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 A⊆ℝ2 under “most” orthogonal projections πθ (to the line through the origin corresponding to θ∈S1) 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 θ∈S1.

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 ℝ 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.

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.