Let S be an orientable surface of finite type. Using Pho-On's infinite unicorn paths, we prove the hyperfiniteness of the orbit equivalence relation coming from the action of the mapping class group of S on the Gromov boundary of the arc graph of S. This is joint work with Marcin Sabok.
The geodesic flow on hyperbolic finite-volume hyperbolic manifolds is a particularly well-studied dynamical system; this is in part due to its connection to other important dynamical systems on the manifold, as well as orbital counting and other number-theoretic problems related to discrete subgroups of orthogonal groups. In recent years, there has been some interest in generalizing many of the properties of the geodesic flow on finite-volume manifolds to the infinite-volume setting. I will discuss joint work with Hee Oh in which we establish exponential mixing of the geodesic flow on infinite-volume geometrically finite hyperbolic manifolds with large enough critical exponent. Patterson-Sullivan densities and Burger-Roblin measures, the Lax-Phillips spectral gap for the Laplace operator on infinite volume geometrically finite hyperbolic manifolds, and complementary series representations are all involved in both the statement and proof of our result, and I will try to explain how these different objects are related in this setting.
I will discuss recent progress on understanding the tmf-based Adams spectral sequence, where tmf = topological modular forms. The idea is to generalize the work of Mahowald and others in the context of bo-resolutions. The work I will discuss is joint with Prasit Bhattacharya, Dominic Culver, and J.D. Quigley.
Together with Alex Degtyarev and Vincent Florence we introduced a new link invariant, called slope, of a colored link in an integral homology sphere. In this talk I will define the invariant, highlight some of its most interesting properties as well as its relationship to Conway polynomials and to the Kojima–Yamasaki eta-function. The stress in this talk will be on our latest computational progress: a formula to calculate the slope from a C-complex.
In this session we will host a Q&A with current researchers who have recently gone through successful applications as well as more senior staff who have been on interview panels and hiring committees for postdoctoral positions in mathematics. The session will be a chance to get varied perspectives on the application process and find out about the different types of academic positions to apply for.
The panel members will be Candy Bowtell, Luci Basualdo Bonatto, Mohit Dalwadi, Ben Fehrman and Frances Kirwan.
During the early growth of the brain, an extraordinary process takes place where axons, neurons, and nerves extend, grow, and connect to form an intricate network that will be used for all brain activities and cognitive processes. A fundamental scientific question is to understand the laws that these growing cells follow to find their correct target.
Most of the talk will be about the Farrell-Jones conjecture from the point of view of an outsider. I'll try to explain what the conjecture is about, why one wants to know it, and how to prove it in some cases. The motivation for the talk is my recent work with Fujiwara and Wigglesworth where we prove this conjecture for (virtually torsion-free hyperbolic)-by-cyclic groups. If there is time I will outline the proof of this result.
Round up, the Oxford Mathematics Annual Newsletter, is a calculated attempt to describe our lives, mathematical and non-mathematical, over the past 12 months. From a summary of some of our research into the Coronavirus to a moving tribute to Peter Neumann by Martin Bridson, via articles on diversity, fantasy football and of course our Nobel Prize winner (pictured), it throws a little light, we hope, on what we did during the year that was 2020.
