Geometry-Topology Archives for Academic Year 2013


The toric geometry of polygons in Euclidean space

When: Mon, September 10, 2012 - 3:00pm
Where: Math 1313
Speaker: John Millson (University of Maryland, College Park)
Abstract: see http://www.math.umd.edu/~karin/seminarabstracts.html

An abstract construction of the maximal extension for globally hyperbolic conformally flat space-times

When: Mon, September 17, 2012 - 3:00pm
Where: Math 1313
Speaker: Clara Rossi Salvemini (University of Avignon)
Abstract: A conformally flat space-time M of dim M>2, is a (G,X)-manifold. The model space X is the space of Einstein,
which is S^n \times R with the conformal class of the metric ds^2-dt^2, where ds^2 is the standard
metric on the sphere et dt^2 on R. The group G is the identity component of the group O(2,n), which is the group
of conformal diffeomorphisms of the Einstein's space.
The causal structure of a Lorentzian manifold is a conformal invariant, so we have a well
defined causal structure on M. We assume that this causal structure is globally hyperbolic.
We will define also the notion of unique maximal extension for these space-times.
We will make an abstract construction of this maximal extension starting from a Cauchy hypersurface of M which use only the (G,X)-structure of M. This allow us to characterize the maximal space-time by their causal boundary.
We will also show some results about the developing map of these space-times. In particular we have:
Every globally hyperbolic conformally flats maximal spaces-time M which have two lightlike geodesics freely homotopic with same ends is a finite quotient of the Einstein's space.

Some complexes of differential operators

When: Mon, September 24, 2012 - 3:00pm
Where: Math 1313
Room: Math 1313
Speaker: Katharina Neusser (Australian National University),
Abstract: We shall present a method for constructing complexes of invariant differential operators on manifolds endowed with various geometric structures. The geometric structures will mainly be certain bracket generating vector distributions, like for example a contact structure. For these structures the constructed complexes will give rise to fine resolutions of the sheaf of locally constant functions and so can serve as an alternative to the de Rham complex. In the case of parabolic geometries we recover the so called BGG complexes associated to the trivial representation.
Joint work with Robert Bryant, Michael Eastwood and Rod Gover.


Field Committee Meeting for Geometry-Topology

When: Mon, October 1, 2012 - 3:00pm
Where: Math 1313
Speaker: Jim Schafer (University of Maryland, College Park)

Dynamics on the PSL(2, C)-character variety of certain hyperbolic 3-manifolds

When: Mon, October 8, 2012 - 3:00pm
Where: Math 1313
Speaker: Michelle Lee (University of Maryland, College Park)
Abstract: The PSL(2, C)-character variety of a hyperbolic 3-manifold M is essentially the set of homomorphisms of the fundamental group of M into PSL(2, C), up to conjugacy. We will discuss the action of the group of outer automorphisms of the fundamental group act on this space. In particular, we will discuss how one can find domains of discontinuity for the action.

Field Committee Meeting for Geometry-Topology

When: Mon, October 15, 2012 - 3:00pm
Where: Math 1313
Speaker: Jim Schafer (University of Maryland, College Park) -

Cellular Automorphisms and Self Duality

When: Mon, October 22, 2012 - 3:00pm
Where: Math 1313
Speaker: Lowell Abrams (George Washington University) -
Abstract: Given a graph G cellularly embedded in a closed surface S, an automorphism of G is called a cellular automorphism of G in S when, loosely speaking, it takes facial boundary walks to facial boundary walks. I will describe how we constructed complete catalogs of all irreducible cellular automorphisms of the sphere, projective plane, torus, Klein bottle, and three-crosscaps surface for a particular notion of reducibility related to taking minors.

We have also determined concrete procedures sufficient for constructing all possible self-dual embeddings in any closed surface S given a catalog of all irreducible cellular automorphisms in S. I will illustrate by way of examples some of these procedures and some resulting self-dual graphs.

Counting Chord Diagrams

When: Tue, October 23, 2012 - 3:30pm
Where: MATH 2300
Speaker: Robert Penner (Aarhus University and California Institute of Technology) -
Abstract: A linear chord diagram on some number b of backbones is a collection of n chords with distinct endpoints attached to the interiors of b intervals.

Taking the intervals to lie in the real axis and the chords to lie in the upper half-plane associates a fatgraph to a chord diagram, which thus has its associated genus g. The numbers of connected genus g chord diagrams on b backbones with n chords are of significance in mathematics, physics and biology as we shall explain. Recent work using the topological recursion of Eynard-Orantin has computed them perturbatively via a closed form expression for the free energies of an Hermitian matrix model with potential V(x)=x^2/2-stx/(1-tx). Very recent work has moreover shown that the partition function satisfies a second order non-linear pde which gives a generalization of the Harer-Zagier equation that arises for one backbone.

