Let F_k(M) denote the ordered k-point configuration space of a connected open manifold M. Work of Church and others shows that for a given manifold, as k increases, this family of spaces exhibits a phenomenon called homological "representation stability" with respect to the natural symmetric group actions. In this talk I will explain what this means, and describe a higher-order "secondary stability" phenomenon among the unstable homology classes. The project is work in progress, joint with Jeremy Miller.

Slow entropy is an useful invariant when dealing with systems of intermediate (polynomial) growth. The most classic examples are: horocycle flows, time changes of nilflows and mixing smooth flows on surfaces (with finitely many fixed points). In the talk we will focus mostly on computing slow entropy for the class of smooth flows on two-torus with one fixed point. As a consequence we get that such flows never are rank one and that the order of degeneracy of the fixed point is an invariant. Moreover, we establish variational principle for slow entropy in this class.

The handlebody group H_g is the subgroup of the mapping class group Mod_g of a surface formed by all those elements which extend to a given handlebody. In this talk we will first show that finite index subgroups of this group are rigid: any inclusion into Mod_g is conjugate to the standard inclusion. We then discuss flexible behaviour: the existence of inclusion of H_g into Mod_h whose image is not conjugate into any handlebody subgroup of Mod_h.

In this talk, I will explain why a typical interval exchange transformation does not commute with any other interval exchanges except for its powers.

A duality group has a pairing exhibiting isomorphisms between its homology and cohomology groups. Many naturally occurring groups fail to be duality groups, but are morally very close. In this talk we make this precise with the notion of a semiduality group and show that the lamplighter group is a semiduality group. We'll finish by stating a conjecture for semiduality of arithmetic groups over function fields and a positive result for arithmetic groups acting on products of trees. This talk covers work joint with Kevin Wortman.

Let X be a negatively curved symmetric space and Gamma a noncocompact lattice in Isom(X). We show that small, parabolic-preserving deformations of Gamma into a negatively curved symmetric space containing X remain discrete and faithful. (The cocompact case is due to Guichard.) This applies in particular to a version of Johnson and Millson's bending deformations, providing for all n infinitely many noncocompact lattices in SO(n,1) which admit discrete and faithful deformations into SU(n,1). We also produce deformations of the figure-8 knot group into SU(3,1), not of bending type, to which the result applies. This is joint work with Sam Ballas and Pierre Will.

For n at least 3, consider a lattice in Sl(n,R). Zimmer’s conjecture asserts that every action of the lattice on a manifold of dimension at most n-2 is finite. Recently, D. Fisher and S. Hurtado, and I established Zimmer’s conjecture under the additional assumption that the lattice is cocompact. I will give some background and motivation for the conjecture. I will outline our proof and explain a number of tools we use: strong Propterty (T), cocycle superrigidity, Ratner’s measure classification theorem, and smooth ergodic theory of Z^d actions.

I will present joint work with Parker and Paupert, that allowed us to exhibit new commensurability classes of non-arithmetic lattices in the isometry group of the complex hyperbolic plane. If time permits, I will also explain close ties between our work and the theory of discrete reflection groups acting on other 2-dimensional complex space forms.

Let V be a finite dimensional vector space over a p-adic field. The affine building of GL(V) can be constructed as the set of all lattice functions on V. Let G be a Chevalley group attached to a simple, split, Lie algebra over the p-adic field. I will explain how the affine building of G can be constructed as the set of (some) lattice functions on the Lie algebra.

The motivation for the talk is the recent result, joint with Vincent Guirardel and Camille Horbez, that Out(F_n) admits a topologically amenable action on a Cantor set. This implies the Novikov conjecture for Out(F_n) and its subgroups. Most of the talk will be an introduction to boundary amenability and ways to prove it for simpler groups.

The statistics of lattice points in Borel sets have been studied extensively, both for single lattices like the integral points in Euclidean space and on average over the space of lattices. The magnitude of the error term in the approximation is related to problems in spectral theory and number theory and good error terms have been obtained for typical lattices using tools from representation theory. However, averages over closed subspaces over the space of all lattices are far less accessible and the only discrepancy results so far have been associated to rank one subgroups of GL(n). In this work, joint with J. Athreya, we provide power savings bounds for the number of lattice points of a typical lattice from the general symplectic ensemble in a nested family of Borel sets. This is the first example of lattice point statistics for a higher rank group other than the full GL(n) and SL(n).

We will discuss some classical results on univalent maps and their applications to the renormalized volume of hyperbolic 3-manifolds. This is joint work with M. Bridgeman and J. Brock.

