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.