Littlewood-Offord Theory

A note on inverse results of random walks in Abelian groups (with Charles Koenig and Amanda Pan, Combinatorics and Number Theory, Vol. 13 (2024), No. 1, 67-92).
Small probability, inverse theorems, and applications (with Van Vu, on the occasion of Paul Erdos' 100th anniversary, Bolyai Society Mathematical Studies, Vol. 25 (2013)).
Inverse Littlewood-Offord problems and the singularity of random symmetric matrices (Duke Mathematics Journal Vol. 161, 4 (2012), 545-586).
Optimal inverse Littlewood-Offord theorems (with Van Vu, Advances in Mathematics, Vol. 226 6 (2011), 5298-5319) (Erratum).
Anti-concentration of inhommogeneous random walks in non-abelian groups (preprint).

Random matrices

Rank fluctuation of matrix products and a moment method for growing groups (with Roger Van Peski, arXiv).
Local and global universality of random matrix cokernels (with Melanie Matchett Wood, Mathematische Annalen, to appear).
Universality for cokernels of random matrix products (with Roger Van Peski, Advances in Mathematics, Volume 438, February 2024, 109451).
Random integral matrices: universality of surjectivity and the cokernel (with Melanie Matchett Wood, Inventiones Mathematicae, 228 (2022), no. 1, 1-76).
Some new results in random matrices over finite fields (with Kyle Luh and Sean Meehan, Journal of the London Mathematical Society, Volume 103, Issue 4 (2021), 1209-1252).
Random matrices: overcrowding estimates for the spectrum (Journal of Functional Analysis, (2018) Volume 275, Issue 8, 2197-2224).
Normal vector of a random hyperplane (with Van Vu, International Mathematics Research Notices, (2018) Issue 6, 1754-1778).
Random matrices: tail bounds for gaps between eigenvalues (with Terence Tao and Van Vu, Probability Theory and Related Fields, (2017), Vol. 167, 3, 777-816).
The elliptic law (with Sean O'Rourke, International Mathematics Research Notices (2015), Vol. 2015, 7620-7689).
On the concentration of random multilinear forms and the universality of random block matrices (with Sean O'Rourke, Probability Theory and Related Fields (2015), Vol. 162, 1, 97-154).
Random doubly stochastic matrices: the circular law (Annals of Probability (2014), Vol. 42, No. 3, 1161-1196).
Random matrices: law of the determinant (with Van Vu, Annals of Probability (2014), Vol. 42, No. 1, 146-167).
On the least singular value of random symmetric matrices (Electronic Journal of Probability 17 (2012), no. 53).
Cokernels of adjacency matrices of random r-regular graphs (with Melanie Matchett Wood, preprint).

Random polynomials

Concentration of the number of real roots of random polynomials (with Ander Aguirre and Jingheng Wang, submitted).
Hole radii for the Kac polynomials and derivatives (with Oanh Nguyen, Stochastic Processes and their Applications (2024), 104386).
Real roots of random orthogonal polynomials with exponential weights (with Yen Do, Doron Lubinsky, Oanh Nguyen, and Igor Pritsker, submitted).
Central Limit Theorem for the number of real roots of random orthogonal polynomials (with Yen Do, Oanh Nguyen, and Igor Pritsker, Annales de l’Institut Henri Poincare, Probabilites et Statistiques, (2024) Vol. 60, No. 3, 2211-2240).
Exponential concentration for the number of roots of random trigonometric polynomials (with Ofer Zeitouni, Annales de l’Institut Henri Poincare, Probabilites et Statistiques, (2024) Vol. 60, No. 2, 1077-1089).
Concentration of the number of intersections of random eigenfunctions on flat tori (Proceedings of the American Mathematical Society 151 (2023), no. 7, 3127-3143).
Universality of Poisson limits for moduli of roots of Kac polynomials (with Nicholas Cook, Oren Yakir, and Ofer Zeitouni, International Mathematics Research Notices, 2023, no. 8).
Random trigonometric polynomials: universality and non-universality of the variance for the number of real roots (with Yen Do and Oanh Nguyen, Annales de l’Institut Henri Poincare, Vol. 58, No. 3 (2022), 1460-1504).
Universality of the minimum modulus for random trigonometric polynomials (with Nicholas Cook, Discrete Analysis, 20 (2021), 45pp).
Random eigenfunctions on flat tori: universality for the number of intersections (with Mei-Chu Chang, Oanh Nguyen, and Van Vu, International Mathematics Research Notices, (2020) Vol. 2020, Issue 24, Pages 9933-9973).
On the number of real roots of random polynomials (with Oanh Nguyen and Van Vu, Communications in Contemporary Mathematics (2016) Vol. 18, 4, 1550052).
Real roots of random polynomials: expectation and repulsion (with Yen Do and Van Vu, Proceedings London Mathematical Society (2015), Vol. 111 (6), 1231-1260).

Structure in large sumsets

A characterization of incomplete sequences in Fpd (with Van Vu, Journal of Combinatorial Theory, Series A, 119 (2012)).
Squares in sumsets (with Van Vu, on the occasion of Endre Szemeredi's 70th birthday , Bolyai Society Mathematical Studies, Vol. 21 (2010)).
A classification theorems for sumsets modulo a prime (with Van Vu, Journal of Combinatorial Theory, Series A, 16 (2009)).
Subset sums in Fp (with Endre Szemeredi and Van Vu, Acta Arithmetica, 131 (2008), 303-316).

Others

A note on the singularity probability of random directed d-regular graphs (with Amanda Pan, European Journal of Combinatorics, Volume 122, pages 104039 (2024)).
Rank of near uniform matrices (with Charles Koenig, Journal of Combinatorics, Vol. 13, No. 3 (2022), pp. 397-436).
Surjectivity of near square random matrices (with Elliot Paquette, Combinatorics, Probability and Computing, 29 (2020), no. 2, 267-292).
Eigenvectors of random matrices of symmetric entry distributions (with Sean Meehan, Proceedings of the American Mathematical Society, (2019) Volume 147, Number 2, 835-847).
Concentration of distances in Wigner matrices (Linear Algebra and its Applications 536 (2018), 52-86).
Asymptotic Lyapunov exponents for large random matrices (Annals of Applied Probability (2017), Vol. 27, No. 6, 3672-3705).
A general law of large permanent (with Jozsef Balogh, Discrete and Continuous Dynamical Systems (Series A), Vol. 37, no. 10, (2017)).
Near invariance of the hypercube (with Scott Aaronson, Israel Journal of Mathematics (2016) 212, 385-417).
On a condition number of random polynomial systems (Mathematics of Computation (2016) 85, 737-757).
A new approach to an old problem of Erdos and Moser (Journal of Combinatorial Theory, Series A 119 (2012) 977-993).
On the distribution of 3-AP in sparse subsets of Fpn (Combinatorics, Probability and Computing 20 (2011), 777-791).
On two-point configurations in a random set (Integers 9 (2009), A3, 41-45).

Summer research for undergraduate students

The Rank of the Sandpile Group of Random Directed Bipartite Graphs (Atal Bhargava, Jack DePascale, Jake Koenig, Annals of Combinatorics 27 (2023) 979-992).