Geometry-Topology Archives for Academic Year 2013

Title: Moment map flows and the Hecke correspondence for quivers

When: Mon, September 9, 2013 - 3:00pm
Where: Math 1313
Speaker: Graeme Wilkin (NUS)
Abstract: Narasimhan and Ramanan's Hecke correspondence for bundles over curves relates bundles of different degree via a Hecke modification over a point on the curve. In the 1990s, Nakajima used an analogous correspondence for quiver varieties (Hecke modifications of bundles over an ALE 4-manifold) to give a geometric construction of representations of affine Kac-Moody algebras and quantum affine algebras. In this talk I will give an analytic interpretation of Nakajima's Hecke correspondence via the gradient flow of the norm-square of a moment map.

An invariant of hyperbolic 3-manifolds of finite volume in Hochschild relative group homology

When: Mon, September 16, 2013 - 3:00pm
Where: Math 1313
Speaker: Jose Luis Cisneros-Molina
Abstract: Given a hyperbolic 3-manifold of finite volume, using classifying spaces for families of subgroups, we construct an element in the
third Hochschild relative group homology group $H_3([PSL(2,\C):P],\Z)$, where $P$ is the subroup of $PSL(2,\C)$ correspondint to
upper triangular matrices with $1$ in the diagonal.

This is join work with José Antonio Arciniega-Nevárez

Schubert varieties as variations of Hodge structure (joint with Lie Groups and Representation Theory)

When: Wed, September 18, 2013 - 2:00pm
Where: Math 1311
Speaker: Colleen Robles (Institute for Advanced Study/Texas A & M) -
Abstract: Variations of Hodge structure (VHS) are constrained by a system of differential equations known as the infinitesimal period relation (IPR), or Griffiths transversality. The IPR is a distinguished homogeneous system defined on a flag variety X = G/P. I will characterize the Schubert varieties that arise as variations of Hodge structure (VHS). I will also discuss the central role that these Schubert VHS play in our study of arbitrary VHS: infinitesimally their orbits under the isotropy action `span' the space of all VHS, yielding a complete description of the infinitesimal VHS. One corollary is that they provide sharp bounds on the maximal dimension of a VHS.

Riemannian manifolds with local symmetry

When: Mon, September 23, 2013 - 3:00pm
Where: Math 1313
Speaker: Wouter van Limbeek (University of Chicago) -
Abstract: In this talk I will discuss the problem of classifying all closed Riemannian manifolds whose universal cover has nondiscrete isometry group. Farb and Weinberger solved this under the assumption that M is aspherical: roughly, they proved that any such M is a fiber bundle with locally homogeneous fibers. However, if M is not aspherical, many new examples and phenomena appear. I will exhibit some of these, and discuss progress towards a classification.

On the Narasimhan-Seshadri correspondence for real and quaternionic vector bundles

When: Mon, September 30, 2013 - 3:00pm
Where: Math 1313
Speaker: Florent Schaffhauser, Universidad de Los Andes (Bogotá)
Abstract: The Narasimhan-Seshadri theorem establishes a correspondence between stable holomorphic vector bundles over a compact Riemann surface X and irreducible projective unitary representations of the fundamental group of that surface. When X represents an algebraic curve defined over the field of real numbers, there is an induced Galois action on stable holomorphic vector bundles, whose fixed points exhibit remarkable algebraic properties: they are either real or quaternionic holomorphic vector bundles and semi-stable such bundles behave very similarly to semi-stable holomorphic vector bundles. In particular, there is a symplectic description of moduli spaces of semi-stable real and quaternionic vector bundles of fixed topological type, which, under the assumption that X has real points, implies that the Narasimhan-Seshadri map is Galois-equivariant.

In this talk, I will discuss the Narasimhan-Seshadri correspondence in the context of real and quaternionic vector bundles and show how it leads to a simple, differential-geometric proof of the existence of stable algebraic vector bundles that are definable over the field of real numbers.

Verified computations for hyperbolic 3-manifolds

When: Mon, October 7, 2013 - 3:00pm
Where: Math 1303
Speaker: Neil Hoffman (MPIM)
Abstract: Given a triangulated 3-manifold M a natural question is:
Does M admit a hyperbolic structure?
While this question can be answered in the negative if M is known to
be reducible or toroidal, it is often difficult to establish a
certificate of hyperbolicity, and so computer methods have developed
for this purpose. In this talk, I will describe a new method to
establish such a certificate via verified computation and compare the
method to existing techniques.
This is joint work with Kazuhiro Ichihara, Masahide Kashiwagi,
Hidetoshi Masai, Shin'ichi Oishi, and Akitoshi Takayasu.