This talk will focus on abelian subgroups of the mapping class group and Out(F_n). After relating some structural results, we'll discuss how the intrinsic Euclidean geometry of abelian subgroups relates to the geometry of the ambient group.

Ricci flow is a geometric PDE that has had a profound impact on 3-dimensional topology. Like many geometric evolution equations, its solutions develop singularities, and their study has been crucial part of the story. Soon after introducing Ricci flow in 1982, Hamilton defined a notion of Ricci flow with surgery, a regularization scheme that allows one to avoid singularities. Building on many contributions of Hamilton, in 2003 Perelman used Ricci flow with surgery to prove Thurston’s Geometrization Conjecture, which includes the 3-dimensional Poincare Conjecture as special case. At the same time, Perelman drew attention to the ad hoc character of Ricci flow with surgery, and conjectured the existence of "Ricci flow through singularities”, which would be a canonical evolution for any Riemannian 3-manifold. Recently, Richard Bamler and I have proven Perelman’s conjecture, and used it to obtain new information about diffeomorphism groups of 3-manifolds.

Does a surface group act properly on a finite product of finite-valence trees? I don't know. I'll discuss two results motivated by this question, joint with David Fisher, Michael Larsen, and Ralf Spatzier. One hints toward a positive answer: faithful representations of surface groups into arithmetic groups in characteristic p. The other result is a structure theorem for proper actions of CAT(0) groups on products of trees that hints toward a negative answer; for example, these methods prove that right-angled Artin groups admitting such proper actions are products of free and free abelian groups.

Using the intersection number between curves, the space of laminations acts as its own dual space. For free groups, the space of currents acts as dual to the closure of outer space via the intersection form defined by Kapovich and Lustig. This intersection form can be used to show that a fully irreducible outer automorphism acts loxodromically on the free factor complex. With the goal of understanding reducible outer automorphism, in this talk I will define relative currents, relative outer space, discuss the intersection form between them and mention an application to the relative free factor complex.

We explore the bounded cohomology of closed surface groups whose actions on hyperbolic 3-space may have unbounded geometry. The isometry types of marked hyperbolic 3-manifolds are classified in terms of their end invariants. We discuss how the classification gives us a criterion for distinguishing bounded classes in degree 3 for surface groups and, more generally, finitely generated Kleinian groups without parabolics.

The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets. First introduced by Edelsbrunner et al., it compresses the components of the level sets of a multivariate mapping and obtains a summary representation of their relationships. A related construction called mapper, and a special case of the mapper construction called the Joint Contour Net have been shown to be effective in visual analytics. Mapper and JCN are intuitively regarded as discrete approximations of the Reeb space, however without formal proofs or approximation guarantees. An open question has been proposed by Dey et al. as to whether the mapper construction converges to the Reeb space in the limit. In this work, we are interested in developing the theoretical understanding of the relationship between the Reeb space and its discrete approximations to support its use in practical data analysis. Using tools from category theory, we formally prove the convergence between the Reeb space and mapper in terms of an interleaving distance between their categorical representations. Given a sequence of refined discretizations, we prove that these approximations converge to the Reeb space in the interleaving distance; this also helps to quantify the approximation quality of the discretization at a fixed resolution. Joint work with Elizabeth Munch, University at Albany – SUNY Albany.

Knots have traditionally been depicted using projections with crossings where two stands cross. But what if we allow three strands to cross at a crossing? Or four strands? Can we find projections of any knot with just one of these multi-crossings? We will discuss these generalizations of traditional invariants to multi-crossing numbers, ubercrossing numbers and petal numbers and their relation to hyperbolic invariants.

An infinite sequence in a compact metric space X is called minimal if it satisfies a weak periodicity condition. A compact metric space Y is called a minimality detector for X if every non minimal sequence in X can be taken to a non minimal sequence in Y by a continuous function. a priori whether Y is a minimality detector for X depends on if we are considering half-infinite or bi-infinite sequences, but I show this not to be the case. The question of when one finite graph is a minimality detector for another finite graph turns out to be the only interesting case, and some necessary conditions in this way will be presented.

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We prove that if G is a finitely generated non-virtually-abelian group, then (G X Z) * Z does not embed into Diff^2(S^1). In particular, the class of subgroups of Diff^r(S^1) is not closed under taking free products for each r >= 2. We complete the classification of RAAGs embeddable in Diff^r(S^1) for each integer r, answering a question in a paper of M. Kapovich. (Joint work with Thomas Koberda)