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. The three projects I have been most involved in recently are:
 Combinatorial neural coding,
 Sample frequency spectra in population genetics,
 Algebraic matroids in applications (specifically, matrix completion and rigidity theory.)
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

On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force
 Nickolas Arustamyan, Christopher Cox, Erik Lundberg, Sean Perry, and Zvi Rosen
 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 on
those numbers.
 Journal of Mathematical Physics 62 (11)
 (arXiv journal)

Sparse moments of univariate step functions and allele frequency spectra
 Zvi Rosen, Georgy Scholten, and Cynthia Vinzant
 We prove sharp bounds on the number of pieces a piecewiseconstant 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.
 Vietnam Journal of Mathematics 50 (2), 523544
 (arXiv journal)

Algebraic Matroids in Action
 Zvi Rosen, Jessica Sidman, Louis Theran
 We give a selfcontained introduction to algebraic matroids together with examples highlighting their potential application.
The American Mathematical Monthly 127 (3), 199216.
 Winner of the 2021 Merten M. Hasse Prize from the MAA.

 (arXiv  journal)

Hyperplane Neural Codes and the Polar Complex
 Vladimir Itskov, Alex Kunin, and Zvi Rosen
 We establish several natural properties of nondegenerate 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 nondegenerate
hyperplane code is shellable and show that all currently known properties of
the hyperplane codes follow from the shellability of the appropriate polar complex.
 Topological Data Analysis 343369, 2020. (Book Chapter)
 (arXiv  book)

Algebraic signatures of convex and nonconvex codes
 Carina Curto, Elizabeth Gross, Jack Jeffries,
Katherine Morrison,
Zvi Rosen, Anne Shiu, and Nora Youngs
 Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are nonconvex. We also provide algebraic signatures for some families of codes that are convex, including the class of intersectioncomplete codes.
Journal of Pure and Applied Algebra, 223(9), 39193940, 2019.

 (arXiv  journal)

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.
 Genetics 210(2). 665682. 2018. (Selected as Highlight)
 (bioRxiv 
journal)

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.
The 50th Anniversary of GrĂ¶bner Bases, 171205, 2018.

 (arXiv  book)

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), 222238, 2017.
 (arXiv  journal)

The geometry of rankone tensor completion
 Thomas Kahle, Kaie Kubjas, Mario Kummer, Zvi Rosen
 The semialgebraic and algebraic geometry of projections of rankone tensors to some of their coordinates is studied, giving insight into the problem of rankone completion of partial tensors.
 SIAM Journal on Applied Algebra and Geometry, 1(1), 200221, 2017
 (arXiv  journal)

Matrix Completion for the Independence Model
 Kaie Kubjas, Zvi Rosen
 We investigate the problem of completing partial matrices to
rankone 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), 121, 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, 2151, 2016.
 (arXiv  journal)

Parameterfree 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; nonparametric tools including algebraic matroids are employed.
 Proceedings of the National Academy of Sciences, 112(9), 26522657, 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 Lazarsfeldâ€™s 60th Birthday.
 (arXiv  book)
Submitted Articles

Oriented Matroids and Combinatorial Neural Codes.
 Alexander Kunin, Caitlin Lienkaemper, and Zvi Rosen
 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 NPhard.
 (arXiv)

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 consecutiveones 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, lowrank matrix completion and determinantal varieties. We define and compute the circuit polynomials associated to the circuits of these matroids.
 (arXiv)