Dr Misha Rudnev

M.Sc.(Moscow Phys.& Engin.Inst.), Ph.D.(Cal.Tech.)

  • BS8 1UG

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Personal profile

Research interests

Combinatorics. The name comes under many appellations, mine are Geometric, Arighmetic, and Additive combinatorics. Geometric combinatorics seeks first and foremost incidence bounds for finite arrangements of geometric obgects in space: points, lines, triangles, simplices, etc. An incidence is when two objects meet. For instance, the famous Szemerédi-Trotter theorem claims that m lines and n points in the real plane have no more than C[ (mn)2/3 + m + n ] incidences, for some absolute constant (somewhat irrelevant, what one is after are the best posisble powers of and n). My best known contribution to incidence theory is the point-plane bound C[ mn1/2 + k n ] for the number of incidences between n<m points and m planes in three dimensions, where k is the maximum number of colinear points. Unlike the Szemerédi-Trotter theorem, the proof of the point-plane theorem has a purely algebraic nature and works over any field, not just reals (in positive characteristic there is an extra constraint n<p2). 


A notable part of geometric combintorics finds ubiquity of finite point configurations in space. The classical example is the Erdös distance problem: what is the minimum number of distinct pairwise distances, determined by a set of points in Rd. It was solved in 2010 by Guth and Katz in dimension d=2, but is wide open in higher dimension. Similar questions can be asked about spaces over other fields. A stronger verison of the question -- what is the maximum number of realisaitons of a given distance in a set of  n points is wide open in the plane as well.  

Incidence theorems have been so far the main tool in Arithmetic combinatorics, which studies interaction between two ring operations: addition and multiplication. There the key open question, where I have done much research in the past years (and currently hold most of the best known partial "world record" results) is the Erdös-Szemerédi conjecture, roughly that a set of  scalars in a field (not too large in positive characteristic) defines "almost n2" (more precisely n2-o(1)hiding invariably present logarithmic terms) distinct pairwise sums or products.  

Additive combinatorics deals with only one operation, say addition in an Abelian group. Its large part focuses on studying the structure of sets, which are near extremal, for instance what does a finite but large set of  n elements in an Abelian group (say, integers, this connects to number theory) look like if the number of distinct pairwise sums it yields is in some sense comparable to n,  say, at most n1+c , for a tiny c. In the past 15 years there has aslo been plenty of reesearch apropos of the same question in non-commutative groups. The central open problem in additive combinatorics is the so-called Polynomial-Freiman-Ruzsa conjecture, roughly that a set of integers yielding only Kn distinct pairwise sums, where K is tiny relative to n, say K~nc, has a large subset, which looks like an arithmtic progression, whose size is not much greater than N and the number of generators is ~log K.

I have supervised 3 PhD's in arithmetic/geometric combinatorics and currently have 3 PhD students. I may take more, the theme of a PhD being a variation of the above questions or similar.