Workshop announcement The National Academy of Sciences of Armenia together American University of Armenia are organizing a memorial workshop on extremal combinatorics, cryptography and coding theory dedicated to the 60th anniversary of the mathematician Levon Khachatrian. Professor Khachatrian started his academic career at the Institute of Informatics and Automation of National Academy of Sciences. From […]

Lecture 7 First passage percolation 1) Models of percolation. We talked about percolation introduced by Broadbent and Hammersley in 1957. The basic model is a model of random subgraphs of a grid in n-dimensional space. (Other graphs were considered later as well.) Here, a grid is a graph whose vertices have integers coordinates and where two […]

Lecture 4 In the third week we moved directly to the course’s “punchline” – the use of Fourier-Walsh expansion of Boolean functions and the use of Hypercontractivity. Before that we started with a very nice discrete isoperimetric question on a graph which is very much related to the graph of the discrete cube. Consider the […]

Borsuk asked in 1933 if every bounded set K of diameter 1 in can be covered by d+1 sets of smaller diameter. A positive answer was referred to as the “Borsuk Conjecture,” and it was disproved by Jeff Kahn and me in 1993. Many interesting open problems remain. The first two posts in the series “Around Borsuk’s Conjecture” […]

Home page of the course. In the first lecture I defined the discrete n-dimensional cube and Boolean functions. Then I moved to discuss five problems in extremal combinatorics dealing with intersecting families of sets. 1) The largest possible intersecting family of subsets of [n]; 2) The largest possible intersecting family of subsets of [n] so […]

After much hesitation, I decided to share with you the videos of my lecture: Open collaborative mathematics over the internet – three examples, that I gave last January in Doron Zeilberger’s seminar at Rutgers on experimental mathematics. Parts of the 47-minutes talk is mathematical, while other parts are about mathematics on the Internet, blogs, the polymath […]

This fall I am giving a course at Berkeley on analysis of Boolean functions Course number: CS 294-92 Title: Analysis of Boolean Functions Lectures: TuTh 5:00-6:30 Location: Room 310 Soda First lecture will be was on Thursday August 29th at 5:00pm. Graduate students in computer science and in mathematics are welcomed. Advanced undergraduates may find it useful as […]

Michal Karonski (left) who built Poland’s probabilistic combinatorics group at Poznań, and a sculpture honoring the Polish mathematicians who first broke the Enigma machine (right, with David Conlon, picture taken by Jacob Fox). I am visiting now Poznań for the 16th Conference on Random Structures and Algorithms. This bi-annually series of conferences started 30 years ago (as […]

Paul Erdős in Jerusalem, 1933 1993 Update: Here is a link to a draft of a paper* based on the first part of this lecture. Some old and new problems in combinatorial geometry I: Around Borsuk’s problem. I just came back from a great Erdős Centennial conference in wonderful Budapest. I gave a lecture on old and […]

Margulis’ paper Ramanujan graphs were constructed independently by Margulis and by Lubotzky, Philips and Sarnak (who also coined the name). The picture above shows Margulis’ paper where the graphs are defined and their girth is studied. (I will come back to the question about girth at the end of the post.) In a subsequent paper Margulis used the girth […]