Research
My papers are listed below by topic.
Additional links to my work: arXiv  MathSciNet  Google Scholar  CV
Research themes
 Graph regularity and additive combinatorics:
Szemerédi’s graph regularity method is a powerful tool in modern graph theory research.
The classical version of the regularity method only works in the dense setting, and my work extends the method to the sparse setting.
Ideas arising from graph regularity can be transferred to the world of additive combinatorics.
For example, our work on sparse graph/hypergraph regularity counting lemmas led to a simplification of the proof the Green–Tao theorem (see exposition).
 Regularity and counting lemmas in sparse graphs
 Arithmetic regularity, and applications to additive combinatorics
 Popular differences
 Property testing and algorithmic applications
 Graph limits
 Graph inequalities:
Many fundamental problems in extremal graph theory can be phrased in terms of inequalities between subgraph densities or graph homomorphism counts.
My work develops new tools to solve such problems.
 Independent sets, colorings, and graph homomorphisms
 Sidorenko’s conjecture and related problems
 Variational problems arising from large deviations in random graphs
 Discrete geometry: I am interested in extremal problems in discrete geometry, e.g., what is the largest size of a geometric configuration satisfying certain properties. Solutions often apply a variety of tools such as linear programming bounds, the polynomial method, spectral graph theory, group representation theory.
 Sphere packing, spherical codes
 Equiangular lines, spherical twodistance sets
 Incidence geometry, e.g., the joints problem
 The geometry of transitive sets, applications to Cayley graphs
 Extension complexity of polytopes
Papers
The GreenTao theorem and a relative Szemerédi theorem

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.
 The GreenTao theorem: an exposition
(with David Conlon and Jacob Fox)
EMS Surveys in Mathematical Sciences 1 (2014), 249–282
 A short proof of the multidimensional Szemerédi theorem in the primes
(with Jacob Fox)
American Journal of Mathematics 137 (2015), 1139–1145
Sparse graph regularity and counting

Extremal results in sparse pseudorandom graphs (with David Conlon and Jacob Fox)
Advances in Mathematics 256 (2014), 206–290 
The regularity method for graphs with few 4cycles (with David Conlon, Jacob Fox, and Benny Sudakov)
Journal of the London Mathematical Society 104 (2021), 2376–2401 
Removal lemmas and approximate homomorphisms (with Jacob Fox)
Combinatorics, Probability and Computing, to appear
 Which graphs can be counted in $C_4$free graphs?
(with David Conlon, Jacob Fox, and Benny Sudakov)
Pure and Applied Mathematics Quarterly 18 (2022), 2413–2432
Algorithmic graph regularity

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
Popular differences

Triforce and corners (with Jacob Fox, Ashwin Sah, Mehtaab Sawhney, and David Stoner)
Mathematical Proceedings of the Cambridge Philosophical Society, 169 (2020), 209–223 
Patterns without a popular difference (with Ashwin Sah and Mehtaab Sawhney)
Discrete Analysis, 2021:8, 30 pp. 
Towertype bounds for Roth’s theorem with popular differences (with Jacob Fox and Huy Tuan Pham)
Journal of the European Mathematical Society, to appear
Arithmetic regularity and applications

Efficient arithmetic regularity and removal lemmas for induced bipartite patterns (with Noga Alon and Jacob Fox)
Discrete Analysis 2019:3, 14 pp 
Induced arithmetic removal: complexity 1 patterns over finite fields (with Jacob Fox and Jonathan Tidor)
Israel Journal of Mathematics 248 (2022), 1–38 
Testing linearinvariant properties (with Jonathan Tidor)
IEEE Symposium on Foundations of Computer Science (FOCS 2020)
SIAM Journal on Computing 51 (2022), 1230–1279
Topics in additive combinatorics

Common and Sidorenko linear equations (with Jacob Fox and Huy Tuan Pham)
The Quarterly Journal of Mathematics 72 (2021), 1223–1234 
A short proof of the canonical polynomial van der Waerden theorem (with Jacob Fox and Yuval Wigderson)
Comptes Rendus Mathématique 358 (2020), 957–959
Hypergraph expanders
 Hypergraph expanders of all uniformities from Cayley graphs (with David Conlon and Jonathan Tidor)
Proceedings of the London Mathematical Society 121 (2020), 1311–1336
Cayley graphs and transitive sets

