Publications & preprints
My research focuses on the intersection of mathematical biology and applied algebra — how tools from algebraic geometry, commutative algebra, and combinatorics can be applied to biological questions. Three projects I have been most involved with recently are combinatorial neural coding, sample frequency spectra in population genetics, and algebraic matroids in applications (matrix completion and rigidity theory).
See my Google Scholar profile, or check the arXiv for recent work. For a detailed description of current and future projects, please read my research statement.
We relate the emerging theory of convex neural codes to the established theory of oriented matroids, both categorically and with respect to geometry and computational complexity. In particular, we use oriented matroids to construct codes for which deciding convexity is NP-hard.
We explore the number of critical points for a potential generated by n Newtonian point masses. We prove that this number is finite for generic parameters, and then use techniques from Morse theory and BKK theory to find concrete bounds.
We prove sharp bounds on the number of pieces a piecewise-constant function must have in order to capture any possible moment vector. This has applications in population genetics, where it describes all possible allele frequency spectra.
A self-contained introduction to algebraic matroids together with examples highlighting their potential application.
We establish natural properties of non-degenerate hyperplane codes in terms of the polar complex, a simplicial complex associated to any combinatorial code. We prove that the polar complex of a non-degenerate hyperplane code is shellable, and show that all currently known properties of hyperplane codes follow.
Using the neural ideal together with its canonical form, we provide algebraic signatures of certain families of codes that are non-convex, and signatures for some convex families including intersection-complete codes.
The sample frequency spectrum (SFS) is a widely used summary statistic in population genetics, with strong dependence on historical demography. We use algebraic and convex geometry to explain difficulties in demographic inference, and characterize the semialgebraic set of possible spectra.
Algebraic techniques to analyze state space models in the areas of structural identifiability, observability, and indistinguishability.
Combinatorial codes are convex if codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. We provide a complete characterization of local obstructions to convexity.
The semialgebraic and algebraic geometry of projections of rank-one tensors to some of their coordinates, giving insight into the problem of rank-one completion of partial tensors.
We investigate the problem of completing partial matrices to rank-one matrices in the standard simplex. For each pattern of specified entries, we give equations and inequalities which are satisfied if and only if an eligible completion exists.
Current methods from computational algebraic geometry and combinatorics applied to analyze the Shuttle model for the Wnt signaling pathway.
We analyze models of the Wnt signaling pathway, involved in adult stem cell tissue maintenance and cancer. Bayesian parameter inference fails to reject models; non-parametric tools including algebraic matroids are employed.
We study curves consisting of unions of projective lines whose intersections are given by graphs. We discuss property Np for their embeddings, and the subspace arrangements associated to their secant varieties.
Follow-up to our expository article "Algebraic Matroids in Action." We provide a detailed proof of Ingleton's theorem that algebraic matroids over algebraically closed field of characteristic 0 are linearizable, and examine what happens in the absence of Ingleton's hypotheses.
We analyze a construction on top of the set of slopes given by an angle constraint system of incidences and angles. We provide a matricial rigidity formulation for colored graphs, an algebro-geometric reformulation, and a combinatorial characterization in a special case.
We study convex neural codes in dimension 1 (i.e. on a line or a circle), using consecutive-ones matrices for structural and algorithmic results, and generating functions for enumerative results.
Algorithms for computing algebraic matroids using numerical algebra and symbolic computation, applied to various examples.
Algebraic matroids whose ground sets are endowed with graph symmetry — motivated by framework rigidity, low-rank matrix completion, and determinantal varieties. We define and compute the circuit polynomials associated to the circuits of these matroids.