
Abstract: Fix a rank g free group F and a connected reductive complex algebraic group G. Let X(F,G) be the G-character variety of F. When the derived subgroup DG in G is simply connected we show that X(F,G) is factorial (which implies it is Gorenstein), and provide examples to show that when DG is not simply connected X(F,G) need not even be locally factorial. Despite the general failure of factoriality of these moduli spaces, using different methods, we show that X(F,G) is always Gorenstein.

Gerbes on G2 manifolds

When: Mon, February 8, 2016 - 3:00pm
Where: Math 1313
Speaker: Goncalo Oliveira (Duke) -
Abstract: On a projective complex manifold, the abelian group of divisors maps surjectively onto that of holomorphic line bundles (the Picard group). I shall explain a funny analogue of this for G2 manifolds using coassociative submanifolds to define an analogue of Div, and a gauge theoretical equation for a connection on a gerbe to define an analogue of Pic.

Nahm transform for spatially periodic instantons

When: Tue, February 16, 2016 - 10:00am
Where: Math 1310
Speaker: Benoit Charbonneau (University of Waterloo) -
Abstract: In joint work with Jacques Hurtubise, we show that the Nahm transform sending spatially periodic instantons (instantons on the product of the real line and a three-torus) to singular monopoles on the dual three-torus is indeed a bijection as suggested by the famous Nahm heuristic. In the process, we show how the Nahm transform intertwines to a Fourier-Mukai transform via Kobayashi-Hitchin correspondences.

Proper affine actions (and Anosov representations)

When: Mon, February 22, 2016 - 3:00pm
Where: Math 1313
Speaker: Nicolaus Treib (U Heidelberg) -
Abstract: In contrast to discrete groups of Euclidean isometries, actions of discrete groups of affine transformations are still not fully understood. Margulis showed that there exist free groups of affine transformations acting properly discontinuously, giving rise to manifolds now known as Margulis spacetimes. We will mainly be concerned with the question of characterizing properly discontinuous affine actions. Namely, in dimension 3, we give a criterion for the action of an affine deformation of a convex cocompact surface group to be proper. We will outline a new proof of this criterion, as well as possible generalizations.

Massive data analysis via differential geometry

When: Mon, February 29, 2016 - 3:00pm
Where: Math 1313
Speaker: Hau-tieng Wu (Toronto) -
Abstract: The exponential growth of massive data streams is nowadays everywhere, and has been attracting increasing interest. In addition to its size, this data is complex and multi-modal. To handle this kind of datasets, of particular importance is an adaptive model, as well as innovative acquisition of intrinsic features/structure hidden in the massive data sets. In this talk, I will discuss how to apply the knowledge in differential geometry to model and analyze massive datasets in different fields. In particular, I will discuss algorithms like graph connection Laplacian and vector diffusion maps, and their theoretical justification based on the spectral geometry. I will also discuss at least one of the following applications: cryo-electron microscope, phase retrieval and sleep dynamics analysis.

Bi-Lagrangian structures and Teichmuller theory

When: Fri, March 4, 2016 - 11:00am
Where: Math 1313
Speaker: Brice Loustau (IMPA) -
Abstract: A bi-Lagrangian structure is given by two transverse Lagrangian foliations in a symplectic manifold. We study these structures in the complex setting, in particular in the complexification of a Kahler manifold. We also show that bi-Lagrangian structures are relevant in studying moduli spaces in Teichmuller theory, in particular quasi-Fuchsian space. This is joint work with Andy Sanders.

The Area of Convex Projective 2-Orbifolds

When: Mon, March 7, 2016 - 3:00pm
Where: Math 1313
Speaker: Ilesanmi Adeboye (Wesleyan University) -
Abstract: The Hilbert metric on a bounded convex set determines an area 2-form. One of the coordinates used by Fock and Goncharov assigns to an ideal triangle T a shape parameter t >0. We obtain a lower bound for the area of T in terms of t.

