Bio:

My undergraduate diploma from Moscow was in Applied Mathematics and Physics. I got my PhD in Caltech, in Applied Maths. My PhD thesis was in the area of Hamiltonian Mechanics, at the crossroads of applied maths and maths physics. During my last postdoc in the University of Missouri I got exposed to new for me, pure mathematical topics in Harmonic Analysis, and following it, already in Bristol,  wrote a couple of papers on these with Alex Iosevich, who introduced me to the area. Many questions of Harmonic Analysis, having been stripped off context and terminology, end up asking about specific putative arrangements of a finite number of geometric objects, thus falling into the domain of Discrete Mathematics and Combinatorics. I have become interested in these questions per se, in particular, because I could understand what they were asking, without being protected by “the formidable dragon of terminology”.

The 2010 spectacular resolution of the celebrated Erdös distinct distance conjecture by Guth and Katz had a formative impact on my vision of my new research area, and I started working on a similar sounding question about the dot products of vectors. That question, however, is still wide open and appears in its essence to be very different, being in particular closely connected to number theory. Adapting the Guth-Katz scope of ideas to it did not quite work. However, as a fortuitous and unexpected byproduct, it resulted in me proving a point-plane incidence theorem that is associated with my name in my research circles. This theorem found many applications in arithmetic questions in positive characteristics, namely when the underlying integer equations are considered modulo a large prime number p, including the major open sum-product conjecture by Erdös and Szemerédi. The latter scope of questions already falls largely into the Number Theory category, and hence I have ever since been invited to quite a few events in number theory, without ever having a systematic course in it.

Title: My favourite questions in Discrete Mathematics

Abstract:

Pure Mathematics is a sport for young people. I came into it late and could only put my head around questions that sounded easy. These are questions of Geometric and Arithmetic Combinatorics that deal with counting equivalence relations on a large finite set of points.

For instance, what is the minimum possible number of distinct pairwise distances generated by a set of N points in the real plane, in terms of the number of points N? This question was asked by Paul Erdös in 1946 and resolved by Larry Guth and Netz Katz in 2010, the answer being roughly N/log N, as is achieved once the points have been arranged in the square grid.

Or is this true that any set of N integers behaves `randomly’ with respect to either adding or multiplying its elements, namely that the number of distinct pairwise sums or products is always roughly N*N? This is known as the Erdös-Szemerédi conjecture, which is wide open, and whose more modest but proven variants have appeared throughout the XXI century mathematics.

Questions of this type have been so far surprisingly difficult, having required fetching concepts and tools from more sophisticated and abstract areas of mathematics. I will try to review what is known so far and based on what.

 

This event will be followed by a drinks reception in the Fry Building Atrium from 5 - 6pm, please do come along and join us! 

Although this event is free to attend we ask that you please book your place via the ticket tailor page to allow us to monitor numbers.

Contact Information 

For enquiries about this event please contact maths-conference-administrator@bristol.ac.uk.

Professor Misha Rudnev