Research
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Research Papers
The GreenTao theorem and a relative Szemerédi theorem slides video

A relative Szemerédi theorem (with David Conlon and Jacob Fox)
Geometric and Functional Analysis 25 (2015), 733–762.
Companion note: Linear forms from the Gowers uniformity norm 
An arithmetic transference proof of a relative Szemerédi theorem
Mathematical Proceedings of the Cambridge Philosophical Society 156 (2014), 255–261. 
A short proof of the multidimensional Szemerédi theorem in the primes (with Jacob Fox)
American Journal of Mathematics 137 (2015), 1139–1145.
Graph regularity method and applications

Extremal results in sparse pseudorandom graphs (with David Conlon and Jacob Fox)
Advances in Mathematics 256 (2014), 206–290. slides 
On regularity lemmas and their algorithmic applications (with Jacob Fox and László Miklós Lovász)
Combinatorics, Probability and Computing 26 (2017), 481–505. 
A fast new algorithm for weak graph regularity (with Jacob Fox and László Miklós Lovász)
Combinatorics, Probability and Computing 28 (2019), 777–790. 
The regularity method for graphs with few 4cycles (with David Conlon, Jacob Fox, and Benny Sudakov) slides
Additive combinatorics

Triforce and corners (with Jacob Fox, Ashwin Sah, Mehtaab Sawhney, and David Stoner)
Mathematical Proceedings of the Cambridge Philosophical Society, to appear. 
Common and Sidorenko linear equations (with Jacob Fox and Huy Tuan Pham)

Patterns without a popular difference (with Ashwin Sah and Mehtaab Sawhney)

Towertype bounds for Roth’s theorem with popular differences (with Jacob Fox and Huy Tuan Pham)

A short proof of the canonical polynomial van der Waerden theorem (with Jacob Fox and Yuval Wigderson)
Arithmetic regularity and applications

Efficient arithmetic regularity and removal lemmas for induced bipartite patterns (with Noga Alon and Jacob Fox) slides
Discrete Analysis 2019:3, 14 pp. 
Induced arithmetic removal: complexity 1 patterns over finite fields (with Jacob Fox and Jonathan Tidor)

Testing linearinvariant properties (with Jonathan Tidor)
Cayley graphs

Quasirandom Cayley graphs (with David Conlon)
Discrete Analysis 2017:6, 14 pp. 
Hypergraph expanders of all uniformities from Cayley graphs (with David Conlon and Jonathan Tidor)

Cayley graphs without a bounded eigenbasis (with Ashwin Sah and Mehtaab Sawhney)
Equiangular lines and spectral graph theory
 Equiangular lines with a fixed angle (with Zilin Jiang, Jonathan Tidor, Yuan Yao, and Shengtong Zhang) blog
Incidence geometry
 Joints tightened (with HungHsun Hans Yu) blog
Independent sets and graph homomorphisms

The number of independent sets in a regular graph
Combinatorics, Probability and Computing 19 (2010), 315–320. poster 
The number of independent sets in a graph with small maximum degree (with David Galvin)
Graphs and Combinatorics 27 (2011), 177–186. 
The bipartite swapping trick on graph homomorphisms
SIAM Journal on Discrete Mathematics 25 (2011), 660–680. poster
 The number of independent sets in an irregular graph (with Ashwin Sah, Mehtaab Sawhney, and David Stoner)
Journal of Combinatorial Theory Series B 138 (2019), 172–195. blog MIT News
 A reverse Sidorenko inequality (with Ashwin Sah, Mehtaab Sawhney, and David Stoner)
Inventiones Mathematicae, to appear. slides blog
Sphere packing and energy minimization

Sphere packing bounds via spherical codes (with Henry Cohn)
Duke Mathematical Journal 163 (2014), 1965–2002. 
Energyminimizing errorcorrecting codes (with Henry Cohn)
IEEE Transactions on Information Theory 60 (2014), 7442–7450. 
Exponential improvements for superball packing upper bounds (with Ashwin Sah, Mehtaab Sawhney, and David Stoner)
Advances in Mathematics, to appear.
Large deviations in random graphs slides

On replica symmetry of large deviations in random graphs (with Eyal Lubetzky)
Random Structures & Algorithms 47 (2015), 109–146. blog 
On the variational problem for upper tails in sparse random graphs (with Eyal Lubetzky)
Random Structures & Algorithms 50 (2017), 420–436. blog 
On the lower tail variational problem for random graphs
Combinatorics, Probability and Computing 26 (2017), 301–320. 
Upper tails and independence polynomials in random graphs (with Bhaswar B. Bhattacharya, Shirshendu Ganguly, and Eyal Lubetzky)
Advances in Mathematics 319 (2017), 313–347. 
Upper tails for arithmetic progressions in a random set (with Bhaswar B. Bhattacharya, Shirshendu Ganguly, and Xuancheng Shao) video
International Mathematics Research Notices. IMRN 2020, 167–213. 
On the upper tail problem for random hypergraphs (with Yang Liu)
Graph limits

Hypergraph limits: a regularity approach
Random Structures & Algorithms 47 (2015), 205–226. 
An $L^p$ theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions (with Christian Borgs, Jennifer T. Chayes, and Henry Cohn)
Transactions of the American Mathematical Society 372 (2019), 3019–3062. 
An $L^p$ theory of sparse graph convergence II: LD convergence, quotients, and right convergence (with Christian Borgs, Jennifer T. Chayes, and Henry Cohn)
Annals of Probability 46 (2018), 337–396. 
On derivatives of graphon parameters (with László Miklós Lovász)
Journal of Combinatorial Theory, Series A 145 (2017), 364–368. 
A counterexample to the BollobásRiordan conjectures on sparse graph limits (with Ashwin Sah, Mehtaab Sawhney, and Jonathan Tidor)
Random matrices
 On the number of Hadamard matrices via anticoncentration (with Asaf Ferber and Vishesh Jain)
Directed graphs and tournaments
 Impartial digraphs (with Yunkun Zhou)
blog
Combinatorica, to appear.
Extremal and Ramsey graph theory
 The critical window for the classical RamseyTurán problem
(with Jacob Fox and PoShen Loh)
Combinatorica 35 (2015), 435–476.
More sums than differences sets slides

Constructing MSTD sets using bidirectional ballot sequences
Journal of Number Theory 130 (2010), 1212–1220. extended version 
Counting MSTD sets in finite Abelian groups
Journal of Number Theory 130 (2010), 2308–2322. 
Sets characterized by missing sums and differences
Journal of Number Theory 131 (2011), 2107–2134.
Other topics

Constructing numerical semigroups of a given genus
Semigroup Forum 80 (2010), 242–254. 
The coefficients of a truncated Fibonacci power series
Fibonacci Quarterly 46/47 (2009), 53–55.
Expositions and surveys

The GreenTao theorem: an exposition (with David Conlon and Jacob Fox)
EMS Surveys in Mathematical Sciences 1 (2014), 249–282. 
Extremal regular graphs: independent sets and graph homomorphisms
American Mathematical Monthly 124 (2017), 827–843. blog
Older expository papers and notes

Biased riffle shuffles, quasisymmetric functions, and the RSK algorithm

Young tableaux and the representations of the symmetric group
Harvard College Mathematics Review 2 (2008), 33–45.