Activities This Week

A universal deformation of Poisson structures was constructed by M.Kontsevich in 90’s. D.Tamarkin explained that the set of universal deformations are in one-to-one correspondence with Drinfeld Associators. On the other hand, we know that all universal deformations of linear Poisson structures are trivial and coincide with universal enveloping algebra. We show that universal deformations of quadratic Poisson structure are as rich as the full set of all deformations.

The first part of the talk will be devoted to the elementary description of Kontsevich Graph complexes and related combinatorics. The relationships with the universal quantization problems of generic and quadratic Poisson structures will be given in the second part of the talk (based on the joint results with Sergei Merkulov https://arxiv.org/abs/2109.07793).

The Arithmetic Fundamental Lemma (AFL) is a local conjecture motivated by decomposing both sides of the Gross—Zagier Formula into local terms using the Relative Trace formula. For each of the local terms, one side is the intersection number in some Rappoport—Zink space. The other side is some orbital integral. To reduce the global computation to local, one needs to consider intersection numbers on both basic and non-basic locus, while the original linear AFL only considers basic locus.

Collaborated with Andreas Mihatsch, we consider the non-basic locus of Unitary Shimura varieties and conjectured a similar version of linear AFL for Rappoport Zink space on non-basic locus parameterizing p-divisible groups with étale extensions. We proved that this version of linear AFL conjecture can be essentially reduced to the linear AFL conjecture for Lubin—Tate spaces, which corresponds to the basic locus parameterizing one-dimensional connected p-divisible groups.

A common method for quantifying the size of sets of Lebesgue measure zero is via Hausdorff or packing dimension. A more delicate question is to determine the value of the corresponding Hausdorff or packing measure at dimension. In this talk we will show a way to approach this question for some simple fractal sets arising from numeration systems.