Spring 2022

Abstract: Functor calculus was introduced by Tom Goodwillie as a means to give meaningful approximations to a homotopy-invariant functor between the categories of spaces or spectra. These approximations assemble into a “Taylor tower” which in many ways resembles the usual notion of a Taylor series in calculus and real analysis. In particular, the Taylor tower of a functor F is in some sense “controlled” by a sequence of spectra called the derivatives of the functor.

In this talk, we will give an overview of the construction of Taylor towers and derivatives for functors of spaces and outline some recent results on the structure inherent to a functor’s derivatives. Particular emphasis will be given to the identity functor—whose Taylor tower is already highly nontrivial—and whose derivatives play a key role in describing certain homotopical properties of the category of spaces.

Abstract: An action of the p-adic integers on a compact metric space X is said to raise dimension if the diminsion of the orbit space is larger than the dimension of X. We will present generalizations of classical results obtained by Williams, Raymond et al. in the 60s on dimension raising actions of the p-adic integers on compact metric spaces.

Abstract: If a finitely generated group acts properly on a contractible manifold, then the group (with a word metric associated to some finite set of generators) admits a coarse embedding into that manifold. However, the converse is not true. For example, all Baumslag-Solitar groups BS(m,n) coarsely embed into some contractible 3-manifold but cannot act properly on a contractible 3-manifold unless |m|=|n|. Kapovich--Kleiner and Hruska--Stark--Tran gave examples of delta-hyperbolic groups which coarsely embed into contractible 3-manifold but don't act properly on any contractible 3-manifold. Schreve showed that k-fold products of some of the above-mentioned groups don't act on contractible 3k-manifolds, although all of them coarsely embed into contractible 3k-manifolds. In my talk, I will describe these groups, give an overview of ideas that go into some of these proofs and time permitting discuss some new examples of this phenomenon.

Abstract: A big question in Coarse Homotopy Theory is "When is any coarse map coarse homotopic to a cone of a continuous map?" This is true for any coarse map into a metric space that is an infinite cone of a space. For example, any two coarse maps of rays into the real plane are coarse homotopic. In this talk, I will first show how you can "straighten" any coarse map of a ray, and then I will go on to show how this generalizes to any coarse map.

Abstract: Given a filling closed geodesic on a hyperbolic surface, one can consider its canonical lift in the projective tangent bundle. Drilling this knot, one obtains a hyperbolic 3-manifold. In this talk we are interested in volume bounds for these manifolds in terms of geometric quantities of the geodesic, such as the hyperbolic length. In particular, we give a volume lower bound in terms of length when the filling geodesic is a closed geodesic approximating the Liouville geodesic current. The bound is given in terms of a counting problem in the unit tangent bundle that we solve by applying an exponential multiple mixing result for the geodesic flow. This is joint work with Didac Martinez Granado, Yannick Krifka and Franco Vargas Pallete.

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Abstract: A topological group is said to have the Rokhlin property if it contains a dense conjugacy class. We will start with explaining how this topological/dynamical property of a group has algebraic consequences when combined with automatic continuity properties. We will then give a classification of the connected, orientable, second-countable 2-manifolds whose mapping class groups have the Rokhlin property and explain our original motivation for considering this problem. This is joint work with Justin Lanier.

Abstract: Closed curves on surfaces come with a wealth of geometric invariants: the self-intersection number, the "hyperbolic" length function, the "extremal" length function, the "dual" cube complex, the "simple-lifting" degree, the "pushing-point" mapping class. Relationships among these invariants are mostly opaque, and a broad research goal is to produce and sharpen bounds pertaining to these quantities. I will discuss ongoing joint work with Tarik Aougab, in which we pursue some such bounds for "random" curves.

Abstract: Let S be an orbifold of negative Euler characteristic and C a collection of closed curves. The set of tangents to C is a link in the tangent bundle UT(S) and drilling this link, we obtain a 3-manifold M_C. Any invariant of M_C is automatically a mapping class group invariant of C. Further, the manifold M_C uniquely determines this mapping class group orbit. In this talk, we will go over results that explain the behavior and provide coarse bounds on the hyperbolic volume of M_C in terms of topological and geometric properties of the family C. For example, when C is a filling pair of simple closed curves, we show that the volume is coarsely comparable to Weil-Petersson distance between strata in Teichmuller space. Lastly, we will explain algorithmic methods and tools for building such links and computing other invariants. This work is joint with Tommaso Cremaschi, Jacob Intrater, and Jose Andres Rodriguez-Migueles.

Abstract: What does a typical quotient of a group look like? Gromov had looked at density model of quotients of free groups. The density parameter d measures the rate of exponential growth of the number of relators compared to the size of the Cayley ball. Using this model, he had proved that for d<1/2 a typical quotient of a free group is non-elementary hyperbolic. Ollivier extended Gromov's result to show that for d<1/2 a typical quotient of even a non-elementary hyperbolic group is non-elementary hyperbolic.

Żuk/Kotowski-Kotowski proved that for d>1/3 a typical quotient of a free group has Property-(T).We show that (in a closely related density model) for 1/3<d<1/2 a typical quotient of a non-elementary hyperbolic group is non-elementary hyperbolic and has Property-(T).

This provides an answer to a question of Gromov (and Ollivier).

Abstract: In this talk we will define what it means for a cube complex to be collapsible. In particular, our definition will apply to the case that the complex is not finite. Then, we will show that all locally-finite CAT(0) cube complexes are collapsible. The process will yield an inverse sequence of compacta, the inverse limit of which will provide a weak Z-structure. Time permitting, we will discuss how this compactification relates to other established compactifications of CAT(0) cube complexes.