Speaker: William Mance (University of North Texas)

Title: Unexpected distribution phenomenon resulting from Cantor series expansions

Abstract: We explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen known ones.

Title: A Case of Anything Goes in Infinite Ergodic Theory

Speaker: Terry Adams, US DoD

Seminar Type:  Ergodic Theory/Probability

Abstract: Dynamical systems are well studied in the finite measure preserving case. Many of the same principles do not apply for infinite measure preserving transformations. As an example, for an invertible finite measure preserving transformation, its Cartesian product is ergodic if and only if it is weak mixing. As a consequence, all products of all non-zero powers are ergodic. In the case of invertible infinite measure preserving transformations, the situation is quite different. We give a class of transformations that demonstrate just about any reasonable behavior when it comes to ergodicity and conservativity of products of powers. Also, we’ll provide background on “weak” mixing notions in infinite measure.

Title: Joint ergodicity along generalized linear functions

Speaker: Younghwan Son, Weizmann Institute (Israel)

Title: Dynamics on random networks

Speaker: Sebastian Van Strien, Imperial College, London

Seminar Type:  Ergodic Theory/Probability

Abstract: Networks in which some nodes are highly connected and others have low connectivity are ubiquitous (they are used to model the brain, the internet, cities etc). In this talk I will consider coupled dynamics on randomly selected networks of this type. The methods used for analyzing this are related to those used to show stochastic stability of certain dynamical systems. This work is joint with Tiago Pereira and Jeroen Lamb (Imperial College)

Title: A Dimension for Subsets of Z^d and a  Related Marstrand Theorem

Speaker: Daniel Glasscock, The Ohio State University

Title: Just so stories (R. Kipling)

Speaker: Dima Burago,  Penn State

Seminar Type:  Ergodic Theory and Probability

Abstract: This is not a usual type of a seminar talk, though I have given several talks with almost the same title lately. Still the talks are different. I made transparencies for more than 20 topics, two to three slides per topic. Necessary definitions, formulations of key results, hints towards proofs, open problems. I choose about 8 topics per talk. The choice depends on the audience, the weather, what I had for breakfast and such. The only thing that unites the topics is that they have been of interest to me in past number of years. They all are related to geometry, PDEs, dynamics, geometric group theory and such.

Title: Gowers Norms and Multiple Recurrence of Sparse Measures on R^d

Speaker: Marc Carnovale, The Ohio State University

Seminar Type:  Ergodic Theory and Probability Seminar

Abstract: It is classical that any positive measure subset of the reals must contain scaled, translated images of any finite configuration of points. Does this still hold for natural classes of ”large” singular sets? A construction of Keleti says that Hausdorff dimension 1 is insufficient to guarantee such a result even for 3-term arithmetic progressions (3APs), while a result of Laba and Pramanik says that the stronger notion of Fourier dimension does yield a result for 3APs, but leaves the case of longer progressions open. Using the notion of intersections of measures from geometric measure theory as a guide, we study a quantity which can be thought of as measuring the multiple recurrence properties under the shift operator of a singular measure, rather than a set, in the torus, and show that it is positive when the measure under question satisfies a certain “higher order Fourier dimension” condition.

Title: Finite Sum Sets and Minimally Almost Periodic Groups

Speaker: Donald Robertson, The Ohio State University

Seminar Type: Ergodic Theory and Probability Seminar

Abstract: A result due to Hindman states that, no matter how the positive integers are finitely partitioned, one cell of the partition contains a sequence and all its sums without repetition. Straus, answering a question of Erdos, later gave an example showing that a density version of Hindman’s result does not hold. He exhibited sets of positive integers with arbitrarily large density, each having the property that no shift contains a sums set of the above kind. In this talk I will present recent joint work with V. Bergelson, C. Christopherson and P. Zorin-Kranich in which we generalize Straus’s example to a class of locally compact, second countable, amenable groups and show, using ergodic theory techniques recently developed by Host and Austin, that positive density subsets of groups outside this class must contains sets with strong combinatorial properties. In particular, this allows us to give a combinatorial characterization of minimally almost periodic, amenable groups.