Spring 2016

May 24, 2016
Simion Filip: Lower bounds on entropy of smooth maps (after Yomdin)
Entropy is a measure of complexity of a self-map of a topological space. In the context of manifolds and reasonably smooth maps, Yomdin proved a lower bound for entropy using the action on cohomology. I will discuss some of the techniques that he used.
Note: A few years ago I gave a FFSS talk on a theorem of Gromov, saying that for holomorphic self-maps of Kahler manifolds, entropy and the lower bound from the action on cohomology actually agree. This talk will be (mostly) disjoint from the previous talk.
May 19, 2016
Nick Salter: Singularities
Singularity theory is the study of critical points of differentiable maps; for the purposes of this talk I will specialize to the setting of a polynomial map f:C^n to C. Following Arnol’d, I will discuss some of the basic tools in the study of singularities, building off of the theorem of Picard-Lefschetz. The main goal will be to understand how an analysis of singularities leads naturally to bilinear forms, Dynkin diagrams, and ADE- type classifications. I will use these ideas to present another perspective on the relationship between the E_8 manifold and the Poincare homology sphere, and possibly discuss some connections with the mapping class group and surface bundles.
May 12, 2016
Lvzhou Chen: Stable Commutator Length
This will be an introductory talk on scl: history, motivating
questions, points of view from surface mappings and quasimorphisms, some
applications, known results and conjectures on how it behaves (or is
expected to behave) on different kinds of groups.
May 5, 2016
Margaret Nichols: A non-smoothable 4-manifold
Not all 4-manifolds admit smooth structures. I will construct an example of one such non-smoothable manifold.
April 28, 2016
Reid Harris: Picard-Deligne-Mostow Theory of Hypergeometric Functions
The hypergeometric differential equation is a family of second order ODEs first proposed by Gauss. In 1872, Schwarz characterized all solutions to these equations which are algebraic. His proof relied heavily on the monodromy group of the differential equation. In order to generalize these results to multivariable hypergeometric functions, Picard developed a theory around the cohomology of a complex manifold with coefficients in a local system. This talk will quickly introduce this theory, introduced by Picard and later developed by Deligne and Mostow, and also show some of it’s consequences in generalizing Schwarz’s result.
April 21, 2016
MurphyKate Montee: Cubulated Groups
Ian Agol concluded the proof of the virtual Haken Conjecture (Theorem) when he proved that all cubulated groups are virtually special. This talk will define cubulated and virtually special groups, and attempt to explain what these have to do with the Virtual Haken Theorem.
April 7, 2016
Andrew Geng: The classification of 5-dimensional geometries
Thurston’s eight homogeneous geometries formed the building blocks of 3-manifolds in the Geometrization Conjecture. Filipkiewicz classified the 4-dimensional geometries in 1983, finding 18 and one countably infinite family. I have recently classified the 5-dimensional geometries. I will review what a geometry in the sense of Thurston is, outline the classification in 5 dimensions, and point out new features that appear as the dimension increases. This includes some wacky examples such as an infinite family of inequivalent homogeneous spaces with the same diffeomorphism type. The classification touches a number of topics including foliations, fiber bundles, representations of compact Lie groups, Lie algebra cohomology, algebraic number fields, and conformal transformation groups. I hope to give some indication of how all of these come into play.
March 31, 2016
Yan Mary He: Basmajian-type Identities and Hausdorff Dimension of Limit Sets
Basmajian’s identity expresses the length of the boundary of a compact hyperbolic surface as a summation over the orthogeodesics on the surface. In this talk, I will discuss Basmajian-type series identities on limit sets associated to familiar one-dimensional complex dynamical systems. In particular, I will show how to extend Basmajian’s identity to certain Schottky groups via analytic continuation and exhibit examples of nontrivial monodromy. I will also introduce Basmajian-type identities for complex quadratic polynomials and discuss Hausdorff dimension of Julia sets.