Seminar 4.9.15

Title: A discrete to continuous framework for projection theorems

Speaker: Daniel Glasscock

Abstract: Projection theorems for planar sets take the following form: the image of a “large” set A2 under “most” orthogonal projections πθ (to the line through the origin corresponding to θS1) is “large”; the set of those directions for which this does not hold is “small.” The first such projection theorem was given by J. M. Marstrand in 1954: assuming the Hausdorff dimension of A is less than 1, the Hausdorff dimension of πθA is equal to that of A for Lebesgue-almost every θS1.

Recent progress has been made on some fundamental problems in geometric measure theory by discretizing and using tools from additive combinatorics.  In 2003, for example, J. Bourgain building on work of N. Katz and T. Tao, used this approach to prove that a Borel subring of  cannot have Hausdorff dimension strictly between 0 and 1 (a result shown independently by G. A. Edgar and C. Miller), answering a question of P. Erd\H{o}s and B. Volkmann. The goal of my talk is to explain a discrete approach to continuous projection theorems.  I will show, for example, how Marstrand’s original theorem and a recent result of Bourgain and D. Oberlin can be obtained combinatorially through their discrete analogues. This discrete to continuous framework connects finitary combinatorial techniques to continuous ones and hints at further parallels between the two regimes.

Seminar 3.31.15 Sharp

Title: Noncommutative geometry and measures on dynamical systems

Speaker: Richard Sharp (University of Warwick)

Abstract: Noncommutative geometry aims to describe a wide range of mathematical objects in terms of C^*-algebras, in particular through the notion of a spectral triple. We will discuss how to recover important classes of invariant measures for certain dynamical systems on Cantor sets.

Seminar Schedule Spring 2015

Here is an overview of our current seminar schedule for this semester.

Feb 5: Brandon Seward (Michigan)

Feb 19: Dan Thompson (Ohio State)

Feb 26: Bill Mance (University of North Texas)

Mar 12: Sergey Bezuglyi  (University of Iowa & Institute for Low Temperature Physics, Ukraine)

Tuesday Mar 31: Richard Sharp (Warwick) [Note unusual day]

Thursday Apr 2: Departmental Colloquium: Richard Sharp (Warwick)

Apr 9: Daniel Glasscock (Ohio State)

Seminar 3.12.15 Bezuglyi

Speaker: Sergey Bezuglyi (University of Iowa & Institute for Low Temperature Physics, Ukraine)

Title: Homeomorphic measures on Cantor sets and dimension groups

Abstract: Two measures, m and m’ on a topological space X are called homeomorphic if there is a homeomorphism f of X such that m(f(A)) = m'(A) for any Borel set A. The question when two Borel probability non-atomic measures are homeomorphic has a long history beginning with the work of Oxtoby and Ulam: they found a criterion when a probability Borel measure on the n-dimensional cube [0, 1]^n is homeomorphic to the Lebesgue measure. The situation is more interesting for measures on a Cantor set. There is no complete characterization of homeomorphic measures. But, for the class of the so called good measures (introduced by E. Akin), the answer is simple: two good measures are homeomorphic if and only if the sets of their values on clopen sets are the same.

In my talk I will focus on the study of probability measures invariant with respect to a minimal (or aperiodic) homeomorphism. These measures are in one-to-one correspondence with traces on the corresponding dimension group. The technique of dimension groups allows us to apply new methods for studying good  traces. A good trace is characterized by its kernel having dense image in the annihilating set of affine functions on the trace space. A number of examples with seemingly paradoxical properties is considered.

The talk will be based on joint papers with D. Handelman and with O. Karpel.

Seminar 2.19.15 Thompson

Speaker: Dan Thompson (Ohio State)

Title: Unique equilibrium states for the robustly transitive diffeomorphisms of Mañé and Bonatti-Viana

Abstract: We establish results on uniqueness of equilibrium states for the well-known Mañé and Bonatti-Viana examples of robustly transitive diffeomorphisms. This is an application of machinery developed by Vaughn Climenhaga and myself, which applies when systems satisfy suitably weakened versions of expansivity and the specification property. The Mañé examples are partially hyperbolic, whereas the Bonatti-Viana examples are not partially hyperbolic but admit a dominated splitting. I’ll explain why these maps satisfy our hypotheses. This is joint work with Vaughn Climenhaga (Houston) and Todd Fisher (Brigham Young).

Seminar 2.5.15 Seward

Speaker: Brandon Seward

Title: Krieger’s finite generator theorem for ergodic actions of countable groups

Abstract: The classical Krieger finite generator theorem states that if a free ergodic probability-measure-preserving action of Z has entropy less than log(k), then the action admits a generating partition consisting of k sets. This was extended to actions of amenable groups independently by Rosenthal and Danilenko–Park. We introduce the notion of Rokhlin entropy which is defined for actions of general countable groups. Rokhlin entropy may be viewed as a natural extension of classical entropy, as when the acting group is amenable the two notions coincide. Using this notion of entropy, we prove Krieger’s finite generator theorem for actions of general countable groups.