Colloquium, no seminar

When: Mon, October 14, 2013 - 3:00pm

Constructing Non-Arithmetic Lattices

When: Mon, October 21, 2013 - 3:00pm
Speaker: John Parker (Durham)
Abstract: A lattice in a semi-simple Lie group is a discrete subgroup of finite
co-volume. An arithmetic group in a Q-algebraic group is a group
commensurable with the integral points. These two notions are related
in the following way. All arithmetic groups are lattices. Except for
subgroups of SO(n,1) and SU(n,1), all lattices are arithmetic.
In the 1980s Deligne and Mostow constructed examples of non-arithmetic
lattices in SU(2,1) and SU(3,1). In this talk I will survey the
history of this problem, and then go on to describe a joint project
with Martin Deraux and Julien Paupert, where we construct new
non-arithmetic lattices in SU(2,1). These are the first new examples
to be constructed since the work of Deligne and Mostow.

The renormalized volume of asymptotically hyperbolic metrics

When: Mon, October 28, 2013 - 3:00pm
Where: Math 1313
Speaker: Otis Chodosh, Stanford
Abstract: We'll discuss asymptotically hyperbolic metrics and how they
differ from asymptotically flat manifolds, including a notion of
renormalized volume and a corresponding scalar curvature volume
comparison result (this is joint work with S. Brendle). This has
interesting consequences for large isoperimetric regions in such
manifolds, which we will indicate.

Quasiconformal instability for isometry groups in hyperbolic 4-space

When: Mon, November 4, 2013 - 3:00pm
Where: Math 1313
Speaker: Youngju Kim (Korea Institute for Advanced Study)
Abstract: We will discuss the geometry of real 4-dimensional hyperbolic space.
In particular, we will talk about the Clifford matrix representations of the isometries
and thrice-punctured sphere groups.

Complex hyperbolic geometry of the figure eight knot

When: Mon, November 11, 2013 - 3:00pm
Where: Math 1313
Speaker: Martin Deraux (University of Grenoble/ICERM)
Abstract: I will explain recent techniques for studying representations of
3-manifold groups into SL(3,C), and apply these techniques to show that the figure eight knot complement admits a spherical CR uniformization (i.e. it occurs as the manifold at infinity of a well chosen discrete subgroup of automorphisms of the ball). This is joint work with Elisha Falbel.

A Spectral Perspective on Neumann-Zagier

When: Mon, November 18, 2013 - 3:00pm
Speaker: Tudor Dimofte, IAS
Abstract: The classic formula of Neumann-Zagier on deformations of the volume of a hyperbolic 3-manifold rested on a key result about symplectic properties of hyperbolic gluing equations. This symplectic result and its generalizations have since played a major role in (e.g.) the quantization of hyperbolic/PGL(2) structures on 3-manifolds, and the study of higher PGL(K) structures. Nevertheless, the basic methods of proof have remained combinatorial and somewhat unintuitive.
In this talk, I relate (and thus re-prove) symplectic properties of gluing equations to elementary facts about homology of branched covers of boundaries of 3-manifolds. The relation is motivated by a "non-abelianization map" of Gaiotto-Moore-Neitzke that constructs nontrivial PGL(K) local systems on a space from GL(1) local systems on a K-fold cover.

Homology of Yang-Baxter operators and knot theory

When: Mon, November 25, 2013 - 3:00pm
Where: Math 1313
Speaker: Josef Przytycki (George Washington University) -
Abstract: Yang-Baxter operators can be used to construct invariants of links
(e.g. Jones or Homflypt polynomial), as demonstrated by Jones, Turaev,
and Kauffman.
In a similar manner quandles can be used to construct link invariants
(e.g. Fox colorings). Additionally homology of quandles gives state sum
invariants of links.
We outline potential relations to Khovanov homology and
categorification, via Yang-Baxter operators.
We use here the fact that Yang-Baxter equation can be thought of as a
generalization of self-distributivity (the main axiom of quandles).

