Activities This Week

Whitney studied the embeddings of (C^\infty) manifolds into R^N. A simple initial idea is: start from a map M-> R^N, and deform it generically. Hopefully one gets an embedding, at least an immersion. This fails totally because of the “stable maps”. They are non-immersions, but are preserved in small deformations. The theory of stable maps was constructed in 50’s-60’s by Thom, Mather and others. The participating groups are infinite-dimensional, and the engine of the theory was vector fields integration. This chained all the results to the real/complex-analytic case. I will discuss the classical case, then report on the new results, extending the theory to the arbitrary field (of any characteristic).

A fascinating question in the geometry of numbers and diophantine approximation pertains to the maximal covering radius of a lattice with respect to a fixed function. An important covering radius is the multiplicative covering radius, since it is invariant under the diagonal group and relates to the Littlewood’s conjecture. Minkowski conjectured that the multiplicative covering radius of a unimodular lattice in $R^d$ is bounded by above by $1/2^d$ and that this upper bound is unique to the diagonal orbit of the standard lattice. Minkowski’s conjecture is known to be true for $d\leq 10$, yet there isn’t a general proof for higher dimensions.

In this talk, I will discuss the function field (positive characteristic) analogue of Minkowski’s conjecture, which we stated and proved for every dimension. The proofs and the results are surprisingly different from the real case and have implications in geometry of numbers and dynamics. This talk is based on joint work with Uri Shapira.

When studying quotients of C-algebras generated by creation and annihilation operators on analogues of Fock space, the question of the existence of a co-universal quotient plays an important role in answering fundamental questions in the theory. The study of co-universal quotients goes back to works of Cuntz, and Cuntz and Krieger, on uniqueness theorems for C-algebras arising from symbolic dynamics, and by now co-universal quotients have been shown to exist in several broad classes of examples of Toeplitz C*-algebras.

When associating Toeplitz C-algebras to random walks on a group $G$, new notions of *ratio-limit space and boundary emerge from searching for their co-universal quotients, and the existence of these co-universal quotients becomes intimately related to the group dynamics on the ratio-limit boundary.

In this talk I will exlain how we extended results of Woess to show that there is co-universal quotient for a large class of symmetric random walks on relatively hyperbolic groups. This sheds light on some questions of Woess on ratio-limits for random walks on relatively hyperbolic groups, and extends a result mine on the existence of co-universal quotients for Toeplitz C*-algebras for random walks.

*This talk is based on joint work with Ilya Gekhtman.