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Max Dehn Seminar

on Geometry, Topology, Dynamics, and Groups

Fall 2019-Spring 2020 3:15-4:15 LCB 225

Date Speaker Title click for abstract (if available)
August 21 Emily Stark
University of Utah

Two far-reaching methods for studying the geometry of a finitely generated group with non-positive curvature are (1) to study the structure of the boundaries of the group, and (2) to study the structure of its finitely generated subgroups. Cannon--Thurston maps, named after foundational work of Cannon and Thurston in the setting of fibered hyperbolic 3-manifolds, allow one to combine these approaches. Mj (Mitra) generalized work of Cannon and Thurston to prove the existence of Cannon--Thurston maps for normal hyperbolic subgroups of a hyperbolic group. These maps can be used to understand the structure of the boundary of such groups. I will explain why similar theorems fail for certain CAT(0) groups. This is joint work with Benjamin Beeker, Matthew Cordes, Giles Gardam, and Radhika Gupta

September 11 Spencer Dowdall
Vanderbilt University

Given a nested decreasing family of targets B_n in a measure space X equipped with a flow phi_t (or transformation), the shrinking target problem asks to characterize when there is a full measure set of points x that hit the targets infinitely often in the sense that $${n \in N : phi_n(x)\in B_n}$$ is unbounded. This talk will examine the discrete shrinking target problem for the Teichmüller flow on the moduli space of unit-area quadratic differentials and show that for any ergodic probability measure, almost every differential will hit a nested spherical targets infinitely often provided the measures of the targets are not summable. Our key tool is an effective mean ergodic theorem stating that the time-average of any L^2 function converges to its space-average at a uniform rate in L^2. As an application, we obtain a logarithm law describing how quickly generic discrete geodesic trajectories accumulate on a given point. Joint with Grace Work.

September 18 Daniel Woodhouse
Oxford University

Gromov's program for understanding finitely generated groups up to their large scale geometry considers three possible relations: quasi-isometry, abstract commensurability, and acting geometrically on the same proper geodesic metric space. A *common model geometry* for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is *action rigid* if any group G' that has a common model geometry with G is abstractly commensurable to G. We show that free products of closed hyperbolic manifold groups are action rigid. As a corollary, we obtain torsion-free, Gromov hyperbolic groups that are quasi-isometric, but do not even virtually act on the same proper geodesic metric space. This is joint work with Emily Stark.

September 25 John Smillie
University of Warwick

A powerful tool for understanding the geometry and dynamics of the torus is to look at dynamics of flows on the moduli space of tori. There are two natural generalisations of this idea. One is to look at higher dimensional tori the other is to flat surfaces of higher genus. In the first case the relevant dynamics are homogeneous dynamics we can apply the powerful results of Ratner and others. The second case involves more exotic dynamics and is more mysterious. I will describe some recent joint work with Jon Chaika and Barak Weiss and some older work with Barak Weiss and explain how it is connected to this question.

October 2 Alexander Rasmussen
Yale University

The curve graph of a finite type surface is a crucial tool for understanding the algebra and geometry of the corresponding mapping class group. Many of the applications that arise from this relationship rely on the fact that the curve graph is hyperbolic. We will describe actions of mapping class groups of infinite type surfaces on various graphs analogous to the curve graph. In particular, we will discuss the hyperbolicity of these graphs, some of their quasiconvex subgraphs, properties of the corresponding actions, and applications to bounded cohomology.

October 9 No seminar (Fall Break)
October 16 Giovanni Forni
University of Maryland

We study cohomological equations and ergodic integrals for twisted translation flows, define as products of a translation flow on a translation surface and a linear flow on a circle. By standard Fourier analysis the questions we consider reduce respectively to non-homogeneous cohomology equations with purely imaginary constant zero-order term (twisted cohomological equation) and to ergodic integrals of functions times an exponential of time with purely imaginary phase (twisted ergodic integrals). The motivation is two-fold: on the one hand we want to understand a simple example of 3-dimensional translation flows, on the other hand there is a well-known close connection between twisted ergodic integrals and spectral measures of translation flows, already exploited in the work of Bufetov-Solomyak. In this respect our aim is to cast their work in more geometric terms and to generalize it. Our main results results are effective weak mixing results for translation flows: lower bounds on the dimension of spectral measures and upper bounds on the speed of weak mixing.

