About

I am an Assistant Professor of Statistics at Harvard. My research is at the interface of probability, mathematical physics, theoretical computer science, and statistics. I am especially fascinated by computational barriers and algorithmic thresholds in random complex systems.

Research topics

• Physics
• Geometry
• Combinatorics
• Mathematical physics
• Mathematics
• Quantum mechanics

Selected publications

• A Counterexample to the Gaussian Completely Monotone Conjecture

  arXiv (Cornell University) · 2026-05-12

  articleOpen accessSenior author

  We provide an explicit probability measure on $\mathbb{R}$ for which the fifth time derivative of the entropy along the heat flow is positive at some time. This disproves the Gaussian completely monotone (GCM) conjecture (Cheng-Geng '15) and therefore also the Gaussian optimality conjecture (McKean '66) and the entropy power conjecture (Toscani '15). Our proof also implies the existence of a log-concave probability measure on $\mathbb{R}$ for which the GCM conjecture fails at some order. The explicit counterexample was found by GPT-5.5 Pro.

  PublisherOA PDF

• Stable algorithms cannot reliably find isolated perceptron solutions

  arXiv (Cornell University) · 2026-03-31

  preprintOpen accessSenior author

  We study the binary perceptron, a random constraint satisfaction problem that asks to find a Boolean vector in the intersection of independently chosen random halfspaces. A striking feature of this model is that at every positive constraint density, it is expected that a $1-o_N(1)$ fraction of solutions are \emph{strongly isolated}, i.e. separated from all others by Hamming distance $Ω(N)$. At the same time, efficient algorithms are known to find solutions at certain positive constraint densities. This raises a natural question: can any isolated solution be algorithmically visible? We answer this in the negative: no algorithm whose output is stable under a tiny Gaussian resampling of the disorder can \emph{reliably} locate isolated solutions. We show that any stable algorithm has success probability at most $\frac{3\sqrt{17}-9}{4}+o_N(1)\leq 0.84233$. Furthermore, every stable algorithm that finds a solution with probability $1-o_N(1)$ finds an isolated solution with probability $o_N(1)$. The class of stable algorithms we consider includes degree-$D$ polynomials up to $D\leq o(N/\log N)$; under the low-degree heuristic \cite{hopkins2018statistical}, this suggests that locating strongly isolated solutions requires running time $\exp(\widetildeΘ(N))$. Our proof does not use the overlap gap property. Instead, we show via Pitt's correlation inequality that after a random perturbation of the disorder, the number of solutions located close to a pre-existing isolated solution cannot concentrate at $1$.

  PublisherDOI

• On Anti-Confinement Estimates for Self-Repelling Random Walks

  arXiv (Cornell University) · 2026-02-16

  articleOpen accessSenior author

  We study a class of $d$-dimensional random walks, including the two-dimensional simple random walk, reweighted by a self-repelling Gibbsian pair potential. We prove lower bounds on the diffusion constant for short-range interactions, and superdiffusive behavior in case the interaction is sufficiently long-range. Finally, we show that in the superdiffusive regime, faster temporal decay can be compensated by stronger spatial repulsion and vice-versa. Our technique combines GKS-based correlation inequalities on path space with recursive multi-scale estimates.

  PublisherOA PDF

• Stable algorithms cannot reliably find isolated perceptron solutions

  ArXiv.org · 2026-03-31

  articleOpen accessSenior author

  We study the binary perceptron, a random constraint satisfaction problem that asks to find a Boolean vector in the intersection of independently chosen random halfspaces. A striking feature of this model is that at every positive constraint density, it is expected that a $1-o_N(1)$ fraction of solutions are \emph{strongly isolated}, i.e. separated from all others by Hamming distance $Ω(N)$. At the same time, efficient algorithms are known to find solutions at certain positive constraint densities. This raises a natural question: can any isolated solution be algorithmically visible? We answer this in the negative: no algorithm whose output is stable under a tiny Gaussian resampling of the disorder can \emph{reliably} locate isolated solutions. We show that any stable algorithm has success probability at most $\frac{3\sqrt{17}-9}{4}+o_N(1)\leq 0.84233$. Furthermore, every stable algorithm that finds a solution with probability $1-o_N(1)$ finds an isolated solution with probability $o_N(1)$. The class of stable algorithms we consider includes degree-$D$ polynomials up to $D\leq o(N/\log N)$; under the low-degree heuristic \cite{hopkins2018statistical}, this suggests that locating strongly isolated solutions requires running time $\exp(\widetildeΘ(N))$. Our proof does not use the overlap gap property. Instead, we show via Pitt's correlation inequality that after a random perturbation of the disorder, the number of solutions located close to a pre-existing isolated solution cannot concentrate at $1$.

