Abstract. The realization of m.p automorphisms as transfer on the space of the paths on the graded graphs allows to use new kind of encoding of one-sided Bernoulli shift. I will start with simple example how to realize Bernoulli shift in the locally finite space (graph) $\prod_n {1,2,\dots n}$ (triangle compact.) Much more complicated example connected to old papers by S.Kerov-Vershik and recent by Romik-Sniady in which one-sided Bernoulli shift is realized as Schutzenberger transfer on the space of infinite Young tableaux with Plancherel Measure. These examples open series of

Abstract: A Markov chain over a finite state space is said to exhibit the total variation cutoff phenomenon if, starting from some Dirac measure, the total variation distance to the stationary distribution drops abruptly from near maximal to near zero. It is conjectured that simple random walks on the family of $k$-regular, transitive graphs with a two sided $\epsilon$ spectral gap exhibit total variation cutoff (for any fixed $k$ and $\epsilon). This is known to be true only in a small number of cases.

First talk: Speaker: Madeleine Weinstein (Berkeley) Title: Voronoi Cells of Varieties Abstract:

Speaker: Eyal Karni (BIU) Title: Combinatorial high dimensional expanders Abstract: An eps-expander is a graph G=(V,E) in which every set of vertices X where |X|<=|V|/2 satisfies |E(X,X^c)|>=eps*|X| . There are many edges that "go out" from any relevant set.

Speaker: Uri Rabinovich (U. Haifa) Title: SOME EXTREMAL PROBLEMS ABOUT SIMPLICIAL COMPLEXES Abstract: We shall discuss the following three issues: * The existence of Hamiltonian d-cycles, i.e., simple d-cycles containing a spanning d-hypertree of a complete d-complex K_n^d; * The existence of a distribution D over spanning d-hypertrees T of K_n^d, so that for ANY (d-1)-cycle C there, the expected size of the d-filling of C with respect to a random T from D is Omega(n^d); * The existence of f(k,d) such that any d-simplex of rank r with > f(k,d)*r d-faces contains

Hilbert's 12th problem (Kronecker's Jugendtraum) is one of the major open problems

Speaker: Doron Puder, TAU Title: Aldous' spectral gap conjecture for normal sets Abstract: Aldous' spectral gap conjecture, proved in 2009 by Caputo, Liggett and Richthammer, states the following a priori very surprising fact: the spectral gap of a random walk on a finite graph is equal to the spectral gap of the interchange process on the same graph.

Speaker: Shira Zerbib (U. Michigan, Iowa State University) Title: Envy-free division of a cake without the “hungry players" assumption Abstract: The fair division theorem due to Stromquist (1980) ensures that under some conditions it is possible to divide a rectangular cake into n pieces and assign one piece to each of n players such that no player strictly prefers a piece that has not been assigned to him.

Speaker: Omri Ben Eliezer, TAU Title: Finding patterns in permutations Abstract: For two permutations sigma and pi, we say that sigma contains a copy of pi, if there is a subset (not necessarily consecutive) of elements in sigma, whose relative order is the same as in pi. For example, if pi = (1,2,3), then a copy of pi in sigma amounts to an increasing subsequence in sigma of length 3. As shown by Guillemot and Marx, a copy of a constant length pi can be found in sigma in linear time. However, how quickly can one find such a

Hilbert's 12th problem (Kronecker's Jugendtraum) is one of the major open problems

Abstract: Any birational geometer would agree that the best algorithm for resolution of singularities should run by defining a simple invariant of the singularity and iteratively blowing up its maximality locus. The only problem is that already the famous example of Whitney umbrella shows that this is impossible, and all methods following Hironaka had to use some history and resulted in more complicated algorithms. Nevertheless, in a recent work with Abramovich and Wlodarczyk we did construct such an algorithm, and an independent description of a similar

Liftings mod p² and the Nygaard filtration

Abstract: I will revisit old results on liftings mod p² and decompositions of de Rham complexes in positive characteristic (Deligne-I.) at the light of relations recently discovered independently by Bhargav Bhatt and myself between cotangent complexes, de Rham-Witt, and derived de Rham complexes.

Abstract: A Markov chain over a finite state space is said to exhibit the total variation cutoff phenomenon if, starting from some Dirac measure, the total variation distance to the stationary distribution drops abruptly from near maximal to near zero. It is conjectured that simple random walks on the family of $k$-regular, transitive graphs with a two sided $\epsilon$ spectral gap exhibit total variation cutoff (for any fixed $k$ and $\epsilon). This is known to be true only in a small number of cases.

For a finitely generated subgroup H of the free group F_r, the Stallings graph of H is a finite combinatorial graph, whose edges are labeled by r letters (and their inverses), so that paths in the graphs correspond precisely to the words in H. Furthermore, there is a map between the graphs of two subgroups H and J, precisely when one is a subgroups of the other. Stallings theory studies the algebraic information which is encoded in the combinatorics of these graphs and maps.

Abstract: In this talk, I shall present a generalization of the lattice point counting problem for Euclidean balls in the context of a certain type of homogeneous groups, the so-called Heisenberg groups.