Speaker: Richard Sharp (Warwick)
Title: Growth and spectra on regular covers
Abstract: Two natural numerical invariants that can be associated to a Riemannian manifold are the bottom of the spectrum of the Laplacian operator and, if the manifold are negatively curved, the exponential growth rate of closed geodesics. Suppose we have a regular cover of a compact manifold. Then, for each of these quantities, we might ask under what circumstances we have equality between the number associated to the cover and the number associated to the base.Â This question becomes non-trivial questions once the cover is infinite. It turns out that the question has a common answer in the two cases and this depends only on the covering group as an abstract group. For the Laplacian, this result was obtained by Robert Brooks in the 1980s, and Rhiannon Dougall and I have recently obtained the analogue for the growth of closed geodesics. I will discuss this work, relating it to random walks and a class of groups introduced by von Neumann in his study of the Banach-Tarski Paradox.
Bio sketch: Richard Sharp is a mathematician at Warwick University, UK. He grew up in London and obtained a BSc in Mathematics from Imperial College, London, in 1987. This was followed by postgraduate study at Warwick, where he was supervised by William Parry, and he obtained his PhD, on the periodic orbit structure of hyperbolic flows, in 1990. After postdocs at IHES, Queen Mary, London, and Oxford, he moved to Manchester University in 1995, before returning to Warwick in 2012. He is mainly interested in ergodic theory and applications to geometry.
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)
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.
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.
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).