פעילויות השבוע

Szemeredi‘s theorem asserts that in every subset of the natural numbers of positive density one can find an arithmetic progression of arbitrary length. In 2001, Gowers gave a quantitative proof for this theorem. A key definition in his work are the Gowers norms which measure the randomness of subsets of the natural numbers. Inspired by Furstenberg‘s ergodic theoretical proof of Szemeredi‘s theorem, Gowers proved the following dichotomy: Either the given set is close to a random set with respect to these norms, or it admits some algebraic structure. Gowers then proved that in each of these cases Szemeredi‘s theorem holds. Later, Host and Kra studied the structure of certain ergodic systems associated with an infinitary version of the Gowers norms. Inspired by their work, Green, Tao and Ziegler improved Gowers‘ structure theorem showing that a function (or a set) with large Gowers norm must correlate with a nilsequence. This result is known as the inverse theorem for the Gowers norms. Recently, Jamneshan and Tao proved (roughly speaking) that a generalization of the Host-Kra theorem for ergodic systems associated with actions of the largest countable abelian group $\mathbb{Z}^\omega$ will imply the most general version of the inverse theorem for the Gowers norms. In this talk I will survey the above in more detail and mention some recent developments about these structure theorems.

A key tool in understanding (complex analytic) hypersurface singularities is to study what properties are preserved under special deformations. For example, the relationship between the Milnor number of an isolated singularity and the number of A_1 points. In this talk we will discuss the transversal discriminant of a singular hypersurfaces whose singular locus is a smooth curve, and how it can be applied in order to generalize a classical result by Siersma, Pellikaan, and de Jong regarding morsifications of such singularities. In addition, we will present some applications to the study of Yomdin-type isolated singularities.

Let G be a higher-rank Lie group (for example, SL_n(R) for n>2). Nevo and Zimmer‘s structure theorem describes certain nonsingular actions that naturally arise when studying lattices. This theorem is very powerful and manifests rigidity phenomena. For example, it implies the celebrated Margulis Normal Subgroup Theorem, which classifies all normal subgroups of irreducible lattices of G. The original proof of Nevo–Zimmer Theorem heavily uses the structure of Lie groups.

In this talk, I will present a new theorem on general groups that immediately implies the Nevo–Zimmer Theorem (when restricting to the higher-rank Lie case). I will also explain how the generality of our theorem allows us to adapt it to the setup of normal unital completely positive maps on von Neumann algebras.

The talk is based on joint work with Uri Bader.