My current projects all involve commutative algebra over Cox rings of projective varieties.
See my prelim and defense presentations for some topics and questions that lead me to this point.

Publications and preprints in Mathematics

• Splitting of vector bundles on toric varieties
  27 December 2024. arXiv:2412.19793 [math.AG]

The vanishing requirements which imply a Horrocks-type splitting criterion.

• Computing Direct Sum Decompositions
  With Devlin Mallory.
  In press at Journal of Symbolic Computation.
  27 December 2024. arXiv:2412.19799 [math.AC]

Decomposing a module is equivalent to block diagonalizing its presentation.

• A short resolution of the diagonal for smooth proj. toric varieties of Picard rank 2
  With Michael K. Brown.
  Algebra & Number Theory 18-10 (2024), 1923-1943.
  7 October 2024. arXiv:2208.00562 [math.AG] [math.AC]

The terms of a resolution of the diagonal for Hirzebruch surface of type a.

• Bounds on Multigraded Regularity
  With Juliette Bruce, Lauren Cranton Heller.
  Submitted.
  23 August 2022. arXiv:2208.11115 [math.AC]

Multigraded regularity of powers of two ideals on the Hirzebruch surface.

• Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces
  With Juliette Bruce, Lauren Cranton Heller.
  Submitted.
  20 October 2021. arXiv:2110.10705 [math.AC] [math.AG]

The terms in Beilinson’s spectral sequence corresponding to linearity and quaslinearity.

Mathematical Software

• Connection Matrices in Macaulay2
  With P. Görlach, J. Koefler, A.-L. Sattelberger, H. Schroeder, N. Weiss, F. Zaffalon.
  Submitted.
  2 April 2025. arXiv:2504.01362 [math.AG] [cs.SC]

Connection matrices of a $D$-ideal satisfy the integtability condition $\partial_i\bullet A_j - \partial_j\bullet A_i = [A_i, A_j]$.

• The Virtual Resolutions Package for Macaulay2
  With Ayah Almousa, Juliette Bruce, Michael C. Loper.
  Journal of Software for Algebra and Geometry, Volume 10 (2020)
  16 May 2019. arXiv:1905.07022 [math.AG] [math.AC]

A virtual resolution for 3 points in $\PP^1\times\PP^1$ as a subcomplex of the minimal free resolution.

• Computations over Local Rings in Macaulay2
  26 October 2017. arXiv:1710.09830 [math.AC]

Quantum Information

• What is the optimal way to prepare a Bell state using measurement and feedback?
  With Leigh S. Martin, K. Birgitta Whaley.
  Quantum Science and Technology, Volume 2, Number 4 (2017)
  2 April 2017. arXiv:1704.00332 [quant-ph]

Interactions of qubits in a remote entanglement generation protocol.

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