Title: Recent advances in combinatorics

November 16 speaker: Nati Linial   

Title: Is there geometry in totally discrete spaces?

Abstract: Even in a totally discrete space X you need to know how to move between distinct points. A path P_{x,y} between two points x,y \in X is a sequence of points in X that starts with x and ends with y. A path system is a collection of paths P_{x,y}, one per each pair of distinct points x, y in X. We restrict ourselves to the undirected case where P_{y,x} is P_{x,y} in reverse. 
Strictly metrical path systems are ubiquitous. They are defined as follows: There is some spanning, connected graph (X, E) with positive edge weights w(e) for all e\in E and P_{x,y} is the unique w-shortest xy path. A metrical path system is defined likewise, but w-shortest paths need not be unique. Even more generally, a path system is called consistent  (no w is involved here) if it satisfies the condition that when point z is in P_{x,y}, then P_{x,y} is P_{x,z} concatenated with P_{z,y}. These three categories of path systems are quite different from each other and in our work we find quantitative ways to capture these differences.
Joint work with Daniel Cizma and with Maria Chudnovsky

P.S. The talks about the Sunflower conjecture and Kahn Kalai conjecture were based on the lecture notes: https://www.ias.edu/sites/default/files/Sunflowers%20Lecture%20Notes_1.pdf