CANCELLED because of university closure (Equivariant K-theory of actions of compact Lie groups with maximal rank isotropy subgroups)

When: Mon, October 29, 2012 - 3:00pm
Where: Math 1313
Speaker: Jose Manuel Gomez (Johns Hopkins University)
Abstract: In this talk we study the equivariant K-theory of a compact
Lie group G acting on a space X with maximal rank isotropy subgroups.
In particular we provide conditions that guarantee freeness over the
representation ring of G. Some applications related
to spaces of representations on Lie groups will be provided.

TBA

When: Mon, November 5, 2012 - 3:00pm
Where: Math 1313
Speaker: Scott Wolpert (UMCP) -

Kahler-Einstein (edge) theory and algberaic geometry

When: Mon, November 12, 2012 - 3:00pm
Where: Math 1313
Speaker: Yanir Rubinstein (University of Maryland, College Park)
Abstract: We highlight some problems in algebraic geometry related to Kahler-Einstein (edge) metrics. I will try to emphasize the ideas and give some examples.

Long-time analysis of 3 dimensional Ricci flow

When: Mon, November 19, 2012 - 3:00pm
Where: Math 1313
Speaker: Richard Bamler (Stanford)
Abstract: It is still an open problem how Perelman's Ricci flow with surgeries behaves for large times. For example, it is unknown whether surgeries eventually stop to occur and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$.

In this talk, I will present new tools to treat this question after providing a quick review of Perelman's results. In particular, I will show that in the case in which the initial manifold satisfies a certain purely topological condition, surgeries do in fact stop to occur after some time and the curvature is globally bounded by $C t^{-1}$. For example, this condition is satisfied by manifolds of the form $\Sigma \times S^1$ where $\Sigma$ is a surface of genus $\geq 1$.


The Sullivan-Riemann field of a simply connected closed manifold

When: Mon, November 26, 2012 - 3:00pm
Where: Math 1313
Speaker: Stephen Halperin (University of Maryland, College Park)
Abstract: posted at http://www.math.umd.edu/~karin/seminarpdfs.html

Skein algebras and the decorated Teichmuller space

When: Mon, December 3, 2012 - 3:00pm
Where: Math 1313
Speaker: Julien Roger (Rutgers University) -
Abstract: We construct the skein algebra of a punctured surface S based on framed
links and arcs in Sx[0,1]. We then describe the relationship between this
algebra and quantization of the decorated Teichmuller space, based on the
study of a collection of geodesic length identities in hyperbolic
geometry. This is joint work with T. Yang.

Long-time Behavior of Ricci Flows and Construction of Einstein Metrics

When: Fri, December 7, 2012 - 2:00pm
Where: Math 3206
Speaker: Richard Bamler (Stanford) -
Abstract: In this talk I will survey current results on the long-time existence and
behavior of Ricci flows in dimensions 2, 3 and higher. Moreover, I will
point out analogies with construction techniques for Einstein metrics.

In dimension 3, the Ricci flow together with a certain surgery process has
been used by Perelman to establish the Poincaré and Geometrization
Conjectures. Despite the depth of Perelman's result, a precise description
of the long-time behavior of this flow still does not exist. For example,
it has been unknown whether it suffices to carry out a finite number of
surgeries or whether the geometric decomposition of the manifold is
exhibited by the flow as $t \to \infty$. I will explain Perelman's result
and recent progress on this problem. I will then present long-time
existence results in dimensions 4 and higher and describe possible further
directions in this field.

Rigidity of actions on CAT(0) cube complexes

When: Mon, December 10, 2012 - 3:00pm
Where: Math 1313
Speaker: Talia Fernos (North Carolina State, Greensboro)
Abstract: Let G be a group acting non-elementarily by automorphisms on a finite dimensional CAT(0) cube complex. In a joint work with Indira Chatterji and Alessandra Iozzi, we prove the non-vanishing of second bounded cohomology of G with geometrically defined coefficients. From this we deduce super rigidity for actions of any irreducible lattice in a nontrivial product of locally compact groups.

Polynomial Pick forms for affine spheres and real projective polygons

When: Wed, January 23, 2013 - 2:00pm
Where: Math 3206
Speaker: Mike Wolf (Rice University) -
Abstract: (Joint work with David Dumas.) Convex real projective structures on surfaces, corresponding to discrete surface group representations into SL(3, R), have associated to them affine spheres which project to the convex hull of their universal covers. Such an affine sphere is determined by its Pick (cubic) differential and an associated Blaschke metric. As a sequence of convex projective structures leaves all compacta in its deformation space, a subclass of the limits is described by polynomial cubic differentials on affine spheres which are conformally the complex plane. We show that those particular affine spheres project to polygons; along the way, a strong estimate on asymptotics is found. As some of the background material is rich but outside the usual Riemannian geometric canon, we will spend substantial time explaining it.

