Fall 2014

 October 2, 2014 Curvature pinching for negatively curved manifolds — Clark Butler I will give a brief overview of some of the famous curvature pinching theorems for positively curved manifolds before explaining how their analogues fail in negative curvature. Then I will talk about what is true.
 October 9, 2014 Topological obstruction to solving polynomials effectively — Weiyan Chen Consider the following problem – given any polynomial with some fixed degree,  find all roots of that polynomial within a small error. A theorem of Smale (1987) says that any algorithm solving this problem must not be too simple. And the reason is topological! We will see how a lower bound of the “complexity” of such algorithm arises from Schwarz genus, category of a space, cup product length, and the homology of braid groups. I will define all the terms above.
 October 16, 2014 Diff is difficult (because it is simple) — Nick Salter A fundamental phenomenon in the theory of manifold bundles is that of “flatness”. We will introduce this notion and briefly discuss some of the few known obstructions to flatness. Flatness is particularly difficult to obstruct in low dimensions, and we will explain why the source of this is the fact that Diff_0(M) is a simple group for any manifold M.
 October 23, 2014 Invariant Components of Flat Surfaces — Kathryn Lindsey The flow in a fixed direction on a flat surface determines a decomposition of the surface into invariant components. Invariant components come in two “flavors” – periodic components, which are maximal cylinders of periodic trajectories, and minimal components, which are closures of recurrent, non-periodic orbits.  How many invariant components can a flat surface in a given stratum (i.e. has a fixed genus and number/type of cone points) have?  How does the number of invariant components a surface has tell you which connected component of a stratum the surfaces belongs to?  I will present results by Y. Naveh and by myself that address these questions.
 October 30, 2014 Characteristic Classes of Foliation — Lei Chen Because of Chern Weil theory, we can use de Rham class to represent cohomology classes. This are all called primary classes. However, if we consider foliation, we have a famous theorem called Bott Vanishing. It’s a bad thing. However, like the definition of Massey product, if some class is zero in cohomology, we can still make use of it by defining what we call secondary classes. Godbillon-Vey is such an example. I hope that I could talk about some examples of computing therm if time allows.
 November 6, 2014 Covers, Homology and Lots of Mysteries — Sebastian Hensel In this talk I will explore the following innocent sounding question: how can we describe the homology of a finite, regular cover of a space? Even if the space is a graph or a surface, this question is richer than one might think. The story starts with a classical theorem of Chevalley-Weil which describes the homology of the cover completely as a representation of the deck group (which I will prove twice). But this is only the beginning: if we add in the mapping class group into the mix, and ask how lifts of mapping class act on the homology of the cover, mysteries abound. I will present some of my favourite questions, some known results by others, and some first new results of my own (joint with Benson Farb) that shed some light on this action in the case of graph covers. Notes are here.
 November 13, 2014 Volume distortion in homotopy groups — Fedor Manin Given a nice compact metric space $X$, how can we use geometry to better understand elements $\alpha \in \pi_n(X)$?  One way is by measuring distortion, that is, how geometric measurements of an optimal representative of $k\alpha$, such as Lipschitz constant or volume, grow as a function of $k$.  The Lipschitz distortion of simply connected spaces is the subject of a conjecture of Gromov; volume distortion has not been previously studied.  We identify three main sources of volume distortion arising from rational homotopy theory, the action of the fundamental group, and its geometry.  This turns out to be enough to characterize those spaces which have no distorted elements.
 November 20, 2014 Gluing tetrahedra and 3-manifold — Saima Anis We tried to construct in a natural way a 3-dimensional manifold together with actions of the group PSL(2, F7) on 4-tuples with entries from PL(F7), where F7  is the finite field with 7 elements.
 December 4, 2014 Critical points of distance functions — Wouter van Limbeek Let M be a Riemannian manifold. Then the distance to a fixed basepoint gives a nice function on M that encodes a lot of the geometry of M (e.g. near the basepoint the gradient trajectories are geodesic segments). Inspired by this fact, Grove and Shiohama developed the beginnings of a Morse theory of these functions (even though they are usually not differentiable!). The theory is still in an early state, but there are already nice applications to due to Grove-Shiohama, Grove-Petersen and Gromov. I will discuss some of the theory and these applications.