About

Cynthia Vinzant is a faculty member in the Department of Mathematics at the University of Washington. She has been recognized with several awards, including the Michael and Sheila Held Prize in January 2025 and the Fellow of the American Mathematical Society in November 2023. Her research interests include the geometry of polynomials, networks, combinatorial optimization, and linear optimization. She has also received the Frontiers of Science Awards alongside Uhlmann. Vinzant teaches various courses related to discrete optimization, linear optimization, and special topics in mathematics. She is actively involved in seminars and research activities within her department and the broader mathematical community.

Research topics

• Mathematics
• Combinatorics

Selected publications

• The convex algebraic geometry of higher-rank numerical ranges

  Journal of Symbolic Computation · 2025-05-19 · 1 citations

  articleSenior author
  PublisherDOI

• Valuated Delta Matroids and Principal Minors of Hermitian matrices

  ArXiv.org · 2025-07-22

  preprintOpen accessSenior author

  In this paper we introduce valuated $Δ$-matroids, a natural generalization of two objects of study in matroid theory: valuated matroids and $Δ$-matroids. We show that these objects exhibit nice properties analogous to ordinary valuated matroids. We also show that these objects arise as the valuations of principal minors of a Hermitian matrix over a valued field, generalizing other forms of $Δ$-matroid representability.

  PublisherOA PDFDOI

• Moments, sums of squares, and tropicalization

  Journal of the London Mathematical Society · 2025-10-01

  articleOpen access

  Abstract We use tropicalization to study the duals to cones of nonnegative polynomials and sums of squares on a semialgebraic set . The truncated cones of moments of measures supported on the set are dual to nonnegative polynomials on , while “pseudomoments” are dual to sums of squares approximations to nonnegative polynomials. We provide explicit combinatorial descriptions of tropicalizations of the moment and pseudomoment cones, and demonstrate their usefulness in distinguishing between nonnegative polynomials and sums of squares. We give examples that show new limitations of sums of squares approximations of nonnegative polynomials. When the semialgebraic set is defined by binomial inequalites, its moment and pseudomoment cones are closed under Hadamard product. In this case, their tropicalizations are polyhedral cones that encode all binomial inequalities on the moment and pseudomoment cones.

  PublisherDOI

• Tropicalizing Principal Minors of Positive Definite Matrices

  arXiv (Cornell University) · 2024-10-15

  preprintOpen access

  We study the tropicalization of the image of the cone of positive definite matrices under the principal minors map. It is a polyhedral subset of the set of $M$-concave functions on the discrete $n$-dimensional cube. We show it coincides with the intersection of the affine tropical flag variety with the submodular cone. In particular, any cell in the regular subdivision of the cube induced by a point in this tropicalization can be subdivided into base polytopes of realizable matroids. We use this tropicalization as a guide to discover new algebraic inequalities among the principal minors of positive semidefinite matrices of a fixed size. We also extend our results to positive semidefinite matrices via taking closures in the tropical semifield $\mathbb{R}\cup\{-\infty\}$.

  PublisherOA PDFDOI

• Higher dimensional Fourier quasicrystals from Lee–Yang varieties

  Inventiones mathematicae · 2024-12-16 · 3 citations

  articleSenior author
  PublisherDOI

• Log-concave polynomials III: Mason’s ultra-log-concavity conjecture for independent sets of matroids

  Proceedings of the American Mathematical Society · 2024 · 23 citations

  Senior authorCorresponding
  • Combinatorics
  • Mathematics

  We give a self-contained proof of the strongest version of Mason’s conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.

  PublisherDOI

• Gap distributions of Fourier quasicrystals with integer weights via Lee–Yang polynomials

  Revista Matemática Iberoamericana · 2024-04-22 · 1 citations

  articleOpen accessSenior author

  Recent work of Kurasov and Sarnak provides a method for constructing one-dimensional Fourier quasicrystals (FQ) from the torus zero sets of a special class of multivariate polynomials called Lee–Yang polynomials. In particular, they provided a non-periodic FQ with unit coefficients and uniformly discrete support, answering an open question posed by Meyer. Their method was later shown to generate all one-dimensional Fourier quasicrystals with \mathbb{N} -valued coefficients ( \mathbb{N} -FQ). In this paper, we characterize which Lee–Yang polynomials give rise to non-periodic \mathbb{N} -FQs with unit coefficients and uniformly discrete support, and show that this property is generic among Lee–Yang polynomials. We also show that the infinite sequence of gaps between consecutive atoms of any \mathbb{N} -FQ has a well-defined distribution, which, under mild conditions, is absolutely continuous. This generalizes previously known results for the spectra of quantum graphs to arbitrary \N -FQs. Two extreme examples are presented: first, a sequence of \mathbb{N} -FQs whose gap distributions converge to a Poisson distribution. Second, a sequence of random Lee–Yang polynomials that results in random \mathbb{N} -FQs whose empirical gap distributions converge to that of a random unitary matrix (CUE).

  PublisherOA PDFDOI

• Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties

  arXiv (Cornell University) · 2024-07-15

  preprintOpen accessSenior author

  In this paper, we construct Fourier quasicrystals with unit masses in arbitrary dimensions. This generalizes a one-dimensional construction of Kurasov and Sarnak. To do this, we employ a class of complex algebraic varieties avoiding certain regions in $\mathbb{C}^n$, which generalize hypersurfaces defined by Lee-Yang polynomials. We show that these are Delone almost periodic sets that have at most finite intersection with every discrete periodic set.

  PublisherOA PDFDOI

• A semidefinite programming characterization of the Crawford number

  arXiv (Cornell University) · 2024-03-13

  preprintOpen accessSenior author

  We give a semidefinite programming characterization of the Crawford number. We show that the computation of the Crawford number within $\varepsilon$ precision is computable in polynomial time in the data and $|\log \varepsilon |$.

  PublisherOA PDFDOI

• The convex algebraic geometry of higher-rank numerical ranges

  arXiv (Cornell University) · 2024-10-29

  preprintOpen accessSenior author

  The higher-rank numerical range is a convex compact set generalizing the classical numerical range of a square complex matrix, first appearing in the study of quantum error correction. We will discuss some of the real algebraic and convex geometry of these sets, including a generalization of Kippenhahn's theorem, and describe an algorithm to explicitly calculate the higher-rank numerical range of a given matrix.

  PublisherOA PDFDOI

Recent grants

• Real algebraic and combinatorial structures in matrix spaces

  NSF · $150k · 2016–2020

• PostDoctoral Research Fellowship

  NSF · $150k · 2012–2016

• CAREER: Determinantal, hyperbolic, and log-concave polynomials in theory and applications

  NSF · $385k · 2021–2026

• CAREER: Determinantal, hyperbolic, and log-concave polynomials in theory and applications

  NSF · $153k · 2020–2021

Frequent coauthors

• Mario Kummer

  51 shared

• Grigoriy Blekherman

  Georgia Institute of Technology

  40 shared

• Cordian Riener

  UiT The Arctic University of Norway

  36 shared

• Markus Schweighofer

  University of Konstanz

  36 shared

• Nima Anari

  16 shared

• Shayan Oveis Gharan

  14 shared

• Daniel Plaumann

  11 shared

• Kuikui Liu

  Massachusetts Institute of Technology

  10 shared

Awards & honors

• Cynthia Vinzant receives Michael and Sheila Held Prize (Janu…
• Cynthia Vinzant named Fellow of the AMS (November 2, 2023)
• Uhlmann and Vinzant recipients of Frontiers of Science Award…