פעילויות השבוע

Let M be a Riemannian manifold with a given volume form and hyperbolic fundamental group. We will explain how to construct coclasses in the cohomology of the group of volume preserving diffeomorphisms (or homeomorphisms) of M. As an application, we show that the 3rd bounded cohomology of those groups is highly non-trivial.

A string graph is the intersection graph of a family of continuous arcs in the plane. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on $n$ vertices can be partitioned into five cliques such that some pair of them is not connected by any edge ($n\rightarrow\infty$). As a corollary, we obtain that almost all string graphs on $n$ vertices are intersection graphs of plane convex sets.

This is a joint work with Janos Pach and Bruce Reed.

פעולת הערך המוחלט על המספרים הממשיים היא בעלת את התכונות הבאות:

• $\lvert x+y\rvert\le\lvert x\rvert+\lvert y\rvert$ לכל שני ממשיים $x$ ו-$y$ (אי-שוויון המשולש)
• $\lvert 2\rvert>1$

מסתבר שיש פעולה נוספת על הממשיים, בעלת תכונות דומות לערך המוחלט, אך מקיימת את התנאים:

• $\lvert x+y\rvert\le\max(\lvert x\rvert,\lvert y\rvert)$ לכל שני ממשיים $x$ ו-$y$
• $\lvert 2\rvert< 1$

פעולה זו חייבת את קיומה לקיומו של הערך המוחלט ה-$2$-אדי על $\mathbb{Q}$. אנחנו נדבר על הערך המוחלט ה-$p$-אדי (לכל ראשוני $p$), אשר מוביל לגאומטריה ולאנליזה שונות מאלה שאנחנו רגילים.

באמצעות הערך המוחלט ה-$2$-אדי על $\mathbb{R}$ נוכיח את משפט מונסקי, הנוגע למספרים הממשיים הרגילים:

לא ניתן לחלק ריבוע למספר אי-זוגי של משולשים, שלכולם אותו שטח.

The reducibility and structure of parabolic inductions is a basic problem in the representation theory of p-adic groups. Of particular interest is its principal series and degenerate principal series representations, that is parabolic induction of 1-dimensional representations of Levi subgroups. In this talk, I will start by describing the functor of normalized induction and its left adjoint the Jacquet functor and by going through several examples in the group SL_4(Q_p) will describe an algorithm which can be used to determine reducibility of such representations. This algorithm is the core of a joint project with Hezi Halawi, in which we study the structure of degenerate principal series of exceptional groups of type En (see https://arxiv.org/abs/1811.02974).

The concept of Benjamini-Schramm convergence can be extended to Riemannian manifolds. In this setup a question frequently studied is whether topological invariants that can be expressed as integers are continuous when normalized by the volume. An example of such an invariant is the Euler characteristic, that also exists for graphs. A vast generalization of the Euler characteristic for Riemannian manifolds are characteristic numbers. I will speak on my results showing continuity of normalized characteristic numbers on a suitable class of random Riemannian manifolds defined by a lower Ricci curvature and injectivity radius bound.