  PublisherOA PDF

• On Anti-Confinement Estimates for Self-Repelling Random Walks

  Open MIND · 2026-02-16

  preprintSenior author

  We study a class of $d$-dimensional random walks, including the two-dimensional simple random walk, reweighted by a self-repelling Gibbsian pair potential. We prove lower bounds on the diffusion constant for short-range interactions, and superdiffusive behavior in case the interaction is sufficiently long-range. Finally, we show that in the superdiffusive regime, faster temporal decay can be compensated by stronger spatial repulsion and vice-versa. Our technique combines GKS-based correlation inequalities on path space with recursive multi-scale estimates.

  DOI

• Short proofs in combinatorics, probability and number theory II

  arXiv (Cornell University) · 2026-04-08

  articleOpen access

  We give a quintet of proofs resulting from questions posed by Erdős. These questions concern ordinary lines in planar point sets, sequences with uniformly small exponential sums, $K_4$-free $4$-critical graphs with few chords in any cycle, a counterexample to a "fewnomial" version of the Erdős--Turán discrepancy bound, and a finiteness theorem for integers $n$ such that $n-a k^2$ is prime for all $k\leq \sqrt{n/a}$ coprime to $n$ (for fixed $a\in\mathbb Z_+$). Each proof is due to an internal model at OpenAI.

  PublisherOA PDF

• A Counterexample to the Gaussian Completely Monotone Conjecture

  arXiv (Cornell University) · 2026-05-12

  preprintOpen accessSenior author

  We provide an explicit probability measure on $\mathbb{R}$ for which the fifth time derivative of the entropy along the heat flow is positive at some time. This disproves the Gaussian completely monotone (GCM) conjecture (Cheng-Geng '15) and therefore also the Gaussian optimality conjecture (McKean '66) and the entropy power conjecture (Toscani '15). Our proof also implies the existence of a log-concave probability measure on $\mathbb{R}$ for which the GCM conjecture fails at some order. The explicit counterexample was found by GPT-5.5 Pro.

  PublisherDOI

• Short proofs in combinatorics, probability and number theory II

  arXiv (Cornell University) · 2026-04-08

  preprintOpen access

  We give a quintet of proofs resulting from questions posed by Erdős. These questions concern ordinary lines in planar point sets, sequences with uniformly small exponential sums, $K_4$-free $4$-critical graphs with few chords in any cycle, a counterexample to a "fewnomial" version of the Erdős--Turán discrepancy bound, and a finiteness theorem for integers $n$ such that $n-a k^2$ is prime for all $k\leq \sqrt{n/a}$ coprime to $n$ (for fixed $a\in\mathbb Z_+$). Each proof is due to an internal model at OpenAI.

  PublisherDOI

• Short proofs in combinatorics and number theory

  ArXiv.org · 2026-03-31

  articleOpen access

  We give a triplet of short proofs, each of which answers a question raised by Erdős. The first concerns the small prime factors of $\binom{n}{k}$, the second concerns whether an additive basis $A$ can always be split into pieces $A_1$ and $A_2$ such that each of $A_i + A_i$ has bounded gaps, and the final concerns whether $\{αp\}$ is "well-distributed" in the sense introduced by Hlawka and Petersen. In each case, the proof is due entirely to an internal model at OpenAI.

  PublisherOA PDF

• Strong topological trivialization of multi-species spherical spin glasses

  The Annals of Probability · 2026-02-26

  articleSenior author

  We study the landscapes of multi-species spherical spin glasses. Our results determine the phase boundary for annealed trivialization of the number of critical points and establish its equivalence with a quenched strong topological trivialization property. Namely, in the “trivial” regime, the number of critical points is constant, all are well conditioned, and all approximate critical points are close to a true critical point. As a consequence, we deduce that Langevin dynamics at sufficiently low temperature has logarithmic mixing time. Our approach begins with the Kac–Rice formula. We characterize the annealed trivialization phase by explicitly solving a suitable multidimensional variational problem, obtained by simplifying certain asymptotic determinant formulas from (Probab. Math. Phys. 3 (2022) 731–789; Ann. Inst. Henri Poincaré Probab. Stat. 60 (2024) 636–657). To obtain more precise quenched results, we develop general purpose techniques to avoid subexponential correction factors and show nonexistence of approximate critical points. Many of the results are new, even in the one-species case.

  PublisherDOI

Frequent coauthors

• Sébastien Bubeck

  43 shared

• Yuanzhi Li

  21 shared

• Yin Tat Lee

  12 shared

• A. El Alaoui

  12 shared

• Brice Huang

  Massachusetts Institute of Technology

  12 shared

• Andrea Montanari

  10 shared

• Yuval Peres

  Beijing Institute of Mathematical Sciences and Applications

  10 shared

• Victoria Kostina

  California Institute of Technology

  7 shared

Education

• Ph.D., Mathematics

  Stanford

Awards & honors

• Outstanding Paper Award at NeurIPS 2021
• Best Paper Award and Best Student Paper Award at SODA 2020