Geometry-Topology Archives for Academic Year 2016


Introduction to geodesic flows and hyperbolicity

When: Tue, August 2, 2016 - 12:30pm
Where: Kirwan Hall 1308
Speaker: Sarah Bray (University of Michigan) - http://www-personal.umich.edu/~brays/
Abstract: In this minicourse, I'll gently introduce the Patterson-Sullivan program for
studying hyperbolic geodesic flows, and survey some modern contexts to which the program has been applied.

Ergodic geometry: Introduction to Sullivan-Patterson densities

When: Thu, August 4, 2016 - 1:30pm
Where: Kirwan Hall 1308
Speaker: Sarah Bray (University of Michigan) - http://www-personal.umich.edu/~brays/
Abstract: In this minicourse, I'll gently introduce the Patterson-Sullivan program for
studying hyperbolic geodesic flows, and survey some modern contexts to which the program has been applied.

Entropy-maximizing measures: examples beyond uniform hyperbolicity

When: Tue, August 9, 2016 - 12:30pm
Where: Kirwan Hall 1308
Speaker: Sarah Bray (University of Michigan) - http://www-personal.umich.edu/~brays/
Abstract: In this minicourse, I'll gently introduce the Patterson-Sullivan program for
studying hyperbolic geodesic flows, and survey some modern contexts to which the program has been applied. The lecture titles are:

1. Introduction to geodesic flows and hyperbolicity

2. Ergodic geometry: Introduction to Sullivan-Patterson densities

3. Entropy-maximizing measures: examples beyond uniform hyperbolicity

Meeting to discuss the MATH 740 syllabus

When: Mon, August 29, 2016 - 3:15pm
Where: Kirwan Hall 1313
Speaker: () -
Abstract: The current syllabus for MATH 730-734 is at
https://www-math.umd.edu/graduate/current-students/examinations.html?id=121 .
There is a proposal from Bill Goldman to change the syllabus for 740. This should be discussed.

Conformal dynamics of simple Lie groups on Lorentzian manifolds

When: Mon, September 12, 2016 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Vincent Pecastaing (University of Maryland ) - http://www.normalesup.org/~pecastai/en/
Abstract: A result of R. Zimmer going back to the 1980's asserts that up to local isomorphism, SL(2,R) is the only non-compact simple Lie group that can act by isometries on a Lorentzian manifold of finite volume. Later, Gromov characterized the geometry of the manifolds where such dynamics occur. In this talk, I will discuss the analogous problem for conformal dynamics of simple Lie groups on compact Lorentzian manifolds. A larger amount of groups appears, and many of them can act on various manifolds. Nevertheless, we will see that the local geometry is prescribed by the existence of a non-compact simple group of conformal transformations. I will also explain the implications of this result on the general form of the conformal group of a compact Lorentzian manifold.


Deformation theory of equivariant holomorphic immersions and rigidity of G-opers

When: Mon, September 19, 2016 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Andrew Sanders (University of Heidelberg) - http://homepages.math.uic.edu/~andysan/
Abstract: Given a compact complex manifold Y, a complex Lie group G, and a G-homogeneous space N, we wish to study the deformation theory of pairs of holomorphic immersions of the universal cover of Y into N which are equivariant for a homomorphism of the fundamental group of Y into G. Interpreting this question in the language of holomorphic, flat principal bundles over Y with a transverse reduction of structure, we compute the space of infinitesimal deformations, which appears as the hypercohomology of a complex of locally free sheaves over Y.

As an application, we will compute the space of infinitesimal deformations of a G-oper, which are certain equivariant immersions of the universal cover of a compact Riemann surface into the variety of complete flags associated to a simple, complex Lie group.
This is joint work with David Dumas.

Schiffer variations and Abelian differentials

When: Mon, September 26, 2016 - 3:15pm
Where: 1313 Kirwan Hall
Speaker: Scott Wolpert (University of Maryland) - http://www.math.umd.edu/~swolpert/
Abstract: The study of the Teichmuller geometry and dynamics of the moduli space of curves has been in a period of high activity for over a decade. I will begin with a description of the Teichmuller metric and deformations of translation surfaces. This will be followed by a description of the Eskin-Mirzakhani-Mohammadi theorem (the main citation for Mirzakhani’s Fields medal). This will be followed by a cut-and-paste (Cech style) description of deformations of translation surfaces. This will be followed by a description of Schiffer’s Cech style argument for the variation of Abelian differentials. I use the latter to present a second order variation formula for the Riemann period matrix.