October 23 Eduard Schesler
Universitaat Bielefeld

Given a finitely generated group G, the Sigma invariants of G consist of geometrically defined subsets Sigma^k(G) of the set S(G) of all characters chi: G -> R of G. These invariants where introduced independently by Bieri-Strebel and Neumann for k=1 and generalized by Bieri-Renz to the general case in the late 80's in order to determine the finiteness properties of all subgroups H of G that contain the commutator subgroup [G,G]. In this talk we determine the Sigma invariants of certain S-arithmetic subgroups of Borelgroups in Chevalley groups. In particular we will determine the finiteness properties of every subgroup G of the group of upper triangular matrices B_n(Z[1/p]) < SL_n(Z[1/p]) that contains the group U_n(Z[1/p]) of unipotent matrices where p is any sufficiently large prime number.

October 30 Benjamin Brück
Universität Bielefeld

Much of the modern treatment of automorphism groups of free groups is motivated by analogies with arithmetic groups. I will present a new family of complexes interpolating between two well-studied objects associated to these classes of groups: the free factor complex and the Tits building of GLn(Q). Each of the new complexes is associated to the automorphism group Aut(AΓ) of a right-angled Artin group and has the homotopy type of a wedge of spheres. The dimension of these spheres forms a new invariant associated to Aut(AΓ). These complexes can also be seen as an Aut(AΓ)-analogue of the curve complex.

November 6 Genevieve Walsh
Tufts University

A group G is called coherent if every finitely generated subgroup of G is finitely presented. We show that free-by-free groups satisfying a particular homological criterion are incoherent. This class is large in nature, including many examples of hyperbolic and non-hyperbolic free-by-free groups. We apply this criterion to finite index subgroups of $$F_2\rtimes F_n$$ to show incoherence of all such groups, and to other similar classes of groups. We also discuss some limitations of our methods. This is joint work with Rob Kropholler.

November 20 Camille Horbez
University of Paris-Sud and CNRS

Motivated by questions concerning the rigidity of certain von Neumann algebras associated to groups or group actions, Boutonnet, Ioana and Peterson recently introduced the notion of proper proximality of a countable group. I will describe this notion and the motivations behind it, and explain how techniques from geometric group theory can be used to show that certain nonpositively curved groups, including rank one CAT(0) groups and mapping class groups, are properly proximal. This is a joint work with Jingyin Huang and Jean Lécureux.

November 25 at 3:15
JWB 335
Note unusual time and place
Chenxi Wu
Rutgers University

Thurston's "master teapot" is a 3 dimensional plot of the roots of Galois conjugates of the entropy of unimodal maps with periodic critical orbit. We found a description of the shape of this teapot using iterated function systems, which provides algorithms for testing if a point belongs to the teapot. We also proposed a conjecture on a Julia-Mandabrot like relationship on the teapot, which might be useful for the study of iterated function systems and symbolic dynamics in general. This is a collaboration with Harrison Bray, Diana Davis and Kathryn Lindsey.

November 27 No seminar (Thanksgiving)
December 6
LCB 215 at 11-12
Note unusual time and place
Polona Durcik
California Institute of Technology

For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain's approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. Joint work with Shaoming Guo and Joris Roos.

January 8 Yuchen Liu
Yale University

A few years ago, Chi Li introduced the notion of local volume of Kawamata log terminal (klt) singularities as the minimum normalized volume of valuation. This invariant carries lots of interesting geometric information of the singularity, for instance: it characterizes smooth points; it detects orbifold order of quotient singularities; it is bounded from above by the minimal log discrepancy. In this talk, I will discuss the conjecture that local volumes of klt singularities in a fixed dimension with finite coefficient set has only accumulation point zero. We confirm this conjecture when ambient singularities are bounded. This is a joint work in progress with Jingjun Han and Lu Qi.

