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My research lies at the intersection of algebraic geometry with discrete mathematics, in particular I study spaces with rich combinatorial structure such as toric varieties and closely related varieties, namely those admitting toric degenerations. My research roughly splits into two parts: 1) fundamental research regarding the construction and comparison of toric degenerations, and 2) applications of toric degenerations.
The tools I use in 1) include algebraic combinatorics, such as cluster algebras, computational algebra such as Gröbner bases and discrete geometry such as tropical geometry, and representation theory. The applications beyond algebraic geometry that I investigate are in theoretical and fundamental physics, such as mirror symmetry and scattering amplitudes, and symplectic geometry specifically moment polytopes and integrable systems.
Publications
Binary geometries from pellytopes, with Máté L. Telek and Hannah Tillmann-Morris, arXiv:2410.08002 [math.AG], Special volume on Positive Geometry, Le Matematiche 80 (1) (2025), Link, arxiv
Tropical initial degeneration for systems of algebraic differential equations, with Sebastian Falkensteiner, Cristhian Garay-López, Marc Paul Noordman, arXiv:2309.10761 [math.AG], Bol. Soc. Mat. Mex. (3) 31 (2025), no. 1, Paper No. 15, 26 pp.
Newton-Okounkov bodies and minimal models for cluster varieties, with Man-Wai Cheung, Timothy Magee, and Alfredo Nájera Chávez, Adv. Math. 447 (2024), Paper No. 109680, 72 pp., arxiv:2305.04903[math.AG]
Following Schubert varieties under Feigin's degeneration of the flag variety, with Martina Lanini, Journal of Algebraic Combinatorics, April 2024, DOI:10.1007/s10801-024-01320-3, Link, arXiv:1802.04320[math.RT], Computational data
Adjacency for scattering amplitudes from the Gröbner fan, with James Drummond and Ross Glew, open access Journal of High Energy Physics, DOI: 10.1007/JHEP11(2023)002, Link, arXiv:2212.08931[hep-th]
Families of Gröbner degenerations, Grassmannians and universal cluster algebras, with Fatemeh Mohammadi and Alfredo Nájera Chávez, SIGMA 17 (2021), 46 pages, DOI:10.3842/SIGMA.2021.059 Link, arXiv:2007.14972[math.AG], Computational data
Birational sequences and the tropical Grassmannian, Journal of Algebra, 10.1016/j.jalgebra.2021.04.028, November 2021, Link, arXiv:1903.12106[math.RT]
Full rank valuations and toric initial ideals, International Mathematics Research Notices, Volume 2021, Issue 10, May 2021, 7715-7763, /doi.org/10.1093/imrn/rnaa071, Link, arxiv:1903.11068[math.AG]
Toric degenerations of cluster varieties and cluster duality, with Bosco Frías-Medina, Timothy Magee, Alfredo Nájera Chávez, Compositio Mathematica, 156(10), 2149-2206. DOI:10.1112/S0010437X2000740X, 2020, Link, arxiv:1809.08369[math.AG]
String cone and Superpotential combinatorics for flag and Schubert varieties in type A, with Ghislain Fourier, J. Combin. Theory Ser. A., Volume 167, October 2019, pp 213-256. https://doi.org/10.1016/j.jcta.2019.04.006, Link, arXiv:1611.06504[math.RT]
Toric degenerations of Gr(2,n) and Gr(3,6) via plabic graphs, with Xin Fang, Ghislain Fourier, Milena Hering, Martina Lanini, Ann. Comb. (2018) Volume 22, Issue 3, pp 491-512, Link, arXiv:1612.03838[math.CO], Computational data
Computing toric degenerations of flag varieties, with Sara Lamboglia, Kalina Mincheva, Fatemeh Mohammadi, Combinatorial Algebraic Geometry, Smith G., Sturmfels B. (eds), Fields Institute Communications, pp.247-281, Vol 80. Springer, New York, NY, November 2017, Link, arXiv:1702.05480[math.AG], Computational data
The degree of the Hilbert-Poincaré polynomial of PBW-graded modules, with Teodor Backhaus, Christian Desczyk, Ghislain Fourier, Comptes Rendus Mathematique, Volume 352, Issue 12, December 2014, 959-963, Link, ariv:1408.0901[math.RT]
Preprints
Singularities of massless scattering and cluster algebras, with James Drummond, Ross Glew, Ömer Gürdogan and Rowan Wright, arXiv:2507.01015[hep-th]
Unveiling Crystal Embeddings: New Perspectives on String Polytopes and Atomic Decompositions, with Jacinta Torres, arXiv:2505.22127[math.RT]
Cluster structures on spinor helicity and momentum twistor varieties, with Jian-Rong Li, arXiv:2408.14956[math.AG]
Infinite matroids in tropical differential algebra, with F. Aroca, S. Falkensteiner, C. Garay Lopez, L. R. Gonzalez-Ramirez, and C. V. Valencia Negrete, arxiv:2305.04784[math.AG]
Maps to toric varieties and toric degenerations, with Takuya Murata, arXiv:2210.13137[math.AG]
Tropical totally positive cluster varieties, arXiv:2208.01723[math.AG]
Other
A survey on toric degenerations of projective varieties, accepted for publication in the Proceedings of the Nottingham Algebraic Geometry seminar, arXiv:2301.02545[math.AG]
Mini-Workshop: Superpotentials in Algebra and Geometry, with Eduardo González, Konstanze Rietsch and Lauren Williams, MFO Report No. 10/2020 DOI: 10.4171/OWR/2020/10
Toric degenerations of tropical Grassmannians, with Megumi Harada and Fatemeh Mohammadi, Oberwolfach Workshop: Toric Geometry, MFO Report No. 43/2019, DOI:10.4171/OWR/2019/43
Positive initial ideals and the FFLV polytope, Oberwolfach Mini-Workshop: Degenerations Techniques in Representation Theory, MFO Report No. 46/2019, DOI: 10.4171/OWR/2019/46
Toric degenerations: a bridge between representation theory, tropical geometry and cluster algebras, PhD thesis, University of Cologne, July 2018, Link, arXiv:1806.02090[math.AG]
Birational sequences and Demazure characters, Oberwolfach Mini-Workshop: PBW Structures in Representation Theory, MFO Report No. 13/2016, DOI: 10.4171/OWR/2016/13
Collaborators
I have had the pleasure to collaborate with the following people:
Fuensanta Aroca, Teodor Backhaus, Mandy Cheung, Christian Desczyk, James Drummond, Sebastian Falkensteiner, Xin Fang, Bosco Frías-Medina, Ghislain Fourier, Cristhian Garay-López, Ross Glew, Laura R. Gonzalez-Ramirez, Ömer Gürdogan, Milena Hering, Sara Lamboglia, Timothy Magee, Martina Lanini, Jian-Rong Li, Kalina Mincheva, Fatemeh Mohammadi, Takuya Murata, Alfredo Nájera Chávez, Marc-Paul Noordmann, Máté L. Telek, Hannah Tillmann-Morris, Jacinta Torres, Carla V. Valencia-Negrete, Rowan Wright.
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