My current projects mostly 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.

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Publications in Mathematics

• Characterizing Multigraded Regularity and Virtual Resolutions on $\PP^{n_1}\times\cdots\times\PP^{n_r}$
  With Juliette Bruce and Lauren Cranton Heller.
  Advances in Mathematics, Volume 500 (2026) 111049.
  27 May 2026. arXiv:2110.10705 [math.AC] [math.AG]

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

• Computing Direct Sum Decompositions
  With Devlin Mallory.
  Journal of Symbolic Computation, Volume 133 (2026) 102486.
  11 August 2025. 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.
  23 October 2024. arXiv:2208.00562 [math.AG] [math.AC]

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

Preprints in Mathematics

• Seshadri Regions and the Asymptotic Shape of Multigraded Regularity
  With Juliette Bruce, Lauren Cranton Heller, Alexandra Seceleanu.
  Submitted.
  4 December 2025. arXiv:2512.05289 [math.AG] [math.AC]

The Seshadri region packages the classical Seshadri constants with respect to every line bundle.

• Computing Global Ext for Complexes
  With Michael K. Brown, Souvik Dey, Guanyu Li.
  Submitted.
  29 September 2025. arXiv:2509.25103 [math.AG] [math.AC]

The Hirzebruch surface of type 2 has a full strong exceptional collection of complexes,
constructed via a spherical twist functor, which are not quasi-isomorphic to any sheaves.

• Splitting of Vector Bundles on Toric Varieties
  Submitted.
  27 December 2024. arXiv:2412.19793 [math.AG]

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

• Bounds on Multigraded Regularity
  With Juliette Bruce, Lauren Cranton Heller.
  Submitted.
  28 February 2025. arXiv:2208.11115 [math.AC]

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

Mathematical Software

• Connection Matrices in Macaulay2
  With P. Görlach, J. Koefler, A.-L. Sattelberger, H. Schroeder, N. Weiss, F. Zaffalon.
  Journal of Software for Algebra and Geometry, Volume 16 (2026).
  25 May 2026. 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).
  19 May 2020. 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]

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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.