Activities This Week

Earlier this year, we discovered a new class of zero-dimensional dynamical systems, which we call “fiberwise essentially minimal”, that are of importance to operator algebras because of the nice properties, in particular K-theoretic classification, of the crossed product. Today, we discuss the Bratteli diagrams associated to these systems, and extend the K-theoretic classification to include a dynamical condition called “strong orbit equivalence”, extending the existing result in the minimal case due to Giordano-Putnam-Skau.

A function at a non-critical point can be converted to a linear form by a local coordinate change. At an isolated critical point one has the weaker statement: higher order perturbations do not change the group orbit. Namely, the function is determined (up to the local coordinate changes) by its (finite) Taylor polynomial.

This finite-determinacy property was one of the starting points of Singularity Theory. Traditionally such statements are proved by vector field integration. In particular, the group of local coordinate changes becomes a ``Lie-type” group.

I will show such determinacy results for maps of germs of (Noetherian) schemes. The essential tool is the “vector field integration” in any characteristic. This equips numerous groups acting on filtered modules with the ``Lie-type” structure. (joint work with G. Belitskii, A.F. Boix, G.M. Greuel.)

I will discuss results and open problems in an emerging theory of ‘canonical’ algebraic cycles for all motives enjoying a certain symmetry. The construction is inspired by theta series, and based on special subvarieties in arithmetic quotients of the complex unit ball. The ‘theta cycles’ seem as pleasing as Heegner points on elliptic curves: (1) their nontriviality is detected by derivatives of complex or p-adic L-functions; (2) if nontrivial, they generate the Selmer group of the motive. This supports analogues of the Birch and Swinnerton-Dyer conjecture. I will focus on (2), whose proof combines the method of Euler systems and the local theta correspondence in representation theory.

Apéry’s proof of the irrationality of ζ(3) used a specific linear recursion that formed a Polynomial Continued Fraction (PCF). Similar PCFs can prove the irrationality of other fundamental constants such as 𝜋 and e. However, in general, it is not known which ones create useful Diophantine approximations and under what conditions they can be used to prove irrationality. Here, we will present theorems and general conclusions about Diophantine approximations created from polynomial recursions. Specifically, we generalize Apéry’s work from his particular choice of PCF to any general PCF, finding the conditions under which a PCF can be used to prove irrationality or to provide an efficient Diophantine approximation. We further propose new conjectures about Diophantine approximations based on PCFs. Our study may contribute to ongoing efforts to answer open questions such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., ζ(5)).