2026
submitted
Four hyperplanes do not always equipartition a mass in ℝ⁴
A smooth positive density in four dimensions that admits no equipartition into sixteen equal parts by four affine hyperplanes.
Research
Geometry, combinatorics, topology.
My research lies in discrete geometry and topological combinatorics. I am interested in how combinatorics, algebraic topology, and linear algebra interact in problems about partitions and intersections.
The topics below overlap: a paper can belong to more than one area. The publication catalog includes the full list; selected papers follow the topic overview.
Tverberg theory
Partitions, convex hulls, and intersection patterns. Colorful, tolerant, quantitative, and topological versions of Tverberg’s theorem.
29 papers
Mass partitions
Fairly dividing measures using hyperplanes and other geometric shapes, with connections to topology and convexity.
16 papers
Helly-type theorems
How local intersection conditions force global conclusions. Colorful, fractional, and quantitative forms of Helly’s theorem.
17 papers
Topological combinatorics
Using topology to organize combinatorial choices: Borsuk–Ulam and KKM theorems, selection structures, and configuration spaces.
15 papers
Combinatorial convexity
Transversals, quantitative intersection theorems, convex sets, and geometric depth.
19 papers
Graph theory
Graph homomorphisms, partition functions, geometric graphs, and crossing numbers.
5 papers
Fair division
Envy-free partitions and fair allocations, studied through geometric and topological methods.
3 papers
Discrepancy & tolerance
Connections between discrepancy, probabilistic methods, and geometric partitions that tolerate deletions.
2 papers
Polytopes & symmetry
Realizing prescribed symmetry groups and automorphisms in convex polytopes.
2 papersA curated selection
Representative papers
2026
submitted
KKM theorems and discrete geometry beyond matroids
Selection structures extend the role of color classes and matroids in KKM theorems and discrete geometry.
2026
submitted
Tverberg cores and Kalai’s cascade conjecture
Topological Tverberg cores, vertex-deletion tolerance, and consequences for Kalai’s cascade conjecture.
2024
published
Generalizations of the Yao–Yao partition theorem and the central transversal theorem
Discrete Comput. Geom. 71 (4), 1381–1402.
2024
published
Complex analogues of the Tverberg–Vrećica conjecture and central transversal theorems
Trans. Amer. Math. Soc. 379 (2026), no. 5, 3665–3691.
2024
2018
published
Robust Tverberg and colorful Carathéodory results via random choice
Combinatorics, Probability and Computing 27 (3), 427–440.