A Personal Roadmap

This page provides a personal mathematical roadmap: a very incomplete and subjective collection of papers in areas that I have made some effort to understand over the past few years, with a link to an online version of the paper and its mathscinet review whenever these are available. The purpose is to give the reader a better idea of where my mathematical interests lie, and to provide a guide to reading material for those new to the subject.

Such a project comes with all the obvious disclaimers: The classification into subject areas is very rough, and the order of authors within them completely arbitrary. I have by no means attempted to select the most "important" papers of any one author, but rather the ones I have most benefited from reading. This includes numerous survey articles. In the case of joint papers, no judgement is intended by linking only to one author's website.

Mathematical blogs

Terry Tao | Tim Gowers | Jordan Ellenberg | Izabella Laba | Richard Borcherds | Luca Trevisan
Emmanuel Kowalski | Gil Kalai

Arithmetic combinatorics

Tim Gowers
 

A new proof of Szemeredi's Theorem for arithmetic progressions of length 4	mathscinet	pdf
Quasirandomness, counting and regularity for 3-uniform hypergraphs	mathscinet	ps
Lower bounds of tower-type for Szemeredi's uniformity lemma	mathscinet	ps
Mathematics: A very short Introduction	mathscinet

Ben Green
 

The primes contain arbitrarily long arithmetic progressions (with T. Tao)		pdf
An inverse theorem for the Gowers U^3 norm, with applications (with T. Tao)	mathscinet	pdf
Finite field models in additive combinatorics	mathscinet	pdf
Montreal lecture notes on quadratic Fourier analysis	mathscinet	pdf

Terry Tao
 

A sum-product estimate in finite fields, and applications (with J. Bourgain and N. Katz)	mathscinet	pdf
A quantitative ergodic theory proof of Szemeredi's Theorem	mathscinet	pdf
The dichotomy between structure and randomness, arithmetic progressions and the primes	mathscinet	pdf

Imre Ruzsa
 

Arithmetic progressions in sumsets	mathscinet	pdf
Difference sets without squares	mathscinet
Generalized arithmetical progressions and sumsets	mathscinet
An infinite Sidon sequence	mathscinet	dvi

Ergodic theory

Sasha Leibman
 

Polynomial extensions of van der Waerden's and Szemeredi's Theorems (with V. Bergelson)	mathscinet	pdf
Convergence of multiple ergodic averages along polynomials of several variables	mathscinet	pdf
Intersective polynomials and the polynomial Szemeredi Theorem (with V. Bergelson and E. Lesigne)		pdf

Bryna Kra
 

Non-conventional ergodic averages and nilmanifolds (with B. Host)	mathscinet	pdf
Montreal lecture notes on ergodic methods in additive combinatorics	mathscinet	pdf
Multiple recurrence and nilsequences (with V. Bergelson, B. Host and an appendix by I. Ruzsa)	mathscinet	pdf

Nikos Frantzikinakis
 

Ergodic averages for independent polynomials and applications (with B. Kra)	mathscinet	pdf
A Hardy field extension of Szemeredi's Theorem (with M. Wierdl)		pdf
Multiple ergodic averages for three polynomials and applications		pdf

Tamar Ziegler
 

The primes contain arbitrarily long polynomial progressions (with T. Tao)		pdf
Universal characteristic factors and Furstenberg averages	mathscinet	pdf

Combinatorics

Noga Alon
 

Combinatorial Nullstellensatz	mathscinet	pdf
Large sets in finite fields are sumsets	mathscinet	pdf
Every monotone graphs property is testable (with A. Shapira)	mathscinet	pdf
Tools from higher algebra	mathscinet	pdf

Benny Sudakov
 

Pseudo-random graphs (with M. Krivelevich)

mathscinet

pdf

Jozsef Solymosi
 

Note on a generalization of Roth's Theorem	mathscinet
A note on a question of Erdos and Graham	mathscinet
Crossing patterns of segments (with J. Pach)	mathscinet	pdf

Janos Pach
 

Thirteen problems on crossing numbers (with G. Toth)	mathscinet	pdf
Crossing families (with B. Aronov, P. Erdos, W. Goddard, D.J. Kleitman, M. Klugerman, L.J. Schulman)	mathscinet	pdf

Classical harmonic analysis

Jean Bourgain
 

On triples in arithmetic progression	mathscinet
On arithmetic progressions in sums of sets of integers	mathscinet
Lambda_p-sets in analysis: results, problems and related aspects	mathscinet

Izabella Laba
 

From harmonic analysis to arithmetic combinatorics	mathscinet	pdf
An improved bound on the Minkowski dimension of Besicovitch sets in R^3 (with N. Katz and T. Tao)	mathscinet	ps

Number theory

Alex Gamburd
 

Sieveing and expanders (with J. Bourgain and P. Sarnak)

mathscinet

Akshay Venkatesh
 

The work of Einsiedler, Katok and Lindenstrauss on the Littlewood Conjecture	mathscinet	pdf
Equidistribution, L-functions and ergodic theory: on some problems of Yu.V. Linnik (with P. Michel)	mathscinet	pdf

Elon Lindenstrauss
 

Invariant measures and arithmetic quantum unique ergodicity	mathscinet	pdf
Invariant measures and the set of exceptions to Littlewood's conjecture (with M. Einsiedler and A. Katok)	mathscinet	pdf

Peter Sarnak
 

Equidistribution and primes

pdf

Trevor Wooley
 

Diophantine problems in many variables: the role of additive number theory	mathscinet	pdf
Waring's problem: a survey (with R.C. Vaughan)	mathscinet	pdf

Kannan Soundararajan
 

Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim

mathscinet

pdf

Computer science

Assaf Naor
 

The two possible values of the chromatic number of a random graph (with D. Achlioptas)	mathscinet	pdf
Euclidean distortion and the sparsest cut (with S. Arora and J.R. Lee)	mathscinet	pdf

Avi Wigderson
 

Expander graphs and their applications (with S. Hoory and N. Linial)	mathscinet	pdf
P, NP and mathematics - a computational complexity perspective	mathscinet	pdf

Alex Samorodnitsky
 

Low degree tests at large distances		ps
Inverse conjecture for the Gowers norm is false (with S. Lovett and R. Meshulam)		pdf

Up-and-coming

Neil Lyall
 

Polynomial configurations in difference sets (with A. Magyar)

pdf

Tom Sanders
 

Boolean functions with small spectral norm (with B.J. Green)		pdf
Appendix to 'Roth's theorem on progressions revisited' by J. Bourgain		pdf
Difference sets and the primes (with I. Ruzsa)	mathscinet	pdf

Olof Sisask
 

A new proof of Roth's Theorem on arithmetic progressions (with E. Croot)		pdf
On the maximal number of three-term arithmetic progressions in subsets of Z_p (with B.J. Green)		pdf

Hoi Nguyen
 

Squares in sumsets (with V. Vu)

pdf

Ciprian Demeter
 

Almost everywhere convergence of series in L^1	mathscinet	pdf
Breaking the duality in the return times theorem (with M. Lacey, T. Tao, C. Thiele)		ps

Derrick Hart
 

Sum product estimates in finite fields via Kloosterman sums (with A. Iosevich and J. Solymosi)

pdf

Boris Bukh
 

No non-trivial solutions to a linear equation in integers

mathscinet

pdf

Other websites

mathscinet search | arxiv search | jstor search | AMS calendar | Cambridge LaTeX | NASA LaTeX | Jahr der Mathematik

Since this page contains a large number of references, it will take some effort to maintain, so I'll be very grateful for any help you can give by notifying me of broken links.