Timelike spaces

When: Mon, October 3, 2016 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Athanase Papadopoulos (University of Strasbourg) - http://www-irma.u-strasbg.fr/~papadop/
Abstract: I will talk about timelike spaces, with a particular stress on the timelike analogues of the Funk and the Hilbert metrics.

Schottky groups and maximal representations

When: Mon, October 10, 2016 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Jean-Philippe Burelle (University of Maryland) - http://www.math.umd.edu/~jburelle/
Abstract: Fuchsian Schottky groups are constructed by choosing disjoint half planes
in the hyperbolic plane and pairing them with isometries. Groups defined this way are
free, act properly discontinuously on H^2, and the quotient is a hyperbolic surface
with non empty boundary. We generalize this construction to the setting of the symplectic
group Sp(2n,R) acting on real projective space (and more generally, semisimple Lie groups
of Hermitian type). We establish a connection between these generalized Schottky groups
and maximal representations of fundamental groups of surfaces with boundary.

The geometry of quasi-Hitchin symplectic Anosov representations

When: Mon, October 24, 2016 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Sara Maloni (University of Virginia) - http://www.people.virginia.edu/~sm4cw/Welcome.html
Abstract: After revising the background theory of symplectic Anosov representations and their domains of discontinuity, we will focus on our joint work in progress with Daniele Alessandrini and Anna Wienhard. In particular, we will describe partial results about the homeomorphism type of the quotient of the domain of discontinuity for quasi-Hitchin representations in Sp(4,C) acting on the Lagrangian space Lag(C^4).

Marked length spectrum rigidity for surfaces with singularities

When: Mon, October 31, 2016 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Dave Constantine (Wesleyan University ) - http://dconstantine.web.wesleyan.edu/
Abstract: The marked length spectrum rigidity question is as follows: Suppose we know for some space the function which assigns to each element of the fundamental group the length of the shortest curve in its free homotopy class. Does this determine the space up to isometry. In this talk I will prove that the answer is yes for closed surfaces carrying metrics of nonpositive curvature and having some cone singularities. The argument involves synthesizing a number of previous results on this problem, so we will get a survey of the question as a whole along the way.

The depth and LS category of a topological space

When: Mon, November 7, 2016 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Steve Halperin (University of Maryland) - http://www.math.umd.edu/~shalper/
Abstract: In analogy , We construct a '"geometric" completion of the rational loop space homology of a connected CW complex X. In analogy with the definition of the depth of an augmented ring via the Ext functor we use this to define the depth of a connected CW complex X and establish the
Theorem: Depth X

The Pressure Metric Geometry of Spaces of Metric Graphs

When: Mon, November 14, 2016 - 3:15pm
Where: Kirwan 1313
Speaker: Lien-Yung Kao (University of Notre Dame) - http://www3.nd.edu/~lkao/Kao/HOME.html
Abstract:In this talk I will first recall the short history of dynamical- system-theoretically defined Riemannian metrics on deformation spaces – pressure metrics. Then I will focus on a moduli space of metric graphs and discuss two pressure metrics on it. These two pressure metrics could be thought of as analogues of Weil-Petersson metrics (or Thurston’s Riemannian metrics) for spaces of metric graphs. In particular,
I will discuss and compare Riemannian geometric features of these two metrics with the “classic” Weil-Petersson metric in Teichm ̈ller theory. This work is motivated by an earlier work of Pollicott and Sharp.


Operads with homological stability

When: Mon, November 21, 2016 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Sarah Yeakel (University of Maryland) - http://www2.math.umd.edu/~syeakel/index.html
Abstract: For a carefully constructed operad M of surfaces, Tillmann showed that algebras over M group complete to infinite loop spaces. This result relies, in part, on Harer's homological stability theorem for mapping class groups of surfaces. We will review Tillmann's result and provide a more general framework which shows that operads satisfying a certain homological stability condition detect infinite loop spaces. This is joint work with M. Basterra, I. Bobkova, K. Ponto, and U. Tillmann.

