Abstract: Let X be a simplicial complex on n vertices without missing faces of dimension larger than d. Let L_k denote the k-Laplacian acting on real k-cochains of X and let μ_k(X) denote its minimal eigenvalue. We study the connection between the spectral gaps μ_k(X) for k ≥ d and μ_{d-1}(X). Applications include:
1) A cohomology vanishing theorem for complexes without large missing faces.
2) A fractional Hall type theorem for general position sets in matroids.
Abstract: Let G be a finitely generated group and let G>G_1>G_2 ... be a sequence of finite index normal subgroups of G with trivial intersection.
We expect that the asymptotic behaviour of various group theoretic invariants of the groups G_i should relate to algebraic, topological or measure theoretic properties of G.
A classic example of this is the Luck approximation theorem which says that the growth of the ordinary Betti numbers of sequence G_i is given by the L^2-Betti number of (the classifying space) of G.
In the first talk we gave a brief outline of the contents of the course. In the rest of the semester we will get deeper into some topics. In the coming lecture ( and the next one) we will discuss Kazhdan property T and its connections with expanders and with first cohomology groups. No prior knowledge will be assumed.
The family of high rank arithmetic groups is a class of groups playing an important role in various areas of mathematics. It includes SL(n,Z), for n>2 , SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more.
A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasi-isometric rigidity.
I will talk about a new Abelian category associated to an open variety with normal-crossings (or more generally, logarithmic) choice of compactification, which behaves in remarkable (and remarkably nice) ways with respect to changes of compactification and duality, and which first appeared in work on mirror symmetry.
The talk is based on the joint work with Yanki Lekili. The associative Yang-Baxter equation
is a quadratic equation related to both classical and quantum Yang-Baxter equations. It appears naturally in connection with triple Massey products in the derived category of
coherent sheaves on elliptic curve and its degenerations. We show that all of its nondegenerate trigonometric solutions are obtained from Fukaya categories of some noncompact surfaces. We use this to prove that any two simple vector bundles on a cycle of projective lines are related by a sequence of spherical twists.
Zilber's trichotomy conjecture, in modern formulation, distinguishes three flavours of geometries of strongly minimal sets --- disintegrated/trivial, modular, and the geometry of an ACF. Each of these three flavours has a classic ``template'' --- a set with no structure, a projective space over a prime field, and an algebraically closed field, respectively. The class of ab initio constructions with which Hrushovski refuted the conjecture features a new flavour of geometries --- non-modular, yet prohibiting any algebraic structure.
Title: The behavior of rational points in one-parameter families
Abstract: How often does a "random" algebraic plane curve f(x,y) = 0
have a solution with rational coordinates? In one-parameter "twist"
families of elliptic curves, Goldfeld conjectured that there should be
a rational point exactly half of the time. Recent progress towards
this conjecture makes use of Selmer groups, and I'll explain the
geometric idea underlying their construction. I'll also describe
results for families of curves of higher genus, and abelian varieties
Dependent theories have now a very solid and well-established collection of results and applications. Beyond first order, the development of "dependency" has been rather scarce so far. In addition to the results due to Kaplan, Lavi and Shelah (dependent diagrams and the generic pair conjecture), I will speak on a few lines of current research around the extraction of indiscernibles for dependent diagrams and on various forms on dependence for abstract elementary classes. This is joint work with Saharon Shelah.
A special class among the countably infinite relational structures is the class of homogeneous structures. These are the structures where every finite partial isomorphism extends to a total automorphism. A countable set, the ordered rationals, and the random graph are all homogeneous.
