Tel Aviv

We study the set $L(G)$ of cycle lengths that appear in a sparse binomial random graph $G(n,c/n)$ and in a random $d$-regular graph $G_{n,d}$. We show in particular that for most values of $c$, for $G$ drawn from $G(n,c/n)$ the set $L(G)$ contains typically an interval $[\omega(1), (1-o(1))L_{\max}(G)]$, where $L_{\max}(G)$ is the length of a longest cycle (the circumference) of $G$. For the case of random $d$-regular graphs, $d\geq 3$ fixed, we obtain an accurate asymptotic estimate for the probability of $L(G)$ to contain a full interval $[k,n]$ for a fixed $k\geq 3$. Similar results are obtained also for the supercritical case $G(n,(1+\epsilon)/n)$, and for random directed graphs.
A joint work with Yahav Alon and Eyal Lubetzky.

Let $G_n$ be a sequence of finite, simple, connected, regular graphs with degrees tending to infinity and let $T_n$ be a uniformly drawn spanning tree of $G_n$. In joint work with Yuval Peres we show that the local limit of $T_n$ is the $\text{Poisson}(1)$ branching process conditioned to survive forever (that is, the asymptotic frequency of the appearance of any small subtree is given by the branching process). The proof is based on electric network theory and I hope to show most of it.

We say that a graph $G$ is $H$-free if $G$ does not contain $H$ as a (not necessarily induced) subgraph. For a positive integer $n$, denote by $\text{ex}(n,H)$ the largest number of edges in an $H$-free graph with $n$ vertices (the Turán number of $H$). The classical theorem of Erdős, Kleitman, and Rothschild states that, for every $r\geq3$, there are $2^{\text{ex}(n,H)+o(n2)}$ many $K_r$-free graphs with vertex set $\{1,…, n\}$. There exist (at least) three different derivations of this estimate in the literature: an inductive argument based on the Kővári-Sós-Turán theorem (and its generalisation to hypergraphs due to Erdős), a proof based on Szemerédi's regularity lemma, and an argument based on the hypergraph container theorems. In this talk, we present yet another proof of this bound that exploits connections between entropy and independence. This argument is an adaptation of a method developed in a joint work with Gady Kozma, Tom Meyerovitch, and Ron Peled that studied random metric spaces.

Color each vertex of an infinite graph blue with probability $p$ and red with probability $1-p$, independently among vertices. For which values of $p$ is there an infinite connected component of blue vertices? The talk will focus on this classical percolation problem for the class of planar graphs. Recently, Itai Benjamini made several conjectures in this context, relating the percolation problem to the behavior of simple random walk on the graph. We will explain how partial answers to Benjamini's conjectures may be obtained using the theory of circle packings. Among the results is the fact that the critical percolation probability admits a universal lower bound for the class of recurrent plane triangulations. No previous knowledge on percolation or circle packings will be assumed.

I will start with briefly describing the HISH ( Holography Inspired Hadronic String) model and reviewing the fits of the spectra of mesons, baryons, glue-balls and exotic hadrons. 

I will present the determination of the hadron strong decay widths. The main decay mechanism is that of a string splitting into two strings. The corresponding total decay width behaves as $\Gamma =\frac{\pi}{2}A T L $ where T and L are the tension and length of the string and A is a dimensionless universal constant. The partial width of a given decay mode is given by $\Gamma_i/\Gamma = \Phi_i \exp(-2\pi C m_\text{sep}^2/T$ where $\Phi_i$ is a phase space factor, $m_\text{sep}$ is the mass of the "quark" and "antiquark" created at the splitting point, and C is adimensionless coefficient close to unity. I will show the fits of the theoretical results to experimental data for mesons and baryons. I will examine both the linearity in L and the exponential suppression factor. The linearity was found to agree with the data well for mesons but less for baryons. The extracted coefficient for mesons $A = 0.095\pm  0.01$  is indeed quite universal. The exponential suppression was applied to both strong and radiative decays. I will discuss the relation with string fragmentation and jet formation. I will extract the quark-diquark structure of baryons from their decays. A stringy mechanism for Zweig suppressed decays of quarkonia will be proposed and will be shown to reproduce the decay width of  states. The dependence of the width on spin and symmetry will be discussed. I will further apply this model to the decays of glueballs and exotic hadrons.

In this talk we present a general method using permanent estimates in order to obtain results about counting and packing Hamilton cycles in dense graphs and oriented graphs. As a warm up we prove that every Dirac graph $G$ contains at least $(reg(G)/e)^n$ many distinct Hamilton cycles, where $reg(G)$ is the maximal degree of a spanning regular subgraph of $G$. We continue with strengthening a result of Cuckler by proving that the number of oriented Hamilton cycles in an almost $cn$-regular oriented graph is $(cn/e)^n(1+o(1))^n$, provided that $c$ is greater than $3/8$. Last, we prove that every graph $G$ of minimum degree at least $n/2+\epsilon n$ contains at least $reg_{even}(G)-\epsilon n$ edge-disjoint Hamilton cycles, where $reg_{even}(G)$ is the maximal even degree of a spanning regular subgraph of $G$. This proves an approximate version of a conjecture made by Osthus and K\"uhn.  Joint work with Michael Krivelevich and Benny Sudakov.

A theorem of Kuyk says that every Abelian extension of a

Hilbertian field is Hilbertian.

We conjecture that for an Abelian variety $A$ defined over

a Hilbertian field $K$

every extension $L$ of $K$ in $K(A_\tor)$ is Hilbertian.

We prove our conjecture when $K$ is a number field.

The proofs applies a result of Serre about $l$-torsion of

Abelian varieties, information about $l$-adic analytic

groups, and Haran's diamond theorem.