Projects per year

## 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

*C*(somewhat irrelevant, what one is after are the best posisble powers of

*m*and

*n*

*)*. My best known contribution to incidence theory is the point-plane bound

*C[ mn*

^{1/2}*+ 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

*p*there is an extra constraint

*n<p*

*).*

^{2}

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 *n *points in *R ^{d}.* 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

*x*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 *n * scalars in a field (not too large in positive characteristic) defines "almost *n ^{2}*" (more precisely

*n*

^{2-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 *A *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 *n ^{1+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

*n*integers yielding only

*Kn*distinct pairwise sums, where

*K*is tiny relative to

*n,*say

*K~n*

^{c}, 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.

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## Projects

## Research Output

## An Energy Bound in the Affine Group

Petridis, G., Roche-Newton, O., Rudnev, M. & Warren, A., 2019, (Submitted) In : SUBMITTED.Research output: Contribution to journal › Article (Academic Journal)

## Bisector energy and pinned distances in positive characteristic

Murphy, B., Rudnev, M. & Stevens, S. C. C., 2019, (Submitted) In : SUBMITTED.Research output: Contribution to journal › Article (Academic Journal)

## New results on sum-product type growth over fields

Murphy, B., Petridis, G., Roche-Newton, O., Rudnev, M. & Shkredov, I. D., 2 Apr 2019, In : Mathematika. 65, 3, p. 588-642 55 p.Research output: Contribution to journal › Article (Academic Journal)