Weil-Petersson class Teichmuller space

When: Mon, March 21, 2016 - 3:00pm
Where: Math 1313
Speaker: David Radnell (Aalto University) -
Abstract: Conformal field theory (CFT) is a mathematically rich two-dimensional quantum field theory that appears in statistical mechanics and string theory. The analytic and geometric structures are deeply tied to the infinite-dimensional Teichmuller theory of bordered Riemann surfaces, which we are using to develop a rigorous analytic setting for CFT.

Moreover, ideas from CFT have led us to various results in Teichmuller theory. In particular, based on the recently developed theory of L^2 Beltrami differentials, we construct a refined Teichmuller space of bordered Riemann surfaces for which the Weil Petersson metric converges.

Chord shortening flow and a theorem of Lusternik-Schnirelman

When: Mon, March 28, 2016 - 3:00pm
Where: Math 1313
Speaker: Martin Li (CUHK) -
Abstract: The celebrated Lusternik-Schnirelman theorem asserts that any Riemannian 2-sphere contains at least three distinct simple closed geodesics. The original proof uses a discrete curve shortening process first introduced by Birkhoff who showed the existence of at least one closed geodesics using min-max argument. For a bounded convex planar domain, Lusternik and Schnirelman asserts that there exist at least two chords meeting the boundary orthogonally. As in the closed case, a serious difficulty is to keep the curve simple (i.e. embedded) under the curve shortening process. In this talk, we introduce a new geometric flow which gives a simple proof of the Lusternik-Schnirelman theorem. This is joint work with Chi-fai Chau.

no seminar due to department review

When: Mon, April 4, 2016 - 3:00pm
Where: Math 1313
Speaker: () -

Subtleties of H^3 and twisted K-theory for compact Lie groups

When: Mon, April 11, 2016 - 3:00pm
Where: Math 1313
Speaker: Jonathan Rosenberg (UMD) -
Abstract: We explicitly compute the map on H^3 induced by a covering of compact simple Lie groups. The result is complicated and quite surprising. We also delve further into the twisted K-theory of compact Lie groups, previously studied by Moore-Maldacena-Seiberg, Hopkins, Braun, and Douglas, and the connection between Langlands duality and T-duality, studied by Daenzer-Van Erp and Bunke-Nikolaus. This is joint work with Mathai Varghese.

Curves, pants, and 3-manifolds

When: Mon, April 25, 2016 - 3:00pm
Where: Math 1313
Speaker: Tarik Aougab (Brown) -
Abstract: For S a surface of negative Euler characteristic, the curve complex of S is a simplicial complex encoding deep information about the algebra of the mapping class group of S, the geometry of the Teichmuller space of S, and properties of hyperbolic 3-manifolds which fiber over the circle with fiber S. For instance, there is a host of useful theorems which relate combinatorial properties of this and other complexes (such as the "pants complex" of S) to various geometric properties of hyperbolic 3-manifolds. Unfortunately, almost all of these theorems involve complicated error terms which grow somewhat mysteriously with the Euler characteristic of S, and this limits their use in many applications. We determine the growth rates of these errors, and as a result we obtain several concrete and effective relationships between these complexes and the mapping class group/Teichmuller space/hyperbolic 3-manifolds. This represents joint work with Samuel Taylor and Richard Webb.

ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface

When: Mon, May 9, 2016 - 3:15pm
Speaker: Lorenzo Foscolo (Stony Brook) -
Abstract: The Kummer construction of Kahler Ricci-flat metrics on the K3 surface provides the prototypical example of the formation of orbifold singularities in non-collapsing sequences of Einstein 4-manifolds. Much less is known about the structure of the singularities forming along sequences of collapsing Einstein metrics. I will describe the construction of large families of Ricci-flat metrics on the K3 surface collapsing to the quotient of a flat 3-torus by an involution. The collapse occurs with bounded curvature away from finitely many points. The geometry around these points is modelled by ALF gravitational instantons.