The thirteenth annual Graduate Student Topology and Geometry Conference will be held at the University of Illinois at Urbana-Champaign on March 28-29, 2015.

The conference aims to expose graduate students in topology and geometry to current research, to provide them with an opportunity to give talks about their own research, and to bring together graduate students to talk to each other and to expert faculty members in their area. Most of the talks at the conference are given by graduate students, with nine others given by distinguished plenary speakers and young faculty.

## Invited speakers

Plenary speakers:

• Kathryn Hess (EPFL)
Homotopy theory of spaces of embeddings — After a brief review of classical results on the homotopy theory of spaces of embeddings, I will introduce the embedding calculus of Goodwillie and Weiss, a particularly powerful tool for analyzing the homotopy type of such spaces. I will then sketch various significant, recent applications of the embedding calculus in work of, among others, Arone, Lambrechts, Turchin, and Volic, as well as of Dwyer and myself.
• Misha Kapovich (UC Davis)
Polyhedral complexes and topology of projective varieties — Polyhedral complexes are obtained by gluing convex polytopes (say, Euclidean or hyperbolic) via isometric maps. In the talk I will explain how to use such complexes to construct irreducible complex-projective varieties with controlled singularities (normal crossings and Whitney umbrellas) and with prescribed fundamental groups. Polyhedral complexes which appear in the construction come from geometric topology of the 1960s and 3-dimensional hyperbolic geometry (Dirichlet fundamental domains of some discrete isometry groups of hyperbolic 3-space). This is partly a joint work with Janos Kollar.
• Daniel Wise (McGill)
Counting cycles in graphs: A rank-1 version of the Hanna Neumann Conjecture — A "$$W$$-cycle" in a labelled digraph $$\Gamma$$ is a closed path whose label is the word $$W$$. I will describe a simple result about counting the number of $$W$$-cycles in a deterministically labelled connected digraph. Namely: the number of $$W$$-cycles in $$\Gamma$$ is bounded by $$|E(\Gamma)| - |V(\Gamma)|+1$$. I will outline the proof which uses left orderable groups. This is joint work with Joseph Helfer and has been proven independently by Lars Louder and Henry Wilton.

Young faculty speakers:

• Anna Marie Bohmann (Northwestern University)
Constructing equivariant spectra — Equivariant spectra determine cohomology theories that incorporate a group action on spaces. Such spectra are increasingly important in algebraic topology but–in contrast to their nonequivariant counterparts–can be difficult to understand or construct. In recent work, Angelica Osorno and I have developed a machine for building such spectra out of purely algebraic data. This method is inspired by classical work of Segal and of Waldhausen. In this talk, I will discuss our construction and how it fits into this classical work.
• Jeff Danciger (University of Texas at Austin)
Introduction to complete 2+1 spacetimes of constant curvature — This will be an introduction to the theory of proper actions by discrete groups on Lorentzian symmetric spaces and the quotient spacetimes those actions define. We will discuss how to make examples and then we will study the deformation space with the aim of describing how flat spacetimes naturally appear at the boundary of the moduli of negatively curved spacetimes. Along the way, we will explore the rich geometry of the group $$PSL(2,\mathbb{R})\times PSL(2,\mathbb{R})$$ and a close connection with the theory of Thurston's Lipschitz metric on Teichmüller space.
• Jo Nelson (Barnard College, Columbia University, Institute for Advanced Study)
Invariants of contact structures and Reeb dynamics — Contact geometry is the study of geometric structures on odd dimensional smooth manifolds given by a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability; these hyperplane fields are called contact structures. The associated one form is called a contact form and uniquely determines a vector field called the Reeb vector field on the manifold. Contact and symplectic geometry are closely intertwined and as in symplectic topology one can make use of J-holomorphic curves provide one with a topological approach known as contact homology to obtaining invariants. Despite the many analytic pitfalls along the way to defining contact homology we explain how recent work of Hutchings and Nelson has managed to redeem this theory in dimension 3 for dynamically convex contact manifolds. Additionally this talk will have lots of cool pictures and animations illustrating these fascinating concepts in contact geometry and many concrete examples will be given.
• Vivek Shende (UC Berkeley)
Knotty character varieties — We study the moduli spaces of constructible sheaves on a surface whose singular support lies in a fixed Legendrian knot in the cosphere bundle. Connections will be made to the theory of cluster varieties and to the Chekanov-Eliashberg theory of Legendrian knots.
• Hiro Lee Tanaka (Harvard University)
Cobordisms, Floer cohomology, and mirror symmetry over ring spectra — Studying sheaves on a complex variety is a fancy version of studying modules over a fixed ring. On the other hand, Floer cohomology is a Morse theory on the space of paths in a symplectic manifold. Mirror symmetry predicts that these two areas of study (put simplistically, the study of modules and of Morse theory on the path space) are largely equivalent. In more concrete terms, Kontsevich's mirror symmetry conjecture predicts that the derived category of coherent sheaves on some algebraic varieties are equivalent to the Fukaya category of mirror symplectic manifolds. In this talk, we'll explain a recent discovery: The theory of cobordisms has a lot to say about the symplectic side of mirror symmetry.
We'll start by describing a stable oo-category of Lagrangian cobordisms associated to certain symplectic manifolds, and show these always admit a highly non-trivial functor to the Fukaya category. For instance, based on the construction of this functor, one can easily deduce that cobordant Lagrangians are equivalent objects in the Fukaya category. As another corollary, we'll see that the cobordism groups of a symplectic manifold map to the K theory of the Fukaya category of that manifold, and hence (by the mirror symmetry conjecture) to the K theory of the mirror algebraic variety. In the case that the mirror variety is affine, this says that cobordism groups can detect algebraic K theory classes of a ring.
• Jing Tao (University of Oklahoma)
Uniform Growth Rate — The principle of evolution, after Sleator, Tarjan and Thurston, says that: If, in an evolutionary system, the rules of mutation are local in nature, then the number of possible descendants of a given object grows exponentially, but at a rate that is independent of the original object. In this talk, I will describe some evolutionary systems and apply this principle to obtain several uniform growth rate results. For example, we will show that the growth rate of the space of homotopy classes of n-vertex triangulations of a surface related by diagonal flips is independent of n and the surface. We will also show that, given the correct choice of generators, $$SL(n,\mathbb{Z})$$, automorphism groups of free groups, and mapping class groups, all grow at a uniform rate.

## Previous conferences

Past editions of this conference were held at:

• 2014 — University of Texas at Austin
• 2013 — University of Notre Dame
• 2012 — Indiana University, Bloomington
• 2011 — Michigan State University
• 2010 — University of Michigan
• 2009 — University of Wisconsin, Madison
• 2008 — University of Illinois at Urbana-Champaign
• 2007 — University of Chicago
• 2006 — Indiana University, Bloomington
• 2005 — Northwestern University
• 2004 — University of Minnesota
• 2003 — University of Notre Dame