Max Dehn Seminar

on Geometry, Topology, Dynamics, and Groups

Fall 2017 Wednesday 3:15 - 4:15, Room LCB 215

Date Speaker Title click for abstract (if available)
August 30
Moon Duchin
Tufts University
Discrete curvature, with applications
September 6
Matthew Smith
University of Utah
Unique Ergodicity for Quadratic Differentials
A quadratic differential on a surface determines a singular foliation equipped with a natural transverse invariant measure. Whether this is the only measure or not has important consequences for the dynamics of the foliation. We will discuss a new sufficient condition for unique ergodicity, which is the case when there is exactly one measure. For special classes of quadratic differentials, we obtain stronger results. We will also discuss the limitations of the theorem and some new examples.
September 13
Alena Erchenko
Pennsylvania State University
Flexibility of some dynamical and geometrical data
We introduce the flexibility program proposed by A. Katok and discuss first results. We show the flexibility of the entropy with respect to the Liouville measure and topological entropy for geodesic flow on negatively curved surfaces with fixed genus and total area (joint with A. Katok). Also, we point out some restrictions which come from additionally fixing a conformal class of metrics (joint with T. Barthelm´e). If time permits, we describe a flexibility result for Lyapunov exponents for smooth expanding maps on a circle of fixed degree.
September 20
Florian Richter
The Ohio State University
The dichotomy between structure and randomness in multiplicative number theory
We will begin the talk by discussing a dichotomy theorem in multiplicative number theory which asserts that any multiplicative function (that satisfies certain minor regularity conditions) is either a (special kind of) almost periodic function or a pseudo-random function. Then we will explore how this phenomenon extends to other classical objects coming from multiplicative number theory. In particular, we will study the combinatorial and dynamical properties of level sets of multiplicative functions and I will present a structure theorem which says that for any level set E of an arbitrary multiplicative function there exists a highly structured superset R such that E is a pseudo-random subset of R.
September 27
Chris Cashen
University of Vienna
The contracting boundary of a group
We construct a bordification of a proper geodesic metric space by adding a ‘contracting boundary’ consisting of equivalence classes of rays satisfying a contraction property enjoyed by rays in a hyperbolic space. We think of these as the distinct ways of going to infinity through hyperbolic directions. The topology we introduce on this contracting boundary is invariant under quasi-isometries and is homeomorphic to the Gromov boundary when the space is hyperbolic. If the space admits a geometric group action then our topology on the boundary is metrizable. This is joint work with John Mackay.
October 18
Marco Lopez
University of North Texas
Dimension of shrinking target sets arising from non-autonomous dynamics.
In analogy to the set of well-approximable numbers in Diophantine approximation, a shrinking target set is defined as the set of points in a metric space, X, whose orbits under a dynamical system on X hit infinitely often a ball of radius shrinking to zero. Using techniques from thermodynamic formalism we establish a formula for the Hausdorff dimension of such sets in the context of non-autonomous iterated function systems.
October 25
Sang-hyun Kim
Seoul National University
Free products in Diff(S1)
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)
November 15
Shariar Mirzadeh
Brandeis University
Dimension estimates for the set of points with non-dense orbit in homogeneous spaces.
In this talk we study the set of points in a homogeneous space whose orbit escapes the complement of a fixed compact subset. We find an upper bound for the Hausdorff dimension of this set. This extends the work of Kadyrov, where he found an upper bound for the Hausdorff dimension of the set of points whose orbit misses a fixed ball of sufficiently small radius in a compact homogeneous space. We can also use our main result to produce new applications to Diophantine approximation. This is joint work with Dmitry Kleinbock.
November 29
Zhenqi Wang
Michigan State University
Smooth Local rigidity of algebraic actions.
At first, we will introduce the background of algebraic actions and give some interesting examples, next, we will review of various smooth rigidity results for higher-rank algebraic actions and recent progress. Finally, we will talk about future directions.
January 8 at 4pm in LCB 219
*Note unusual time and place*
Laura Schaposnik
University of Illinois at Chicago
On Cayley and Langlands type correspondences for Higgs bundles
The Hitchin fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing the type of questions and methods considered in the area, we shall dedicate this talk to the study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations. Through Cayley and Langlands type correspondences, we shall provide a geometric description of these objects, and consider the implications of our methods in the context of representation theory, Langlands duality, and within a more generic study of symmetries on moduli spaces.
January 10
Jennifer Wilson
Stanford University
Stability in the homology of Torelli groups
The Torelli subgroups of mapping class groups are fundamental objects in low-dimensional topology, through some basic questions about their structure remain open. In this talk I will describe these groups, and how to use tools from representation theory to establish patterns their homology. This project is joint with Jeremy Miller and Peter Patzt. These “representation stability” results are an application of advances in a general algebraic framework for studying sequences of group representations.
January 31
Barak Weiss
Tel Aviv University
New examples for the horocycle flow on the moduli space of translation surfaces
The moduli space of translation surfaces is a space on which SL(2,R) acts via its action on the complex plane, viewed as the plane R^2. A folklore and imprecise conjecture is that this action has many features in common with actions of Lie groups generated by unipotent elements on homogeneous spaces, which were shown to be rigid in fundamental work of Ratner. Work of Eskin, Mirzakhani and Mohammadi, concerning action of the entire group SL(2,R), justified this expectation. The restriction of the action to upper triangular unipotent matrices is called the horocycle flow. For this flow a similar folklore conjecture could be made. We provide new examples showing that the horocycle flow exhibits features absent from analogous homogeneous flows, e.g. orbits which are not generic for any measure, and orbit closures with non-integer Hausdorff dimension. Joint work with Jon Chaika and John Smillie.
