Alex Cohen

I am an assistant professor at Courant Institute of Mathematical Sciences, NYU. I currently hold a Clay Research Fellowship.

I am interested in Fourier analysis, PDEs, combinatorics, and related fields.

My CV can be found here. My email is alexcohen at nyu dot edu. My office is WWH 613.

Research

• Moderate-doubling sets in F_2^n intersect subspaces
  with Dmitrii Zakharov, Submitted. (arXiv).

• Counterexample to the two-ends Furstenberg conjecture in R^3 (pdf)

• A simple bound on fluctuations in the 3D Coulomb gas
  with Felipe Hernández, Submitted. (arXiv).

• Sylvester-Gallai configurations on algebraic curves in C^2
  Submitted. (arXiv).

• Lower bounds for incidences
  with Cosmin Pohoata and Dmitrii Zakharov, Inventiones Mathematicae (2025). (journal, arXiv, Vimeo).

• Clustering in typical unit-distance avoiding sets
  with Nitya Mani, Acta Mathematica Hungarica (2025). (journal, arXiv).

• A new upper bound for the Heilbronn triangle problem
  with Cosmin Pohoata and Dmitrii Zakharov, Submitted. (arXiv, Quanta, Blog post, YouTube).

• Fractal uncertainty in higher dimensions
  Annals of Mathematics (2025). (journal, arXiv, YouTube).

• Every real-rooted exponential polynomial is the restriction of a Lee-Yang polynomial
  with Lior Alon and Cynthia Vinzant, Journal of Functional Analysis (2024). (journal, arXiv).

• Fractal uncertainty for discrete 2D Cantor sets
  Analysis and PDE (2025). (journal, arXiv).

• Partition and Analytic Rank are Equivalent over Large Fields
  with Guy Moshkovitz, Duke Mathematical Journal (2023). (journal, arXiv, Quanta, YouTube).

• Uniqueness of excited states to $-\Delta u + u - u^3 = 0$ in three dimensions
  with Zhenhao Li and Wilhelm Schlag, Analysis & PDE (2024). (journal, arXiv, GitHub).

• Structure vs. randomness for bilinear maps
  with Guy Moshkovitz, Discrete Analysis (2022), presented at Symposium on Theory of Computing (2021). (journal, arXiv, YouTube).

• A Sylvester-Gallai result for concurrent lines in the complex plane
  Discrete and Computational Geometry (2020). (journal, arXiv).

• A Sylvester-Gallai theorem for cubic curves
  with Frank de Zeeuw, European Journal of Combinatorics (2022). (journal, arXiv).

• Poissonian correlation of higher order differences
  Journal of Number Theory (2021). (journal, arXiv).

• Sinkhorn limits in finitely many steps
  with Melvyn Nathanson. Linear Algebra and its Applications (2019). (journal, arXiv).

Expository

• Talk notes on discrete and continuous projections in R^2 (notes).
• Lecture notes on the Kakeya conjecture (notes, pset).
  See here for a rigorous proof following these lecture notes (pdf).
• Conjecture about p-adic fractal uncertainty (note)
• PhD Thesis on Fractal Uncertainty Principle (thesis)

Slides

• Branching functions in phase space (slides)
  
• Turing's theory of morphogenesis (slides)
  
• Incidence lower bounds (slides)
  
• Higher dimensional fractal uncertainty (slides)
  
• Heilbronn's triangle problem and connections to projection theory (slides)
  
• Fractal uncertainty for discrete 2D Cantor sets (slides)
• Partition rank vs. analytic rank of tensors (slides)
• A Sylvester-Gallai theorem for concurrent lines in the complex plane (slides)
• A circle of ideas around the Gauss-Bonnet theorem (slides)

Here is a list of properties of the Fourier transform.