Crooked surfaces and anti-de Sitter geometry

When: Mon, January 28, 2013 - 3:00pm
Where: Math 1313
Speaker: Bill Goldman
Abstract: Minkowski space is flat spacetime and is the Lorentzian analog of Euclidean
space; anti-de Sitter space has constant negative curvature and is analogous to
hyperbolic space. Crooked planes were defined by Drumm to bound fundamental
polyhedra in Minkowski space for Margulis spacetimes, and the analogous polyhedra
were defined by Danciger, Gueritaud and Kassel in anti-de Sitter geometry.
In this talk we leisurely expound anti-de Sitter geometry in terms of the group SL(2,R)
and show how the anti-de Sitter crooked planes are just the conformal extensions
(developed by Frances, Barbot, Charette, Drumm, Melnick and myself) invariant under the
involution defining anti-de Sitter geometry.

The Kunneth Theorem in equivariant K-theory

When: Mon, February 4, 2013 - 3:00pm
Where: Math 1313
Speaker: Jonathan Rosenberg (UMCP)
Abstract: First we explain Hodgkin's Kunneth Theorem for equivariant K-theory K^*_G (for G a connected compact Lie group) and then explain why it fails dramatically for G finite. Then we show how to correct the theorem when G is cyclic of order 2. This result will appear soon in Algebraic & Geometric Topology.

Talk canceled

When: Mon, February 11, 2013 - 3:00pm
Where: Math 1313
Speaker: Babak Modami
Abstract: The Weil-Petersson (WP) metric is an incomplete Riemannian metric on
the moduli space of Riemann surfaces with negative sectional
curvatures which are not bounded away from $0$. Brock, Masur and
Minsky introduced a notion of "ending lamination" for WP geodesic rays
which is an analogue of the vertical foliations of Teichm\"{u}ller
geodesics. In this talk we show that these laminations and the
associated subsurface coefficients can be used to determine the
itinerary of a class of WP geodesics in the moduli space. As a result
we give examples of closed WP geodesics staying in the thin part of of
the moduli space, geodesic rays recurrent to the thick part of the
moduli space and diverging geodesic rays. These results can be
considered as a kind of symbolic coding for WP geodesics.

Cutting sequences on Bouw-Möller surfaces

When: Mon, February 18, 2013 - 3:00pm
Where: Math 1313
Speaker: Diana Davis, Brown
Abstract: We will investigate a dynamical system that comes from geodesic trajectories on flat surfaces. We will start with the square torus and the regular octagon surface, and then discuss new results for Bouw-Möller surfaces, made from many polygons.

Homological stability of moduli spaces of manifolds

When: Mon, February 25, 2013 - 3:00pm
Where: Math 1313
Speaker: Soren Galatius, Stanford
Abstract: The moduli space of Riemann surfaces M_g parametrizes bundles of genus g surfaces. A classical theorem of J. Harer implies that the homology H_k(M_g) is independent of g, as long as g is large compared to k. In joint work with Oscar Randal-Williams, we establish an analogue of this result for manifolds of higher dimension: The role of the genus g surface is played by the connected sum of g copies of S^n \times S^n.

On holomorphic maps between ball quotients

When: Mon, March 4, 2013 - 3:00pm
Where: Math 1313
Speaker: Vincent Koziarz, Bordeaux
Abstract: We will expound on a joint result with N. Mok stating that holomorphic
surjective maps between compact ball quotients must have singular fibers
(in the non-equidimensional case). Examples of such surjective maps are
very rare and we will give a few leads towards their classification in
the case when fibers are 1-dimensional.

Totally geodesic curves and cohomology of complex hyperbolic manifolds

When: Mon, March 11, 2013 - 3:00pm
Where:
Speaker: Matthew Stover, Michigan
Abstract: Let M be a compact complex hyperbolic 2-manifold. I will discuss applications of Lefschetz-type theorems to the geometry and cohomology of M. When M is arithmetic and contains holomorphically immersed close totally geodesic curves, this gives a nice structure theorem for the cohomology of M reminiscent of results of Gelbart and Rogawski on cohomology coming from the theta correspondence, and our results have the added bonus of working in the noncongruence setting too. This is joint work with Ted Chinburg.

Poncelet's Porism and Algebraic Geometry

When: Mon, March 25, 2013 - 3:00pm
Where:
Speaker: Jean-Philippe Burelle (UMD)
Abstract: This talk is going to be an exposition of part of a paper by Griffiths and Harris using algebraic geometry to solve a classical geometry problem first investigated by Poncelet. This problem is easy to state but hard to solve using classical methods :

Given two conics in the real projective plane, is there a polygon that is inscribed in one and circumscribed to the other?

