Geometry and Topology Seminar 2019-2020

The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.
For more information, contact Richard Kent.

Fall 2013

date	speaker	title	host(s)
September 6
September 13, 10:00 AM in 901!	Alex Zupan (Texas)	Totally geodesic subgraphs of the pants graph	Kent
September 20
September 27
October 4
October 11
October 18	Jayadev Athreya (Illinois)	Gap Distributions and Homogeneous Dynamics	Kent
October 25	Joel Robbin (Wisconsin)	GIT and [math]\displaystyle{ \mu }[/math]-GIT	local
November 1	Anton Lukyanenko (Illinois)	Uniformly quasi-regular mappings on sub-Riemannian manifolds	Dymarz
November 8	Neil Hoffman (Melbourne)	Verified computations for hyperbolic 3-manifolds	Kent
November 15	Khalid Bou-Rabee (Minnesota)	TBA	Kent
November 22	Morris Hirsch (Wisconsin)	Common zeros for Lie algebras of vector fields on real and complex 2-manifolds.	local
Thanksgiving Recess
December 6
December 13

Fall Abstracts

Alex Zupan (Texas)

Totally geodesic subgraphs of the pants graph

Abstract: For a compact surface S, the associated pants graph P(S) consists of vertices corresponding to pants decompositions of S and edges corresponding to elementary moves between pants decompositions. Motivated by the Weil-Petersson geometry of Teichmüller space, Aramayona, Parlier, and Shackleton conjecture that the full subgraph G of P(S) determined by fixing a multicurve is totally geodesic in P(S). We resolve this conjecture in the case that G is a product of Farey graphs. This is joint work with Sam Taylor.

Jayadev Athreya (Illinois)

Gap Distributions and Homogeneous Dynamics

Abstract: We discuss the notion of gap distributions of various lists of numbers in [0, 1], in particular focusing on those which are associated to certain low-dimensional dynamical systems. We show how to explicitly compute some examples using techniques of homogeneous dynamics, generalizing earlier work on gaps between Farey Fractions. This works gives some possible notions of `randomness' of special trajectories of billiards in polygons, and is based partly on joint works with J. Chaika, J. Chaika and S. Lelievre, and with Y.Cheung. This talk may also be of interest to number theorists.

Joel Robbin (Wisconsin)

GIT and [math]\displaystyle{ \mu }[/math]-GIT

Many problems in differential geometry can be reduced to solving a PDE of form

[math]\displaystyle{ \mu(x)=0 }[/math]

where [math]\displaystyle{ x }[/math] ranges over some function space and [math]\displaystyle{ \mu }[/math] is an infinite dimensional analog of the moment map in symplectic geometry. In Hamiltonian dynamics the moment map was introduced to use a group action to reduce the number of degrees of freedom in the ODE. It was soon discovered that the moment map could be applied to Geometric Invariant Theory: if a compact Lie group [math]\displaystyle{ G }[/math] acts on a projective algebraic variety [math]\displaystyle{ X }[/math], then the complexification [math]\displaystyle{ G^c }[/math] also acts and there is an isomorphism of orbifolds

[math]\displaystyle{ X^s/G^c=X//G:=\mu^{-1}(0)/G }[/math]

between the space of orbits of Mumford's stable points and the Marsden-Weinstein quotient.

In September of 2013 Dietmar Salamon, his student Valentina Georgoulas, and I wrote an exposition of (finite dimensional) GIT from the point of view of symplectic geometry. The theory works for compact Kaehler manifolds, not just projective varieties. I will describe our paper in this talk; the following Monday Dietmar will give more details in the Geometric Analysis Seminar.

Anton Lukyanenko (Illinois)

Uniformly quasi-regular mappings on sub-Riemannian manifolds

Abstract: A quasi-regular (QR) mapping between metric manifolds is a branched cover with bounded dilatation, e.g. f(z)=z^2. In a joint work with K. Fassler and K. Peltonen, we define QR mappings of sub-Riemannian manifolds and show that: 1) Every lens space admits a uniformly QR (UQR) mapping f. 2) Every UQR mapping leaves invariant a measurable conformal structure. The first result uses an explicit "conformal trap" construction, while the second builds on similar results by Sullivan-Tukia and a connection to higher-rank symmetric spaces.

Neil Hoffman (Melbourne)

Verified computations for hyperbolic 3-manifolds

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.

Khalid Bou-Rabee (Minnesota)

TBA

Morris Hirsch (Wisconsin)

Common zeros for Lie algebras of vector fields on real and complex 2-manifolds.

Spring 2014

date	speaker	title	host(s)
January 24
January 31
February 7
February 14
February 21
February 28
March 7
March 14
Spring Break
March 28
April 4	Matthew Kahle (Ohio)	TBA	Dymarz
April 11
April 18
April 25
May 2
May 9

Spring Abstracts

Matthew Kahle (Ohio)

TBA

Archive of past Geometry seminars

2012-2013: Geometry_and_Topology_Seminar_2012-2013

2011-2012: Geometry_and_Topology_Seminar_2011-2012

2010: Fall-2010-Geometry-Topology