Activities This Week

Random walks on lattices have been studied for decades and are by now very well understood. In this talk we will define a random walk on the primitive points of a lattice and discuss its properties. The random walk is obtained in a similar manner to the classical one with the difference that one divides by the gcd at each step. Subject to suitable conditions on the measure generating the walk, we will see how these random walks correspond to positive recurrent Markov chains. In particular we will see that there is a unique stationary distribution for these random walks.

I will describe a link between tame geometry and diophantine geometry that has been unfolding in the past decade following the fundamental theorem of Pila-Wilkie in the theory of o-minimal structures. In particular I will describe how this theorem has been used in proofs of the Manin-Mumford conjecture (by Pila-Zannier), the Andre-Oort conjecture for modular curves (by Pila) and many other questions of “unlikely intersections” in diophantine geometry. I will then discuss questions related to effectivity of the Pila-Wilkie theorem and its implications for the diophantine applications. In particular I will discuss our recent proof (joint with Novikov) of the restricted form of Wilkie’s conjecture, and more recent results on effectivity for the larger class of semi-Noetherian sets.

The Birch and Swinnerton-Dyer conjecture predicts that the group of rational points on an elliptic curve E over Q has rank equal to the order of vanishing of the L-function of E. A generalization of this conjecture to all geometric Galois representations V was formulated by Bloch and Kato. I will explain a proof of a version of the Bloch-Kato conjecture in p-adic coefficients, when V is attached to a p-ordinary Hilbert modular form of any weight and the order of vanishing is 1. The case of elliptic curves corresponds to classical modular forms of weight two, and was treated by Perrin-Riou in 1987 using the modular points on E(Q) constructed by Heegner. The proof in the general case relies on the universal p-adic deformation of Heegner points and a formula for its height.

I will briefly discuss the general things that Magdalena Georgescu, Joav Orovitz, and I determined one needs to take into consideration for constructing inverse sequences of groupoids with Haar systems such that the pullback morphism induce a directed sequence of groupoid C*-algebras (to be clear, the groupoid C*-algebra of the inverse limit groupoid is the direct limit of the induced directed system of groupoid C*-algebras). Then I will proceed to discuss a variety of examples of how to create, in a simple way, groupoids whose groupoid C*-algebras are matrix algebras, UHF-algebras, infinite tensor powers of direct sums of such things, and dimension drop algebras $Z_{m,n}$ where $m$ and $n$ are natural or even supernatural numbers. I will briefly discuss my work with Atish Mitra on our current project for making the Jiang-Su algebra as a groupoid C*-algebra of an inverse limit groupoid (which, I believe is much more understandable and geometric than other groupoids which have Jiang-Su algebra as groupoid C*-algebra that show up in the literature). I will also discuss my project with Magdalena Georgescu on taking inverse limits of sigma-compact groupoids by second countable groupoids as a way to bootstrap known results about second countable groupoids to sigma-compact groupoids.