- 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 two-distance sets
- Incidence geometry, e.g., the joints problem
- The geometry of transitive sets, applications to Cayley graphs
- Extension complexity of polytopes
The Green-Tao theorem and a relative Szemerédi theorem
An arithmetic transference proof of a relative Szemerédi theorem
Mathematical Proceedings of the Cambridge Philosophical Society 156 (2014), 255–261.
- The Green-Tao 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 4-cycles (with David Conlon, Jacob Fox, and Benny Sudakov)
Journal of the London Mathematical Society, to appear.
Removal lemmas and approximate homomorphisms (with Jacob Fox)
- Which graphs can be counted in $C_4$-free graphs? (with David Conlon, Jacob Fox, and Benny Sudakov)
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.
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.
Tower-type bounds for Roth’s theorem with popular differences (with Jacob Fox and Huy Tuan Pham)
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, to appear.
Testing linear-invariant properties (with Jonathan Tidor)
IEEE Symposium on Foundations of Computer Science (FOCS) 2020.
Topics in additive combinatorics
Common and Sidorenko linear equations (with Jacob Fox and Huy Tuan Pham)
The Quarterly Journal of Mathematics, to appear.
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 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, to appear.
The cylindrical width of transitive sets (with 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 195 (2022), to appear.
Spherical two-distance sets and eigenvalues of signed graphs (with Zilin Jiang, Jonathan Tidor, Yuan Yao, and Shengtong Zhang)
Graphs with high second eigenvalue multiplicity (with Milan Haiman, Carl Schildkraut, and Shengtong Zhang)
Joints tightened (with Hung-Hsun Hans Yu)
Joints of varieties (with Jonathan Tidor and Hung-Hsun Hans Yu)
Extension complexity and nonnegative rank
- Extension complexity of low-dimensional polytopes (with Matthew Kwan and Lisa Sauermann)
- Exploring a planet, revisited,
American Mathematical Monthly, to appear.
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.
Energy-minimizing error-correcting 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
- Enumerating $k$-SAT functions
(with Dingding Dong and Nitya Mani)
ACM-SIAM Symposium on Discrete Algorithms (SODA) 2022.
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.
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 Bollobás-Riordan conjectures on sparse graph limits (with Ashwin Sah, Mehtaab Sawhney, and Jonathan Tidor)
Combinatorics, Probability and Computing 30 (2021), 796–799.
- On the number of Hadamard matrices via anti-concentration (with Asaf Ferber and Vishesh Jain)
Combinatorics, Probability and Computing, to appear.
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)
Extremal and Ramsey graph theory
- The critical window for the classical Ramsey-Turán problem
(with Jacob Fox and Po-Shen Loh)
Combinatorica 35 (2015), 435–476.
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.
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
Young tableaux and the representations of the symmetric group
Harvard College Mathematics Review 2 (2008), 33–45.