Counting problems in algebraic geometry over an algebraically closed field have been studied for centuries. More recently, it was discovered that there are interesting counting problems over the real numbers. Topology took the place of algebraic closedness. However, the question remained whether there are interesting counting problems over more general fields where the tools of classical topology are not available. I will describe some results in this direction.
Counting problems in algebraic geometry over an algebraically closed field have been studied for centuries. More recently, it was discovered that there are interesting counting problems over the real numbers. Topology took the place of algebraic closedness. However, the question remained whether there are interesting counting problems over more general fields where the tools of classical topology are not available. I will describe some results in this direction.
Manchester Building (Hall 2), Hebrew University Jerusalem
I will discuss some recent advances in combinatorics, among them the disproof of Hedetniemi conjecture by Shitov, the proof of the sensitivity conjecture by Huang, the progress on the Erdos-Rado sunflower conjecture by Alweiss, Lovett, Wu, and Zhang, and the progress on the expectation threshold conjecture by Frankston, Kahn, Narayanan, and Park.
The basic idea is to associate with a combinatorial object Xan algebraic structure A(X), and derive from algebraic properties of A(X)combinatorial consequences for X. For example, Stanley's proof of the UpperBound Theorem for simplicial spheres uses the Cohen-Macaulay property of theface ring associated with a simplicial complex.
We will review the basics of Stanley's theory, illustrate themon examples, and time permitting, discuss more recent advances of this theory.
(All needed terms and background will be given in thetalk.)
Abstract: Let M be an orientable hyperbolic surface without boundary and let c be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of c in the hyperbolic plane is shorter than c. The talk will be presented for advanced undergraduate and beginning graduate students.
This semester will be devoted to resolution of singularities -- a process that modifies varieties at the singular locus so that the resulting variety becomes smooth. For many years this topic had the reputation of very technical and complicated, though rather elementary.
In fact, the same resolution algorithm can be described in various settings, including schemes, algebraic varieties or complex analytic spaces.
Gibbs measures vs. random walks in negative curvature
The ideal boundary of a negatively curved manifold naturally carries two types of measures.
On the one hand, we have conditionals for equilibrium (Gibbs) states associated to Hoelder potentials; these include the Patterson-Sullivan measure and the Liouville measure. On the other hand, we have stationary measures coming from random walks on the fundamental group.
Abstract: I will explain what measure distal transformations are
and describe some new constructions obtained with Eli Glasner.
These answer, inter alia, a question recently raised by Ibarlucia and Tsankov
concerning the existence of strongly ergodic non compact distal actions of the
free group.
Title: Some recent results on sublinear algorithms for graph related properties
Abstract: I will describe property testing of (di)graph properties in bounded degree graph models, and talk about a characterization of the 1-sided error testable monotone graph properties and the 1-sided error testable hereditary graph properties in this model. I will introduce the notion of configuration-free properties and talk about some graph theoretic open problems.
Speaker: Boaz Slomka (Open U.) Title: On Hadwiger's covering problem
Abstract: A long-standing open problem, known as Hadwiger’s covering problem, asks what is the smallest natural number N(n) such that every convex body in R^n can be covered by a union of the interiors of N(n) of its translates.
In this talk, I will discuss some history of this problem and its close relatives, and present more recent results, including a general upper bound for N(n).
Abstract: When are two elements in a given group conjugate? We solve this problem for the group of tree almost-automorphisms. These are homeomorphisms of the tree boundary which locally look like tree automorphisms. The solution is tied with the dynamics of the group action on the tree boundary. This work is joint with W. Lederle.