Curve shortening flow and smooth projective planes

When: Mon, December 2, 2013 - 3:00pm
Where: Math 1313
Speaker: Yu-Wen Hsu (Yale University) -
Abstract: The manifold RP^2 admits an infinite dimensional family of pairwise nonisomorphic smooth projective plane structures. The homotopy theory of this space was not known well. In this talk, we will discuss our approach of using curve shortening flow to construct a smooth homotopy between any smooth projective plane structures.

Spaces of surface group representations

When: Mon, December 9, 2013 - 3:00pm
Where: Math 1313
Speaker: Kathryn Mann (University of Chicago) -
Abstract: Let G be a group of homeomorphisms of the circle and Gamma the fundamental group of a closed surface. The representation space Hom(Gamma, G) is a basic example in geometry and topology: it parametrizes circle bundles over the surface with structure group G, and actions of Gamma on the circle with regularity given by G.
A theorem of Goldman says that when G = PSL(2,R), the components of Hom(\Gamma, G) are completely determined by the Euler number, a classical invariant. By contrast, Hom(\Gamma, Homeo+(S^1)) is relatively unexplored -- in fact it is an open question whether it has finitely many or infinitely many components.

In my talk, I'l report on recent work and new tools to distinguish connected components of Hom(Gamma, Homeo+(S^1)). In particular, we give a new lower bound on the number of components, show that there are multiple components on which the Euler number takes the same value -- in contrast to the PSL(2,R) case -- and show that "geometric" representations are surprisingly rigid. A key technique is the study of rigidity phenomena in rotation numbers, using recent ideas of Calegari-Walker.

Isoperimetric Inequality and Q-curvature

When: Mon, January 20, 2014 - 2:00pm
Where: Math 3206
Speaker: Yi Wang (Stanford) -
Abstract: A well-known question in differential geometry is to prove the
isoperimetric inequality under intrinsic curvature conditions. In
dimension 2, the isoperimetric inequality is controlled by the integral of
the positive part of the Gaussian curvature. In my recent work, I prove
that on simply connected conformally flat manifolds of higher dimensions,
the role of the Gaussian curvature can be replaced by the Branson's
Q-curvature. The isoperimetric inequality is valid if the integral of the
Q-curvature is below a sharp threshold. Moreover, the isoperimetric
constant depends only on the integrals of the Q-curvature. The proof
relies on the theory of $A_p$ weights in harmonic analysis.

Twisted cohomology theories and algebraic K-theory

When: Mon, February 3, 2014 - 3:00pm
Where: Math 1313
Speaker: John Lind (Johns Hopkins University) -
Abstract: Twisted K-theory is a cohomology theory whose cocycles are like vector bundles but with locally twisted transition functions. This subject has garnered much attention for its connections to conformal field theory. While twisted K-theory can be defined in terms of the geometry of vector bundles, there is a homotopy-theoretic formulation using the language of parametrized spectra. In fact, from this point of view we can define twists of any multiplicative generalized cohomology theory R, not just K-theory. The aim of this talk is to explain how this works, and to present classification results describing all possible twists of R-theory. This will allow a description of the cocycles of the algebraic K-theory K(R) of R in terms of twisted versions of R-theory.

Rigidity Results for Hermitian-Einstein manifolds

When: Mon, February 17, 2014 - 3:00pm
Where: Math 1313
Speaker: Tommy Murphy (McMaster),
Abstract: Motivated by work of Alfred Gray, we study a differential operator naturally defined on the unit sphere bundle of any Riemannian manifold. The coefficients of this differential operator are determined by the sectional curvatures of the base manifold. This enabled Gray to prove rigidity results characterising the Hermitian symmetric spaces amongst Kaehler-Einstein manifolds. We will extend the study of his operator to various classes of Hermitian-Einstein manifolds, focusing in particular on nearly Kaehler manifolds and on Hermitian-Einstein surfaces. This is joint with Stuart Hall.

Conformally flat circle bundles over surfaces

When: Mon, February 24, 2014 - 3:00pm
Speaker: Son Lam Ho (UMD)
Abstract: The conformal 3-sphere is the natural boundary of hyperbolic 4-space. Manifolds modeled on this 3-sphere are said to be conformally flat. In this talk I will recall results about existence and non-existence of flat conformal geometry on circle bundles, as well as a conjecture by Gromov-Lawson-Thurston. I will also present a partial result related to this conjecture.

