Current projects
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Given a family of sets in Euclidean space, we call a specific pair of sets in the family secretive if their intersection contains a point that is not contained in any other set in the family. (This point is, of course, the secret they share.) We call the family clandestine if every pair of sets in it is secretive. What is the maximum size of a clandestine family of balls in \mathbb{R}^d?
If d=2, then any clandestine family of disks has at most 4 members. János Pach and I are investigating this and related questions for d\geq 3.
Helly’s theorem is one of the most fundamental results in combinatorial convex geometry. Jiří Matoušek observed that “Helly’s theorem [has] inspired a whole industry of Helly-type theorems”, and I am one of many cogs in that industry. Right now, I’m thinking about problems related to quantitative versions of Helly’s and Krasnoselskii’s theorems.
Given a set of points in Euclidean space, imagine removing the vertices of its convex hull, looking at what remains, removing the vertices of its convex hull, and repeating. Each step peels away a layer of the set; thus, the number of layers a set has—its layer number—is a measure of the depth of the set’s innermost points. My current work, with Gergely Ambrus, is on improving the current best bounds on the layer number of the grid \{1,2,\dots,n\}^d.
Exponential multivalued forbidden configurations (with A. Sali). Discrete Mathematics & Theoretical Computer Science 23 (2021).
An inverse problem for the collapsing sum. Australasian Journal of Combinatorics 79 (2021): 183–192.
A combinatorial interpretation of Gaussian blur. Minnesota Journal of Undergraduate Mathematics 5 (2020).
Discrete and Convex Geometry
A near-optimal upper bound for the layer number of the grid (with N. Varadarajan), submitted.
A mélange of diameter Helly-type theorems (with P. Soberón). SIAM Journal on Discrete Mathematics 35 (2021): 1615–1627. [arXiv]
Discrete quantitative Helly-type theorems with boxes. Advances in Applied Mathematics 129 (2021): 102217. [arXiv]
Dynamical Systems
Dynamics and entropy of S-graph shifts. Discrete and Continuous Dynamical Systems 42 (2022): 5637–5663. [arXiv]
Number Theory
An average of generalized Dedekind sums (with S. Gaston). Journal of Number Theory 212 (2020): 323–338. [arXiv]
Expository notes
Proof of G.E. Andrew’s theorem that the number of vertices in a lattice polytope P in \mathbb{R}^d is at most O\big(\operatorname{Vol}(P)^{\frac{d-1}{d+1}}\big).
Course notes
An introduction to graph limits. One-semester course on dense graph limits at the advanced senior undergraduate level.
Survey of enumerative combinatorics. Notes from a one-semester topics course taught by Alex Postnikov.
Algebraic topology. Notes from a one-semester course.
Rational choice theory. A quick tour through some aspects of the theories of social choice and games.
Teaching, mentoring, and outreach
In 2023 and 2024, I am an MIT PRIMES research mentor.
In summer 2023, I taught classes at Canada/USA Mathcamp on finite difference calculus, probabilistic methods in combinatorics, linear algebraic methods in discrete geometry, the connection between hat games and non-measurable sets, infinite random graphs, and transcendental numbers. I also advised a project on writing short stories and wrote one of my own.
In summer 2022, I taught classes at Canada/USA Mathcamp on symbolic dynamics, enumerative combinatorics, Diophantine approximation and continued fractions, linear algebra in combinatorics, and convex/discrete geometry.
In summer 2023, I was also a semi-organizer at Maths Beyond Limits.
Graphs, Groups, Infinity: Three stories in mathematics is a book I wrote over my last year as an undergraduate, with the goal of bringing the experience of doing mathematics to a general audience. You can read an excerpt here.