A central problem in the study of countable Borel equivalence relations is a question of Weiss, which asks whether every action of an amenable group gives rise to a Borel hyperfinite orbit equivalence relation. Although we are very far from resolving this problem in full generality, a major breakthrough occurred 5 years ago with the introduction of Borel asymptotic dimension by Conley, Jackson, Marks, Seward, and Tucker-Drob. In their work, they show that if a group $G$ admits a normal series of a certain form (e.g., is polycyclic), any (free) action of $G$ has finite Borel asymptotic dimension and thus its orbit equivalence relation is hyperfinite. We extend their results, and, in particular, show that this holds for any free action of a solvable group with finite (standard) asymptotic dimension. Based on a joint work with Qingyuan Chen, Alon Dogon, and Brandon Seward.
Recent advances in AI development made it good enough to solve hard mathematical problems. This summer, several AI companies, including our startup Harmonic, claimed that their system solved 5 out of 6 problems at the International Math Olympiad. Except Harmonic, all other companies that achieved the same results are technological giants like Google and OpenAI.
It is well-known that AI hallucinates to convince itself that it achieved the goal, so how could we trust the AI-produced solutions? Our answer is to make AI produce proofs that a computer can read and verify. Our system Aristotle takes a problem stated in Lean - a special programming language with capabilities to verify mathematical proofs - and produces proofs in the same language. This way, a human doesn't have to read all the details of the computer-generated proof, just double check that the theorem is stated correctly.
I am going to talk about Lean and its mathematical library Mathlib. I will also show our system Aristotle in action. I assume no prior knowledge about computer proof assistants or AI.
We discuss two related problems pertaining to the dynamics of polygonal billiards. First, we discuss joint work with Chaika and Forni describing how chaos, i.e., sensitivity to initial conditions, arises in the setting of rational billiards. Second, we discuss joint work with Athreya and Forni studying quantizations of these systems. No previous knowledge on the subject will be assumed.
Abstract:
I will explain how the theory of contracting self-similar groups can be extended to inverse semigroups. Contracting self-similar inverse semigroups are naturally associated with spaces admitting hyperbolic homeomorphisms with totally disconnected stable direction. We will discuss some basic examples such as the Fibonacci adic transformation and tiling semigroups.
You are invited to the joint meeting of three seminars: Groups&Dynamics, Geometry, and Noncommutative Geometry, on Wednesday August 27, 10-11:00 am.
The talk is by zoom. The talk will be broadcasted in BLOC 302, feel free to attend in person or via zoom.
Our speaker is Christopher-Lloyd Simon, Pennsylvania State University.
Title: Quasi-characters of the modular group from linking numbers of modular knots
Abstract:
The modular group PSL(2;Z) acts on the hyperbolic plane with quotient the modular surface M, whose unit tangent bundle U is a 3-manifold homeomorphic to the complement of the trefoil knot in the sphere. The hyperbolic conjugacy classes of PSL(2;Z) correspond to the closed oriented geodesics in M. Those lift to the periodic orbits for the geodesic flow in U, which define the modular knots.
The linking number between a modular knot and the trefoil is well understood. Indeed, E. Ghys showed in 2006 that it is given by the Rademacher function, a homogeneous quasi-morphism of the modular group which he recognised with J. Barge in 1992 as half the primitive of its bounded Euler class. This shed light on M. Atiyah 1987 work identifying Rademacher's invariant with no less than that six other important functions appearing in diverse areas of mathematics.
What about the linking number between two modular knots? We show that the linking number with a modular knot minus that with its inverse yields a homogeneous quasi-morphism on the modular group, and explain how to extract out of these a Schauder basis for the topological vector space of all quasi-morphisms (for this, we prove that the linking pairing is non-degenerate).
We also present a geometric linking function over the character variety of the modular group, whose value at the boundary recovers the topological linking number.
Address your questions to Nataliya Goncharuk, natasha_goncharuk AT tamu DOT edu