Quasirandom Cayley graphs (with David Conlon)
Discrete Analysis 2017:6, 14 pp 
Cayley graphs without a bounded eigenbasis (with Ashwin Sah and Mehtaab Sawhney)
International Mathematics Research Notices. IMRN 2022 (2022), 6157–6185 
The cylindrical width of transitive sets (with Ashwin Sah and Mehtaab Sawhney)
Israel Journal of Mathematics 253 (2023), 647–672 
Quantum Unique Ergodicity for Cayley graphs of quasirandom groups (with Michael Magee and Joe Thomas)

Cayley graphs that have a quantum ergodic eigenbasis (with Assaf Naor, Ashwin Sah, and Mehtaab Sawhney)
Israel Journal of Mathematics, to appear
Equiangular lines and eigenvalue multiplicities

Equiangular lines with a fixed angle (with Zilin Jiang, Jonathan Tidor, Yuan Yao, and Shengtong Zhang)
Annals of Mathematics 194 (2021), 729–743 
Spherical twodistance sets and eigenvalues of signed graphs (with Zilin Jiang, Jonathan Tidor, Yuan Yao, and Shengtong Zhang)
Combinatorica, to appear 
Graphs with high second eigenvalue multiplicity (with Milan Haiman, Carl Schildkraut, and Shengtong Zhang)
Bulletin of the London Mathematical Society 54 (2022), 1630–1652.
Incidence geometry

Joints tightened (with HungHsun Hans Yu)
American Journal of Mathematics, to appear 
Joints of varieties (with Jonathan Tidor and HungHsun Hans Yu)
Geometric and Functional Analysis 32 (2022) 302–339
Extension complexity and nonnegative rank
 Extension complexity of lowdimensional polytopes (with Matthew Kwan and Lisa Sauermann)
Transactions of the American Mathematical Society 375 (2022), 4209–4250
Plank problem
 Exploring a planet, revisited,
American Mathematical Monthly 129 (2022), 678–680
Independent sets and graph homomorphisms
 Extremal regular graphs: independent sets and graph homomorphisms
American Mathematical Monthly 124 (2017), 827–843

The number of independent sets in a regular graph
Combinatorics, Probability and Computing 19 (2010), 315–320 
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
 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
 A reverse Sidorenko inequality (with Ashwin Sah, Mehtaab Sawhney, and David Stoner)
Inventiones Mathematicae 221 (2020), 665–711
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, 365 (2020), 107056
Asymptotic enumeration

Enumerating $k$SAT functions (with Dingding Dong and Nitya Mani)
ACMSIAM Symposium on Discrete Algorithms (SODA 2022) 
On the number of error correcting codes (with Dingding Dong and Nitya Mani)
Combinatorics, Probability and Computing, to appear 
Nearly all $k$SAT functions are unate (with József Balogh, Dingding Dong, Bernard Lidický, and Nitya Mani)
ACM Symposium on Theory of Computing (STOC 2023)
Large deviations in random graphs

On replica symmetry of large deviations in random graphs (with Eyal Lubetzky)
Random Structures & Algorithms 47 (2015), 109–146 
On the variational problem for upper tails in sparse random graphs (with Eyal Lubetzky)
Random Structures & Algorithms 50 (2017), 420–436 
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)
International Mathematics Research Notices. IMRN 2020, 167–213 
On the upper tail problem for random hypergraphs (with Yang Liu)
Random Structures & Algorithms 58 (2021), 179–220
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)
Combinatorics, Probability and Computing 30 (2021), 796–799
Random matrices
 On the number of Hadamard matrices via anticoncentration (with Asaf Ferber and Vishesh Jain)
Combinatorics, Probability and Computing (2022), no. 3, 455–477
Extremal subgraph density problems in directed graphs and tournaments
 Impartial digraphs (with Yunkun Zhou)
Combinatorica 40 (2020), 875–896
 Paths of given length in tournaments
(with Ashwin Sah and Mehtaab Sawhney)
Combinatorial Theory, 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 
Setcoloring Ramsey numbers and errorcorrecting codes near the zerorate threshold (with David Conlon, Jacob Fox, and Huy Tuan Pham)
Intersecting families of graphs
 $K_4$intersecting families of graphs (with Aaron Berger)
More sums than differences sets

Constructing MSTD sets using bidirectional ballot sequences
Journal of Number Theory 130 (2010), 1212–1220 
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
Miscellaneous 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
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