February 2 at 3:05 in LCB 219
*Note unusual time and place*
Sebastian Hurtado
University of Chicago
Burnside problem on diffeomorphism groups
Suppose G is a finitely generated group such that every element has finite order. Must G be a finite group? This is known as the burnside problem, it was formulated around 1902 by Burnside himself and it was central in the development of group theory during the 20th century. The answer in general turned out to be negative, G might be infinite. Nonetheless, if one restricts G to be a linear group (group of matrices), the answer is positive (Schur, 1911). The problem remains open if we assume G is a group of homeomorphisms of a surface or a manifold in general. I will talk about the case where G is a group of diffeomorphisms of a surface.
February 21
Dmitry Kleinbock
Brandeis University
Hyperbolic dynamics and intrinsic Diophantine approximation
Dynamics on homogeneous spaces of Lie groups has been a useful tool in solving many previously intractable Diophantine problems. In this talk I will describe some existing connections between homogeneous dynamics and Diophantine approximation, and then show how a similar approach can help quantify the density of rational points on quadric hypersurfaces (intrinsic approximation problems). The case of spheres is reduced to dynamics on hyperbolic manifolds. Joint with Lior Fishman, Keith Merrill and David Simmons.
March 14
Brian Collier
University of Maryland
Special deformations of Fuchsian representations into SO(p,q)
The space of discrete and faithful representations of a closed surface group into Isom(H^2)=SO(1,2) defines a connected component of the character variety. While deformations of such representations into SO(1,3) define interesting hyperbolic 3-manifolds, the space of such geometrically interesting representations is not closed in the SO(1,3) character variety. In fact, such representations can be continuously deformed to have compact Zariski closure. In this talk we will define analogous connected components of the SO(p,p+1) character variety and consider their deformations into the SO(p,p+2) character variety (and more generally SO(p,q)). Unlike the p=1 case, such representations cannot be deformed compact representations. This leads to a dichotomy for the connected components of the SO(p,q) character variety which we will relate to recent notions of positivity introduced by Guichard and Wienhard.
March 7 Starting at 3pm
Autumn Kent
University of Wisconsin
Spacious knots
Brock and Dunfield showed that there are integral homology spheres whose thick parts are very thick and take up most of the volume. Precisely, they show that, given R big and r small, there is an integral homology 3-sphere whose R-thick part has volume (1 − r)vol(M). Purcell and I find knots in the 3-sphere with this property, answering a question of Brock and Dunfield.
March 28
Joel Moreira
Northwestern University
Multiple recurrence along sparse sequences over thick sets
Khintchine’s recurrence theorem states that the set of optimal return times in a measure preserving dynamical system is syndetic, i.e., has bounded gaps. In 1977, as part of his ergodic-theoretic proof of Szemeredi’s theorem on arithmetic progressions, Furstenberg established a partial extension of Khintchine’s result by showing that the set of multiple return times is also syndetic. Multiple recurrence has since been established along many different types of sequences, including polynomial sequences and sequences derived from functions in a Hardy field. However, they don’t always lead to syndetic return time sets. In my talk I will describe joint work with Vitaly Bergelson and Florian Richter where we establish that for a general class of non-polynomial sparse sequences, the set of return times still possesses interesting combinatorial properties, and in particular it satisfies a weak form of syndeticity and is thick, i.e. contains arbitrarily long intervals of integers. Via Furstenberg’s correspondence principle our work leads to novel variants of Szemeredi’s theorem.
April 4
Roger Baker
Bringham Young University
Dependent variables that miss fixed targets.
Let n(1), n(2),... be a strictly increasing sequence of integers. The dependent variables in question are n(1)x, n(2) x,... where x is in the interval (0,1]. As for missed targets, an interval modulo one never entered by the fractional parts of the dependent variables is such a target. Now things depend in a rather baffling fashion on the particular sequence of integers. The set of x for which a target is missed is certainly of measure zero, but does it have dimension 1, or some dimension between 0 and 1, or indeed is it finite and can we bound the number of x in this set? I will survey what is known about this and propose some conjectures.
April 11
Donald Robertson
University of Utah
The Erdos sumset conjecture
Erdos conjectured that every set of natural numbers with positive density contains B+C for some infinite sets B,C of natural numbers. In this talk I will describe joint work with J Moreira and F. Richter on resolving this conjecture using ideas from ergodic theory.
April 18
Kelly Yancey
Institute for Defense Analyses
Rigid Substitutions
Rigidity is an important property of a large set of dynamical systems; in fact the property is generic. During this talk I will discuss substitution systems that are rigid. I will introduce a matrix and show how to determine rigidity based on the spectrum of the matrix for constant length substitutions. I will also link this with cutting and stacking transformations. This is joint work with Jon Fickenscher.
April 25
Peter Smillie
Entire spacelike surfaces of constant curvature in Minkowski 3-space
We prove that every regular domain in Minkowski 3-space which is not a wedge contains a unique entire spacelike surface with constant intrinsic curvature equal to -1. This completes the classification of such surfaces in terms of their domains of dependence, for which partial results were obtained by Li, Guan-Jian-Schoen, and Bonsante-Seppi. Using this result, we obtain an analogous classification of entire spacelike surfaces with constant mean curvature (CMC). We'll apply these ideas to the Minkowski problem of prescribed curvature and to the construction CMC times in 2+1 relativity, and we'll see what we can say about the problem of deciding when the induced hyperbolic metric on an entire surface is complete. Everything is joint with Francesco Bonsante and Andrea Seppi.

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Max Dehn Seminar is organized by Mladen Bestvina, Ken Bromberg, Jon Chaika,
Donald Robertson, Domingo Toledo, and Kevin Wortman.

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