A new proof of Bowen's theorem on quasi-circles

When: Mon, April 1, 2013 - 3:00pm
Where: Math 1313
Speaker: Andy Sanders, UMD
Abstract: In 1979, Bowen proved that the Hausdorff dimension of the limit set of a quasi-Fuchsian group is equal to 1 if and only if the group is Fuchsian. Since then, many proofs and sweeping generalizations of this result have been given. We will present a new proof of this result which relies on the existence of equivariant minimal surfaces in hyperbolic 3-space.

Ending laminations for Weil-Petersson geodesics

When: Mon, April 8, 2013 - 3:00pm
Where: Math 1313
Speaker: Babak Modami, Yale
Abstract: The Weil-Petersson (WP) metric is an incomplete Riemannian metric on
the moduli space of Riemann surfaces with negative sectional
curvatures which are not bounded away from $0$. Brock, Masur and
Minsky introduced a notion of "ending lamination" for WP geodesic rays
which is an analogue of the vertical foliations of Teichm\"{u}ller
geodesics. In this talk we show that these laminations and the
associated subsurface coefficients can be used to determine the
itinerary of a class of WP geodesics in the moduli space. As a result
we give examples of closed WP geodesics staying in the thin part of of
the moduli space, geodesic rays recurrent to the thick part of the
moduli space and diverging geodesic rays. These results can be
considered as a kind of symbolic coding for WP geodesics.

Complete flat Lorentzian three-manifolds (joint with Dynamics)

When: Thu, April 18, 2013 - 2:00pm
Where: Math 1311
Speaker: Jeff Danciger (University of Texas, Austin) -
Abstract: A complete flat Lorentzian three-manifold is the quotient of the (2+1)-dimensional Minkowski space by a discrete group acting properly by affine O(2,1) transformations. In the interesting cases, the group acting is a free group and the quotient manifold is called a Margulis space-time. I will describe work in progress toward classifying the topology of Margulis space-times. In particular, when the O(2,1) part of the group action does not contain parabolics, we prove that the quotient manifold is a handle-body. The proof depends on a new properness criterion for free groups acting on Minkowski space and draws on ideas from anti de Sitter (AdS) geometry. This is joint work with François Guéritaud and Fanny Kassel.

Some aspects of three dimensional Lorentz dynamics

When: Mon, April 22, 2013 - 3:00pm
Where: Math 1313
Speaker: Charles Frances (University of Paris XI, Orsay) -
Abstract: It is a consequence of Gromov's theory of rigid geometric
structures that rigid structures with an automorphism group having
"complicated" dynamics are often locally homogeneous, at least on some open
dense subset. This principle is nicely illustrated by a theorem of Zeghib
describing all compact Lorentz 3-manifolds whose isometry group has a
noncompact connected component. The purpose of the talk is to show how new
phenomenas appear when the isometry group has an infinite discrete part.

Annular Twists and Bridge Numbers of Knots

When: Mon, April 29, 2013 - 3:00pm
Where:
Speaker: Ken Baker, Miami
Abstract: Parametrized by rational numbers, the various Dehn surgeries on a knot in
the 3-sphere each produce a new knot in another manifold. Generically, the
new knot is as "simple as possible" in this new manifold, though that's
not always the case. One measure of this simplicity is bridge number
(with respect to a Heegaard splitting). In this talk we'll survey recent
works with Cameron Gordon and John Luecke that these bridge numbers may
behave quite differently for integral and non-integral Dehn surgeries.

Forms, flows and physics

When: Mon, May 6, 2013 - 3:00pm
Where: Math 1313
Speaker: Ted Jacobson (UMCP, physics)
Abstract: Differential forms provide the ideal mathematical language
for aspects of physics that do not involve the spacetime metric.
I will discuss examples from Hamiltonian mechanics, relativistic
electrodynamics, and plasma physics, with the dual aim of
explaining some physics to mathematicians, and illustrating
how the physics is most easily and naturally understood using
this language.

Homological and Homotopical Dehn Functions Are Different

When: Mon, May 13, 2013 - 3:00pm
Where: Math 1313
Speaker: Noel Brady (University of Oklahoma and National Science Foundation)
Abstract: One of the fundamental problems introduced by Max Dehn in the study of finitely presented groups is the word problem. Given a finitely presented group and a word in the generators is there a procedure to determine if the word represents the identity element of the group. An isoperimetric function of a finitely presented group provides an upper bound on the number of relations that must be used to show that a word in the generators represents the identity. A Dehn function is an optimal isoperimetric function.

An isoperimetric function can be interpreted geometrically as providing an upper bound on the combinatorial area of a least area disk, which is bounded by a given loop in the Cayley complex of the finitely presented group. A homological version of this notion can be defined where one considers least area 2-chains, which are bounded by a given loop. We shall describe how to construct finitely presented groups whose homological Dehn functions are strictly smaller than their ordinary Dehn functions. This is joint work with Aaron Abrams, Pallavi Dani and Robert Young.