Activities This Week

We will present a model for construction of random fractals which is called fractal percolation. The main result that will be presented in this talk states that a typical fractal percolation set E intersects every set which is winning for a certain game that is called the “hyperplane absolute game”, and the intersection has the same Hausdorff dimension as E. An example of such a winning set is the set of badly approximable vectors in dimension d. In order to prove this theorem one may show that a typical fractal percolation set E contains a sequence of Ahlfors-regular subsets with dimensions approaching the dimension of E, where all the subsets in this sequence are also “hyperplane diffuse”, which means that they are not concentrated around affine hyperplanes when viewed in small enough scales. If time permits, we will sketch the proof of this theorem and present a generalization to a more general model for random construction of fractals which is given by projecting Galton-Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane.

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.