Activities This Week

A Hamiltonian decomposition of $\Gamma$ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki’s theorem from the 19th century, showing that a complete graph $K_n$ on an odd number of vertices $n$ has a Hamiltonian decomposition. This result was recently greatly extended by Kuhn and Osthus. They proved that every $r$-regular $n$-vertex graph $\Gamma$ with even degree $r=cn$ for some fixed $c>1/2$ has a Hamiltonian decomposition, provided $n=n(c)$ is sufficiently large. In this talk we address the natural question of estimating $H(\Gamma)$, the number of such decompositions of $\Gamma$. The main result is that $H(\Gamma)=r^{(1+o(1))nr/2}$. In particular, the number of Hamiltonian decompositions of $K_n$ is $n^{(1+o(1))n^2/2}$.

Joint work with Zur Luria and Benny Sudakov.

Dilation theory is a paradigm for understanding a general class of objects in terms of a better understood class of objects, by way of exhibiting every general object as ``a part of” a special, well understood object. In the first part of this talk I will discuss both classical and contemporary results and applications of dilation theory in operator theory. Then I will describe a dilation theoretic problem that we got interested in very recently: what is the optimal constant $c = c_{\theta,\theta’}$, such that every pair of unitaries $U,V$ satisfying $VU = e^{i\theta} UV$ can be dilated to a pair of $cU’, cV’$, where $U’,V’$ are unitaries that satisfy the commutation relation $V’U’ =e^{i\theta’} U’V’$?

I will present the solution of this problem, as well as a new application (which came to us as a pleasant surprise) of dilation theory to the continuity of the spectrum of the almost Mathieu operator from mathematical physics.

Based on a joint work with Malte Gerhold.

בהרצאה נדבר על פתרון משוואות (מהבבלים ועד ימינו) ובמיוחד נתמקד במשפט של אבל-רופיני ובמתמטיקה שהוא יצר. ההרצאה תכלול את ההוכחה של Arnold למשפט Abel.

A classical Springer theory is an important ingredient in the classification of representations of finite groups of Lie type, completed by Lusztig.

The first result of this theory is the assertion that the so-called Grothendieck-Springer sheaf is perverse and is equipped with an action of the Weyl group. Our main result asserts that an analogous result also holds in the affine (infinite-dimensional) case.

In the first of my talk I will recall what are perverse sheaves, and why the Grothendieck-Springer sheaf is perverse. In the rest of the talk I will outline how to extend all this to the affine setting.

We believe that this should have applications to the representations theory of p-adic groups.

This is a joint work with Alexis Bouthier and David Kazhdan