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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 SzemerédiTrotter 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 pointplane 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 SzemerédiTrotter theorem, the proof of the pointplane 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 ErdösSzemerédi conjecture, roughly that a set of n scalars in a field (not too large in positive characteristic) defines "almost n^{2}" (more precisely n^{2o(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 noncommutative groups. The central open problem in additive combinatorics is the socalled PolynomialFreimanRuzsa 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.

An Energy Bound in the Affine Group
Petridis, G., RocheNewton, O., Rudnev, M. & Warren, A., 3 Jun 2020, In: International Mathematics Research Notices. 0, rnaa130.Research output: Contribution to journal › Article (Academic Journal) › peerreview
Open Access 
Stronger sumproduct inequalities for small sets
Rudnev, M., Shkredov, I. & Shakan, G., 6 Jan 2020, In: Proceedings of the American Mathematical Society. 148, p. 14671479 13 p.Research output: Contribution to journal › Article (Academic Journal) › peerreview
Open Access 
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) › peerreview