Knot contact homology, knot group representations, and the A-polynomial

When: Mon, March 3, 2014 - 3:00pm
Where: Math 1313
Speaker: Christopher Cornwell (Duke)
Abstract: We discuss augmentations in knot contact homology and their relationship to certain GL_n(C) representations of the fundamental group of the knot complement.
In the study of contact homology, the set of augmentations has been a useful tool for approaching the complicated algebraic structure of underlying differential graded algebras. In the knot contact homology of a knot K there are augmentations that may be associated to a flat connection on the complement of K. We show that all augmentations arise this way. As a consequence, a polynomial invariant of K called the augmentation polynomial represents a generalization of the classical A-polynomial. A recent conjecture, similar to the AJ conjecture, relates a 3-variable augmentation polynomial to colored HOMFLY-PT polynomials. Our results can be seen as motivation for this conjecture having an augmentation polynomial in place of the A-polynomial.

Degeneration of convex real projective structures on surfaces

When: Mon, March 10, 2014 - 3:00pm
Where: Math 1313
Speaker: Tengren Zhang (Michigan)
Abstract: Let S be a closed orientable surface of negative Euler characteristic, and let C(S) be the deformation space of convex real projective structures on S. I will present new results about how some of the geometric properties of the convex projective structure degenerate as one deforms the structure along the internal parameters of the Goldman parameterization of C(S).

Inequalities in topology (special seminar)

When: Thu, March 13, 2014 - 11:00am
Where: ENG 1102 (note different location)
Speaker: Domingo Toledo (Utah)
Abstract: This will be an elementary historical survey of inequalities for characteristic classes of flat bundles. The precursor of such inequalities is Kneser's inequality for the degree of a continuous map f from a surface of genus g to one of genus h, where g,h >1, namely |deg(f) | \le ( g-1)/(h-1). Moreover equality holds if and only if f is homotopic to a covering map. Kneser's paper was submitted in October 1929. That same month Hopf submitted a famous paper pioneering the use of the cohomology ring and duality, where he proves the special case g = h and ends the paper lamenting that he cannot derive Kneser's more general inequality.

In 1958 Milnor published his famous paper on flat plane bundles over surfaces, with a sharp inequality on their Euler classes, which implies Kneser's inequality. In the late 1970's Gromov effectively used bounded cohomology as a universal explanation of these inequalites. In 1980 Goldman characterized the case of equality in the Milnor-Wood inequality. These basic results motivated many other developments. We will outline these developments as much as possible.

Spring break, GEAR retreat

When: Mon, March 17, 2014 - 3:00pm

A role of the Grothendieck-Teichmueller group in quantization

When: Mon, March 24, 2014 - 3:00pm
Where: Math 1313
Speaker: Vasily Dolgushev (Temple)
Abstract: Deformation quantization is a procedure which
assigns a formal deformation of the associative algebra of
functions on a variety to a Poisson structure on this variety.
Such a procedure can be obtained from a formality quasi-isomorphism
for Hochschild cochains. One such formality quasi-isomorphism
was proposed in 1997 by Maxim Kontsevich, however, it is
known that there are infinitely many homotopy inequivalent formality
quasi-isomorphisms. I propose a framework in which all homotopy classes of
formality quasi-isomorphisms for Hochschild cochains can be described.
This description is based essentially on the Grothendieck-Teichmueller
group which was introduced by Vladimir Drinfeld in 1990. This group has
interesting links to the absolute Galois group of rational numbers, moduli of algebraic curves, solutions of the Kashiwara-Vergne problem, and theory of motives.

Manifolds modeled on rank 1 Lie groups (special seminar)

When: Thu, March 27, 2014 - 11:00am
Where: ENG 1102
Speaker: Nicolas Tholozan (Nice Sophia-Antipolis)
Abstract: I will try to give an overview of results about manifolds modeled on simple rank 1 Lie groups, first in a general case and then in the case of PSL(2,R).