The dynamics of classifying geometric structures

When: Mon, November 28, 2016 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Bill Goldman (University of Maryland) - http://math.umd.edu/~wmg
Abstract: The general theory of locally homogeneous geometric
structures (flat Cartan connections) was first formulated by
Ehresmann in 1936.
Their classification leads to interesting dynamical systems.
For example, classifying Euclidean geometries on the torus leads to
the familiar action of the modular group on the upper half-plane
which is dynamically uninteresting.
In contrast, classification of other simple geometries on on the torus
leads to the standard linear action of SL(2,Z) on the plane,
with chaotic dynamics and a pathological quotient space.
Other examples arise from number theory,
where the moduli space is described by the nonlinear
symmetries of cubic equations like Markoff’s famous Diophantine
equation x^2 + y^2 + z^2 = x y z.
Both trivial and chaotic dynamics arise simultaneously here,
and relate to possibly singular hyperbolic metrics on surfaces of Euler
characteristic equals ­1 (joint work with Ser Peow Tan, Greg McShane
and George Stantchev).
I will also describe a general program, bringing in Teichmueller
dynamics, which is recent joint work with Giovanni Forni).

SO(p,q) surface group representations and Higgs bundles

When: Mon, December 5, 2016 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Brian Collier (University of Maryland) -
Abstract:

Equivariant A-theory

When: Mon, January 30, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Mona Merling (Johns Hopkins) - http://www.math.jhu.edu/~mmerling/
Abstract: Waldhausen’s A-theory is an extension of Quillen's K-theory of rings and exact categories to categories of spaces. A-theory is central to the classification of diffeomorphisms of manifolds -- for a manifold M, its A-theory A(M) encodes its pseuso-isotopy theory. I will talk about joint work with C. Malkiewich on a program on equivariant A-theory which we expect to be related to equivariant pseudo-isotopies of G-manifolds.

Nahm transformation for parabolic harmonic bundles

When: Mon, February 20, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Szilard Szabo (Budapest University of tecnology and economics) - http://math.bme.hu/~szabosz/
Abstract: After an introduction to parabolic harmonic bundles on curves
and their moduli spaces, we will describe an integral transform for
parabolic harmonic bundles on the projective line and explain some of its
properties.

Compactifications and Compactifying Spaces of Locally Compact Hausdorff Spaces

When: Mon, February 27, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Marshall Cohen (Cornell University) - https://www.math.cornell.edu/m/People/bynetid/mmc25
Abstract: If the locally compact Hausdorff space X is a (necessarily open) subspace of the compact Hausdorff space Y then Y is a compactification of X and Y \ X is a compactifying space of X. We return to the classical question: "What are the possible compactifying spaces of a given space X?" We give a unified treatment of this question and indicate some surprising outcomes. (As a warmup, ask yourself what the compactifying spaces are of the half-open interval [0, 1). ) This is joint work with Ethan Akin of CUNY.

On the equivalence between Bowditch's Q-condition and the primitive stable condition

When: Mon, March 6, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Binbin Xu (KIAS) -
Abstract: The group Out(F_2) of outer-automorphisms of the rank 2 free group acts naturally on the PSL(2,C)-character variety of F_2. To study the dynamical property of Out(F_2)-action, Bowditch's Q-condition and the primitive stable condition on a representation from F_2 to PSL(2,C) have been introduced by Bowditch (and generalized by Tan-Wong-Zhang) and by Minsky, respectively. Each condition characterizes an open subset of the character variety on which Out(F_2) acts properly discontinuously. These two open sets are both candidates for the maximal domain of discontinuity for the Out(F_2)-action. In a joint work with Jaejeong Lee, we show that these two conditions are equivalent to each other.

The geometry of maximal representations in SO(2,n+1)

When: Mon, March 13, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Jeremy Toulisse (University of Southern California) - http://dornsife.usc.edu/jtoulisse/
Abstract: Let S be a closed oriented surface of genus at least 2. The notion of maximal representations of the fundamental group of S into SO(2,n+1) naturally generalizes Fuchsian representations into PSL(2,R). Given such a maximal representation, we show the existence of a unique equivariant maximal space-like embedding of the universal cover of S into the pseudo-hyperbolic space H^{2,n}. As a consequence, we prove the existence of a unique equivariant minimal surface in the associated Riemannian symmetric space. The proof relies on the theory of Higgs bundles and the non-abelian Hodge correspondance. If time permits, I'll explain how this construction gives a way to construct geometric structures associated to maximal representations. This is a joint work with Brian Collier and Nicolas Tholozan.



Large scale geometry of homeomorphism groups

When: Mon, March 27, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Christian Rosendal (University of Maryland ) - http://homepages.math.uic.edu/~rosendal/
Abstract: We show how every topological group is canonically equipped with a coarse geometric structure. For a compact manifold M, the coarse structure on the identity component of the homeomorphism group, Homeo_0(M), is induced by the so called fragmentation metric. We connect the non-triviality of the geometry to the size of the fundamental group of M and also show how it relates to the group of lifts to a normal cover. (Part of the talk is based on joint work with K. Mann.)

