Max Dehn Seminar

on Geometry, Topology, Dynamics, and Groups

Fall 2017 Wednesday 3:15 - 4:15, Room LCB 222

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
February 21
Dmitry Kleinbock
Brandeis University

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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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