Preferential attachment random graphs evolve in time by sequentially adding vertices and edges randomly so that connections to vertices having high degree are favored. There has been an explosion of research on these models since Barabasi and Albert popularized them in 1999 to explain the so-called power law behavior observed in real networks such as the graph of the Internet where Web pages correspond to vertices and hyperlinks between them correspond to edges. In this talk we will show how…

Given N I/O requests to a disk drive, how long will it take the disk drive to service them? How long does it take to board an airplane? We will show that such questions are modeled by the same math that models the universe in relativity theory, namely, space-time (Lorentzian) geometry.As a result of this connection we will try to understand what is the best way to service I/O, what is the best airplane boarding policy and what makes free…

We consider a switched (queueing) network in which there are constraints on which queues may be served simultaneously; such networks have been used to effectively model input-queued switches and wireless networks. The scheduling policy for such a network specifies which queues to serve at any point in time, based on the current state or past history of the system. In the main result, we provide a new class of online scheduling policies that achieve optimal average queue-size scaling for a…

The talk is devoted to the classical problem of estimating the Van der Waerden number $W(n,r)$. The famous Van der Waerden theorem states that, for any integers $nge 3$, $rge 2$, there exists the smallest integer $W(n,r)$ such that, for every $Nge W(n,r)$, in any $r$-coloring of the set ${1,2,ldots,N}$ there exists a monochromatic arithmetic progression of length $n$. The best upper bound for $W(n,r)$ in the general case was obtained by W.T. Gowers in 2001 who proved thatW(n,r)≤22r22n+9. Our…

In my talk I will present our research groups in Yandex which is the main search engine in Russia, and Moscow Institute of Physics and Technology which is a kind of Russian MIT.

In the past 20 years, there has been a lot of work concerning modeling real networks such as Internet, WWW, social networks, biological networks, etc. In our talk, we will mainly concentrate on models devoted to incorporating properties of the web graph. We will give a survey of different such models including those of Bollob'as--Riordan, Buckley--Osthus, Cooper--Frieze, etc. We will also present some new related mathematical results obtained by our group in Yandex, Moscow Institute of Physics and Technology, Moscow…

Consider the following situation. Alice knows string x and Bob knows string y. There is a limited capacity information channel from Alice to Bob. Alice wants to transmit a message that is sufficient for Bob to recover x. What is a minimum capacity that allows her to do it? Obviously, there is a theoretical minimum: the amount of information in x that is not contained in y. Astonishing enough, this is a precise estimate: Alice can produce an optimal code…

We consider the random graph model $H_{a,m}^n$. An explicit construction of this model was suggested by Buckley and Osthus. For any fixed positive integer $m$ and positive real $a$ we define a random graph $H_{a,m}^n$ in the following way. The graph $H_{a,1}^1$ consists of one vertex and one loop. Given a graph $H_{a,1}^{t-1}$ we transform it into $H_{a,1}^{t}$ by adding one new vertex and one new random edge $(t,i)$, where $i in {1, dots, t }$ and $Prob(i=t) = frac{a}{(a+1)t-1}$,…

DNA sequencing is the basic workhorse of modern biology and medicine. Shotgun sequencing is the dominant technique used: many randomly located short fragments called reads are extracted from the DNA sequence, and these reads are assembled to reconstruct the original DNA sequence. Despite major effort by the research community, the DNA sequencing problem remains unresolved from a practical perspective: it is currently not known how to produce a good assembly of most read data-sets. By drawing an analogy between the…

We study particle systems corresponding to highly connected queueing networks. We examine the validity of the so-called Poisson Hypothesis (PH), which predicts that the Markov process, describing the evolution of such particle system, started from a reasonable initial state, approaches the equilibrium in time independent of the size of the network. This is indeed the case in many situations. However, there are networks for which the relaxation process slows down. This behavior reflects the fact that the corresponding infinite system…