Cartan geometry, with an application to conformal pseudo-riemannian geometry

When: Mon, March 31, 2014 - 3:00pm
Where: MTH 1313
Speaker: Dominik Francoeur (Sherbrooke)
Abstract: Cartan geometry generalizes Klein geometry (the geometry of homogeneous spaces) in the same way that Riemannian geometry generalizes Euclidean geometry. In this talk, I will give the definition of a Cartan geometry and use this theoretical framework to show a classical result of conformal pseudo-riemannian geometry, namely that a lightlike geodesic on a pseudo-riemannian manifold is an unparametrized geodesic for any metric in the same conformal class.


When: Mon, April 7, 2014 - 3:00pm
Where: Math 1313
Speaker: Todd Drumm
Abstract: TBA

Geometrically and diagrammatically maximal knots

When: Mon, April 14, 2014 - 3:00pm
Speaker: Abhijit Champanerkar (CUNY)
Abstract: We present several new results and conjectures about
geometrically and diagrammatically maximal sequences of knots,
which maximize the hyperbolic volume per crossing and determinant
per crossing, respectively. A weaving knot is an alternating knot
with the same projection as a torus knot. We prove that weaving
knots are both geometrically and diagrammatically maximal. We
provide asymptotically correct volume bounds for weaving knots
using angle structures, and show that their geometric limit is
the complement of the infinite weave (alternating square
lattice). Finally, we prove a result, known to I. Agol, about
generating sequences of geometrically maximal knots from the
infinite weave. This is joint work with Ilya Kofman and Jessica

Hitchin harmonic maps are immersions

When: Mon, April 21, 2014 - 3:00pm
Speaker: Andy Sanders (UIC)
Abstract: In 1992, Hitchin used his theory of Higgs bundles to construct an important family of representations of closed surface groups into the split real form of a complex, simple Lie group. These Hitchin representations comprise a component of the space of conjugacy classes of such representations; this component can also be defined as those representations which are continuous deformations of the Fuchsian representations which uniformize the surface. For any choice of complex structure on the surface and any Hitchin representation, we show that the corresponding equivariant harmonic map from the universal cover into the symmetric space is an immersion. This is a generalization of the analogous statement in dimension two, due to Schoen-Yau, that equivariant harmonic maps for Fuchsian representations are immersions. If time permits, we will display the utility of this theorem by giving a new lower bound on the exponential growth rate of orbits of a Hitchin representation acting on the associated symmetric space.

Geometric invariants of highly twisted hyperbolic pretzel knots

When: Mon, April 28, 2014 - 3:00pm
Where: Math 1310
Speaker: Christian Millichap (Temple)
Abstract: A finite volume hyperbolic 3-manifold M has a number of interesting geometric invariants. Two such invariants are the volume of M and the length spectrum of M, which is the set of all lengths of closed geodesics in M counted with multiplicities. It is natural to ask how bad are these invariants at distinguishing hyperbolic 3-manifolds and how do these invariants interact with one another. In this talk, we shall construct large families of non-commensurable hyperbolic pretzel knot complements with the same volume and the same initial length spectrum. This construction will rely on mutating pretzel knots along four-punctured spheres, and then showing that such mutations often preserve the volume and short geodesic lengths of a hyperbolic knot complement.

Cohomology of moduli spaces of manifolds

When: Mon, May 5, 2014 - 3:00pm
Where: Math 1313
Speaker: Soren Galatius (Stanford)
Abstract: I will discuss the cohomology of spaces of the form BDiff(W). If W is a smooth compact manifold, Diff(W) denotes the group of diffeomorphisms (relative to the boundary), and BDiff(W) is the classifying space of this topological group. If W is an oriented 2-manifold, the space BDiff(W) is closely related to the moduli space of Riemann surfaces, and its cohomology was calculated by Madsen and Weiss, in a range of degrees depending on the genus of W. I will discuss recent joint work with O. Randal-Williams on a generalization to manifolds of higher dimensions.

Conformal field theory and the Kähler geometry of unitary character varieties

When: Mon, May 12, 2014 - 3:00pm
Where: Math 1313
Speaker: Claudio Meneses-Torres (Stony Brook)
Abstract: An interesting property of the Liouville theory on punctured 2-spheres is that the evaluation of the Liouville action functional at its extrema (the hyperbolic metrics) gives rise to a Kähler potential for the Weil-Petersson metric on their Teichmüller space. In this talk I will explain how this result can be reproduced on a unitary character variety, corresponding to the moduli space of parabolic bundles over the sphere, by means of a suitable interpretation of the WZNW action functional.