# Faculty Activities

*Abstract:*

In 1941, Turan proved the famous Turan theorem, i.e., If G is a graph which does not contain Kr+1 as its subgraph, then the edge number of G is no more than the Turan graph Tn,r, which started the extremal theory of graphs. In this talk, we will introduce the spectral Turan-Type results which are associated with the adjacency matrix, signless Laplacian matrix. Moreover, some open problems in this field are proposed.

*Announcement:*

**פרופ' רון אהרוני**

הפקולטה למתמטיקה

טכניון

**Prof. Ron Aharoni**

The Faculty of Mathematics

Technion

**Math Club 20.6.18**

**טופולוגיה וחתונות:** איך אפשר להיעזר בטופולוגיה לצורך שידוכים.

**Topology and weddings:** Surprisingly, topology can help you find a matching

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*Abstract:*

Abstract is available here: https://noncommutativeanalysis.files.wordpress.com/2018/06/abstract.pdf

*Abstract:*

NOTE THE SPECIAL TIME.

TheElisha Netanyahu Memorial Lectures

*Abstract:*

* Abstract: *Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them?

It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting flow yield such a partition—with exactly equal areas, no matter how the points are distributed. (See http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. This has an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere. I will also describe open problems regarding greedy and electrostatic matching (Joint work with Nina Holden and Alex Zhai) Another topic where local and global optimization sharply differ appears starts from the classical overhang problem: Given n blocks supported on a table, how far can they be arranged to extend beyond the edge of the table without falling off? With Paterson, Thorup, Winkler and Zwick we showed ten years ago that an overhang of order cube root of n is the best possible; a crucial element in the proof involves an optimal control problem for diffusion on a line segment and I will describe generalizations of this problem to higher dimensions (with Florescu and Racz).

*Abstract:*

How can $d+k$ vectors in $\mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $\theta(d,k):=\min_X\max_{x\neq y\in X}|\langle x,y\rangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $X\subseteq\mathbb{R}^d$. In this talk, we focus on the case where $k$ is fixed and $d\to\infty$. In establishing bounds on $\theta(d,k)$, we find an intimate connection to the existence of systems of ${k+1\choose 2}$ equiangular lines in $\mathbb{R}^k$. Using this connection, we are able to pin down $\theta(d,k)$ whenever $k\in\{1,2,3,7,23\}$ and establish asymptotics for general $k$. The main tool is an upper bound on $\mathbb{E}_{x,y\sim\mu}|\langle x,y\rangle|$ whenever $\mu$ is an isotropic probability mass on $\mathbb{R}^k$, which may be of independent interest. (Joint work with Boris Bukh)

*Abstract:*

**Advisor: **Orr Shalit

**Abstract: **In this talk I will give a brief survey on my Ph.D. thesis which mainly focus on certain types of operator-algebras. The talk, correspondingly to my thesis, is divided into two parts.

The first part is about subalgebras (and also other subsets) of graph C*-algebras. I will present some results from a joint work with Adam Dor-On, in which we studied maximal representations of graph tensor algebra. I will first provide a complete description of these maximal representations and then show some dilation theoretical applications, as well as a characterization of a certain rigidity phenomenon, called hyperrigidity, that may or may not occur for a subset of a C*-algebra. I will then present an independent follow-up work in which I studied, in addition to hyperrigidity, other types of rigidity of other types of subsets of graph C*-algebras and obtained some more delicate results.

The second part is devoted to operator-algebras arising from noncommutative (nc) varieties and is based on a joint work with Orr Shalit and Eli Shamovich. The algebra of bounded nc functions over a nc subvariety of the nc ball can be identified as the multiplier algebra of a certain reproducing kernel Hilbert space consisting of nc functions on the subvariety. I will try to answer the following question: in terms of the underlying varieties, when are two such algebras isomorphic? Along the way, if time allows, I will show that while in some aspects the nc and the classical commutative settings share a similar behavior, the first enjoys – and also suffers from – some unique noncommutative phenomena.