On type preserving representations of the four punctured sphere

When: Mon, April 3, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Tian Yang (University of Stanford) - http://web.stanford.edu/~yangtian/
Abstract:
We give counterexamples to a conjecture of Bowditch that if a non-elementary type-preserving representation of a punctured surface group into PSL(2,R) sends every non-peripheral simple closed curve to a hyperbolic element, then ρ must be Fuchsian. The counterexamples come from relative Euler class ±1 representations of the four-punctured sphere group. As a related result, we show that the mapping class group action on each non-extremal component of the character space of type-preserving representations of the four-punctured sphere group is ergodic, confirming a conjecture of Goldman in this case. The main tool we use is the lengths coordinates of the decorated character spaces defined by Kashaev.

Representations of surface groups with finite mapping class group orbits.

When: Mon, April 10, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Ramanujan Santharoubane (University of Virginia) -
Abstract: This talk deals with point on character/representation variety of surface groups which have finite orbit under the mapping class group. A simple remark is that a point on the character/representation which has finite image (as a morphism), has finite orbit under the mapping class group. Hence we study the following questions :
1)Can we have a point on the character variety of surface group which have finite orbit under the mapping class group but with infinite image?
2)Can we have a point on the representation variety of surface group which have finite orbit under the mapping class group but with infinite image?
I will show that TQFT representations of surface groups give a positive answer to Question 1). Then I will show that Question 2) has a negative answer.
This is a joint work with Indranil Biswas, Thomas Koberda and Mahan MJ.

Positively ratioed representations and their properties

When: Mon, April 17, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Giuseppe Martone (University of Southern California) -
Abstract: For a hyperbolic metric on a surface, a classical construction of Bonahon describes the length of closed geodesics in terms of their geometric intersection number with a certain geodesic current. We generalize this picture to higher Teichm\"uller theory and to Anosov representations satisfying an additional positivity property. We use this tool to relate the entropy and the systole length of such a positively ratioed representation. This is joint work with Tengren Zhang.


Square roots of Thompson's group F

When: Mon, April 24, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Thomas Koberda (University of Virginia) - http://faculty.virginia.edu/Koberda/
Abstract: I will discuss square roots of Thompson's group F, which are certain two-generator subgroups of the homeomorphism group of the interval, the squares of which generate a copy of Thompson's group F. We prove that these groups may contain nonabelian free groups, they can fail to be smoothable, and can fail to be finitely presented. This represents joint work with Y. Lodha.

Transversals to horocycle flow on the moduli space of translation surfaces

When: Mon, May 1, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Grace Work (Vanderbilt University) -

Abstract: Computing the distribution of the gaps between slopes of saddle connections is a question that was studied first by Athreya and Cheung in the case of the torus, motivated by the connection with Farey fractions, and then in the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps between slopes of saddle connections into return times under horocycle flow on the space of translation surfaces to a specific transversal. We show how to use this strategy to explicitly compute the distribution in the case of the octagon, how to generalize the construction of the transversal to the general Veech case (both joint work with Caglar Uyanik), and how to parametrize the transversal in the case of a generic genus 2 translation surface.

Fock-Goncharov coordinates for rank 2 Lie groups.

When: Mon, May 8, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Christian Zickert (UMD) -
Abstract: We discuss the higher Teichmuller space A_{G,S} defined by Fock
and Goncharov. This space is defined for a punctured surface S with
negative Euler characteristic, and a semisimple, simply connected Lie group
G. There is a birational atlas on A_{G,S} with a chart for each ideal
triangulation of S. Fock and Goncharov showed that the transition functions
are positive, i.e. subtraction-free rational functions. We will show that
when G has rank 2, the transition functions are given by explicit quiver
mutations.

Equivariant Morse theory on singular spaces

When: Mon, May 15, 2017 - 3:15pm
Where: Kirwan 1313
Speaker: Graeme Wilkin (National University of Singapore) - http://www.math.nus.edu.sg/~graeme/
Abstract: In this talk I will show that the Main Theorem of Morse Theory holds for a large family of functions on singular spaces. An important class of examples is when the function is the norm-square of a moment map on the affine variety of representations of a quiver with relations. In this case I will explain how to use this theorem to compute topological invariants of moduli spaces of representations of certain quivers with relations.