January 29 Jonathan Campbell
Duke University

The Dehn invariant is a classical invariant on 3 dimensional polytopes that governs scissors congruence classes. These have traditionally been defined homologically, but I'll present a new definition that exposes significant underlying structure and allows for an approach to some conjectures of Goncharov about motivic complexes. The relevant invariant and theorems fall out of a deeper understanding of the classical Solomon-Tits theorem combined with techniques from homotopy theory. In this talk I'll give an introduction to some of these techniques and indicate the proof of a very surprising theorem relating Goncharov's complex to the homology of Lie groups made discrete.

Feb 3 at 3:15 pm
LCB 121
Note unusual place and day
Filippo Mazzoli
University of Luxembourg

In this talk I will describe how constant Gaussian curvature (CGC) surfaces interpolate the structures of the pleated boundary of the convex core and of the boundary at infinity of a quasi-Fuchsian manifold, and I will present a series of consequences of this phenomenon, such as a description of the renormalized volume in terms of the CGC surfaces foliations, and a generalization of McMullen’s Kleinian reciprocity theorem.

February 5 Kurt Vinhage
Pennsylvania State University

The study of high complexity measure-preserving flows has one numerical invariant which characterizes it: Kolmogorov-Sinai entropy. Entropy measures the exponential growth rate of orbit types segments relative to their lengths. For low-complexity systems, if the number of orbit types is uniformly bounded, the system is equivalent to an isometry. We will discuss ``slow'' entropy type invariants, which measure subexponential growth rates, why they are appropriate for a natural class of parabolic systems and how they help to classify such systems.

Feb 6 at 2 pm
LCB 222
Note unusual place, day, and time
Matt Clay
University of Arkansas

I will explain a structure theorem for groups acting on projection complexes under certain hypotheses. I will also mention some applications to normal subgroups of mapping class groups. This is joint work with Johanna Mangahas and Dan Margalit.

Feb 10 at 3 pm
LCB 323
Note unusual place, day, and time
Matt Durham
UC Riverside

The study of the coarse geometry of the mapping class group and Teichmuller space has recently seen an influx of ideas coming from the world of CAT(0) cubical complexes. Perhaps most remarkably, Behrstock-Hagen-Sisto recently proved that the coarse convex hulls of finite sets of points in these spaces are coarsely modeled by cube complexes. Using work of Bestvina-Bromberg-Fujiwara-Sisto, we improve their construction to build modeling cube complexes which are coarsely stable under perturbation of the relevant data. As initial applications, we build a bicombing of the mapping class group and Teichmuller space, and prove that finite sets in these spaces admit coarse barycenters. This is joint work with Yair Minsky and Alessandro Sisto.

February 12 Barak Weiss
Tel Aviv University

We obtain new upper bounds on the minimal density of lattice coverings of R^n by dilates of a convex body K. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice L satisfies L + K = R^n. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem. I will not assume any prior knowledge of lattice coverings. Joint with Or Ordentlich and Oded Regev.

February 19 Javier de la Nuez Gonzales
Universidad del Pais Vasco (Bibao)

I will discuss joint work in preparation with Valentina Disarlo and Thomas Koberda in which we study the curve graph from the point of view of logic. In particular, we prove that it is an omega-stable structure, which means that there is a well-behaved notion of dimension, it's Morley rank, that measures the degrees of freedom of any set definable with parameters (a notion that generalizes Krull dimension for constructible sets in algebraically closed fields). I will address the notion of (bi)-interpretability, which allows to compare the logical strength of theories in different languages, its role in the proof and potential strategies for finding obstructions to the mutual (bi)-interpretability of different geometric complexes.

March 25 Nicholas Miller
UC Berkeley


April 8 Elizabeth Field


April 15 Mark Pengitore
Ohio State University


April 22 Michal Marcinkowski
Polish Academy of Sciences (IMPAN)


April 29 Harrison Bray
University of Michigan


April 30 Dave Futer
Temple University


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You may also be interested in the RTG Seminar
Max Dehn Seminar is organized by Mladen Bestvina, Ken Bromberg, Jon Chaika, Osama Khalil,
Priyam Patel, Emily Stark, Domingo Toledo, and Kevin Wortman.

This web page is maintained by Priyam Patel.