Research

The Hodge-theoretic revolution in matroid theory, initiated by June Huh, turned long-standing combinatorial conjectures into statements about intersection theory on a variety. I work on what that machinery can and cannot reach. White’s conjecture — that the toric ideal of a matroid is generated by symmetric exchanges — is a recurring target: with Michał Lasoń I proved it scheme-theoretically; with Kangjin Han and Julian Weigert I established new cases through inner projections; and with Spencer Backman, Nathan Cheung, Michał Lasoń and Gaku Liu we showed that its Gröbner-basis strengthening is false.

Two projects with June Huh and Botong Wang run in a different direction: realizations of homology classes and projection areas, with Daoji Huang and Shouda Wang; and the linear operators that preserve volume polynomials, with Lukas Grund and Hendrik Süss. With Leonid Monin and Botong Wang I study the Chow polynomials of vertex posets.

How fast can two matrices be multiplied? The question is equivalent to understanding the border rank of one particular tensor, and geometry supplies the sharpest known obstructions. With J. M. Landsberg I gave explicit tensors of border rank greater than 2.02m — the problem of producing, rather than merely proving the existence of, hard tensors. With Petteri Kaski I constructed a universal sequence of tensors for the asymptotic rank conjecture, reducing a question about all tensors to one explicit family.

A statistical model is a variety, and fitting it is an optimisation problem whose difficulty is measured by the maximum likelihood degree — an intersection-theoretic invariant. With Carlos Améndola, Rodica Dinu and Martin Vodička I studied ML degrees for Gaussian graphical models, and with Austin Conner and Kangjin Han described the Sullivant–Talaska ideal of the cyclic model. In phylogenetics, the 3-Kimura model leads to toric varieties attached to finite groups — the setting of the Sturmfels–Sullivant conjectures.

Toric varieties make the dictionary between geometry and combinatorics exact: a lattice polytope is a projective variety with a torus action. Central questions ask when the algebra is as simple as the geometry suggests — normality, the integer decomposition property, and Oda’s question on ample and nef line bundles. My work here includes constructions of non-normal very ample polytopes and answers to several questions of Haase, Hibi and Maclagan.

Classical enumerative geometry keeps turning out to answer modern questions. Characteristic numbers of complete quadrics compute maximum likelihood degrees; the algebraic degree of coupled oscillators, joint with Paul Breiding, Leonid Monin and Simon Telen, counts steady states of a nonlinear physical system. With physicists at Konstanz and ETH Zürich we used these methods to give a topological classification of driven-dissipative nonlinear systems.