Research

Research Interests

My research focuses on the intersection of mathematical biology and applied algebra. In particular, I think about how tools from algebraic geometry, commutative algebra, and combinatorics can be applied to biological questions. I also work on either side of this intersection: I study problems related to algebraic matroids and coordinate projections in other realms, and I have computational biology projects in translation dynamics and gene regulatory networks.

See my profile at Google Scholar, or check the arXiv for my recent work. For a detailed description of current and future projects, please read my research statement.

Published Articles

  • Algebraic tools for the analysis of state space models

  • Nicolette Meshkat, Zvi Rosen, and Seth Sullivant
  • We present algebraic techniques to analyze state space models in the areas of structural identifiability, observability, and indistinguishability. (To appear in Proceedings of Mathematical Society of Japan)
  • (arXiv | journal)
  • What makes a neural code convex?

  • Carina Curto, Elizabeth Gross, Jack Jeffries, Katherine Morrison, Mohamed Omar,
    Zvi Rosen, Anne Shiu, and Nora Youngs
  • 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.
  • SIAM Journal on Applied Algebra and Geometry, 1(1), 222-238, 2017.
  • (arXiv | journal)
  • The geometry of rank-one tensor completion

  • Thomas Kahle, Kaie Kubjas, Mario Kummer, Zvi Rosen
  • The semialgebraic and algebraic geometry of projections of rank-one tensors to some of their coordinates is studied, giving insight into the problem of rank-one completion of partial tensors.
  • SIAM Journal on Applied Algebra and Geometry, 1(1), 200-221, 2017
  • (arXiv | journal)
  • Matrix Completion for the Independence Model

  • Kaie Kubjas, Zvi Rosen
  • 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.
  • Journal of Algebraic Statistics, 8(1), 1-21, 2017.
  • (arXiv | journal)
  • Algebraic systems biology: a case study for the Wnt pathway.

  • Elizabeth Gross, Heather A. Harrington, Zvi Rosen, Bernd Sturmfels
  • Current methods from computational algebraic geometry and combinatorics are applied to analyze the Shuttle model for the Wnt signaling pathway.
  • Bulletin of Mathematical Biology, 78, 21-51, 2016.
  • (arXiv | journal)
  • Parameter-free methods distinguish Wnt pathway models and guide design of experiments

  • Adam L. MacLean, Zvi Rosen, Helen M. Byrne, Heather A. Harrington
  • We analyze models of the Wnt signaling pathway, which is involved in adult stem cell tissue maintenance and cancer. Bayesian parameter inference fails to reject models; non-parametric tools including algebraic matroids are employed.
  • Proceedings of the National Academy of Sciences, 112(9), 2652-2657, 2015.
  • (arXiv | journal)
  • Line arrangements modeling curves of high degree: equations, syzygies and secants

  • Gregory Burnham, Zvi Rosen, Jessica Sidman, Peter Vermeire
  • 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.
  • LMS Lecture Notes Series 417: Recent Advances in Algebraic Geometry A Volume in Honor of Rob Lazarsfeld’s 60th Birthday.
  • (arXiv | book)

Submitted Articles

  • Hyperplane Neural Codes and the Polar Complex

  • Vladimir Itskov, Alex Kunin, and Zvi Rosen
  • We establish several natural properties of non-degenerate hyperplane codes, in terms of the polar complex of the code, 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 the hyperplane codes follow from the shellability of the appropriate polar complex.
  • (arXiv)
  • Geometry of the sample frequency spectrum and the perils of demographic inference

  • Zvi Rosen*, Anand Bhaskar*, Sebastien Roch, Yun S. Song
  • The sample frequency spectrum (SFS) is a widely used summary statistic in population genetics, with strong dependence on the historical population demography. This paper uses algebraic and convex geometry to explain difficulties that arise in demographic inference, and to characterize the semialgebraic set of possible spectra.
  • (bioRxiv)
  • Convex Neural Codes in Dimension 1

  • Zvi Rosen, Yan X. Zhang
  • We study convex neural codes in dimension 1 (i.e. on a line or a circle). We use the theory of consecutive-ones matrices to obtain some structural and algorithmic results; we use generating functions to obtain enumerative results.
  • (arXiv)
  • Computing Algebraic Matroids

  • Zvi Rosen
  • We present algorithms for computing algebraic matroids using numerical algebra and symbolic computation. We use these to compute various examples.
  • (arXiv)
  • Algebraic Matroids with Graph Symmetry

  • Franz J. Király, Zvi Rosen, Louis Theran
  • We study algebraic matroids whose ground sets are endowed with graph symmetry. These results are 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.
  • (arXiv)