We show that the generation problem in Thompson group F is decidable, i.e., there is an algorithm which decides if a finite set of elements of F generates the whole F. The algorithm makes use of the Stallings 2-core of subgroups of F, which can be defined in an analogue way to the Stallings core of subgroups of a free group. An application of the algorithm shows that F is a cyclic extension of a group K which has a maximal elementary amenable subgroup B. The group B is a copy of a subgroup of F constructed by Brin.
Abstract: In the early 2000s Sela proved that all non-abelian free groups share a common first-order theory. Together with R. Sklinos, we use tools developed in his work to show that no infinite field is definable in this theory. In this talk we will survey the line of proof for a formal solution theorem for a simple sort of definable sets, that have a structure of a hyperbolic tower, and use it to characterize definable sets that do not carry a definable structure of an abelian group.
It is known that theessential spectrum of aSchrödinger operator H on\ell^2(\mathbb{N}) is equal to the union of the spectra of right limits ofH. The naturalgeneralization of this relation to \mathbb{Z}^n is known to hold as well.In this talk we study thepossibility of generalizing this characterization of \sigma_{ess}(H) tographs. We show that the general statement fails, while presenting natural families of models where it still holds.
Manchester Building (Hall 2), Hebrew University Jerusalem
Abstract:
In the past two decades the entropy method has been successfully employed in the study of repeated games. I will present a few results that demonstrate the relations between entropy and memory. More specifically: a finite game is repeated (finitely or infinitely) many times. Each player $i$ is restricted to strategies that can recall only the last $k_i$ stages of history. The goal is to characterize the (asymptotic) set of equilibrium payoffs. Such a characterization is available for two-player games, but not for three players or more.
Related papers:
The Birch and Swinnerton-Dyer conjecture predicts that the group of rational points on an elliptic curve E over Q has rank equal to the order of vanishing of the L-function of E. Generalisations of this conjecture to motives M were formulated by Belinson and Bloch-Kato. I will explain a proof of a version of the Bloch-Kato conjecture in p-adic coefficients, when M is attached to a p-ordinary Hilbert modular form of any weight and the order of vanishing is 1.
(joint work with Françoise Dal'Bo and Andrea Sambusetti)
Given a finitely generated group G acting properly on a metric space X,
the exponential growth rate of G with respect to X measures "how big"
the orbits of G are. If H is a subgroup of G, its exponential growth
rate is bounded above by the one of G. In this work we are interested in
the following question: what can we say if H and G have the same
exponential growth rate? This problem has both a combinatorial and a
geometric origin. For the combinatorial part, Grigorchuck and Cohen
Abstract: A permutation representation of a group G is called highly transitive if it is transitive on k-tuples of points for every k. Until just a few years ago groups admitting such permutation representations were thought of as rare. I will focus on three rather recent papers: G-Garion, Hall-Osin, Gelander-G-Meiri (in preparation) showing that such groups are in fact very common.
The Littlewood conjecture is an open problem in simultaneous Diophantine approximation of two real numbers. Similar problem in a field K of formal series over finite fields is also still open. This positive characteristic version of problem is equivalent to whether there is a certain bounded orbit of diagonal semigroup action on Bruhat-Tits building of PGL(3,K).
