Mathematics · science · research
Hello, and welcome. Here I collect and share my research in mathematics, which has always been a great passion of mine. I use AI to explore my favorite topics and have managed to resolve several open questions, including conjectures of my own. By day, I consult on AI and algorithms for high-tech companies. Outside work and mathematics, I enjoy playing chess. For updates on the Minc’s list project, see here.
Research interests Combinatorics · Matrix theory · Representation theory · High-energy physics
Selected work
Ordered by original publication or posting date.
A proof of Foregger’s conjecture that per A ≥ 2¹⁻ⁿ for every nearly decomposable doubly stochastic n × n matrix A, with equality only for ½(Iₙ + Pₙ) up to permutations of rows and columns.
An extension of Dittert’s permanent conjecture, proved in 2026, to every homogeneous real stable polynomial with nonnegative coefficients, with the sharp constant n!/nⁿ and the equality case. In particular, it gives an independent proof of Dittert’s conjecture in every order.
A proof of my 2018 conjecture giving the sharp maximum of a diagonal product of a stochastic matrix of bounded rank, with all equality cases.
We give counterexamples to an inequality for permanents of positive semidefinite matrices that Zhang stated as a consequence of Drury’s 2016 permanent conjecture: a complex one of order twelve and rank two, and a real one of order twenty-one and rank three.
A proof of Kopotun’s 1996 conjecture that σₖ(A) ≥ σₖ((nJₙ + A)/(n + 1)) for every doubly stochastic A of order n and every 2 ≤ k ≤ n, where σₖ is the sum of all k × k subpermanents, with equality only at Jₙ.
We give a counterexample to a conjecture of Johnson, Marijuán, and Pisonero: we show that the spectrum {1, 1/2, ±i/√3} is realizable and spectrally simple, but has no irreducible nonnegative realization. More generally, we prove their nonexistence conjecture for an order-four boundary family.
An analytic proof that the close-packed dimer entropy per site of the simple cubic lattice is at least ½ log(3125/1296) + 47/10000 = 0.444775842629…, above the previous best published lower bound, obtained by Csikvári of 0.441075842629…
Lower bounds for the maximal cp-rank pₙ in every order n ≥ 5 that leave only a linear gap to the known upper bound, so that pₙ = n²/2 + O(n).
A quadratic permanent inequality for positive semidefinite matrices, proved through order five, with the order-four case of Soules’ permanent-on-top conjecture settled.
A proof of Wang’s 1977 conjecture that per A ≥ per((nJₙ + A)/(n + 1)) for every doubly stochastic A, with equality only at Jₙ.
A proof of the 1967 Marcus–Minc conjecture that per A ≥ per((nJₙ − A)/(n − 1)) for every doubly stochastic A, with all equality cases.
A uniform exponential lower bound for permanents of 6-regular circulant matrices that beats the general constant 3125/1296, and the optimal constant, resolving the circulant question in Minc’s Problem 10.
A proof of my 2018 conjecture giving the sharp maximum permanent of a stochastic matrix of bounded rank, with all equality cases.
The Foregger-Sinkhorn tie-point conjecture asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero is larger than its permanent, then that zero is a tie point. We give a counterexample in dimension eight.
A proof of the 1981 Kim–Roush conjecture for the maximum of per(I − A) in odd order, with a classification of all maximizers.
Two conjectures on sharp permanent and diagonal-product bounds for stochastic matrices of bounded rank.