My research interests are (1) algebraic geometry, including tropical geometry, Gromov-Witten theory, moduli spaces, and in general topics that are combinatorial, enumerative, or related to theoretical physics; and (2) critical education theory in postsecondary mathematics, including critical pedagogy, anti-oppressive education, rehumanization, queer theory, and critical race theory, all in the setting of postsecondary mathematics education.

Tropical Geometry

I'm interested in tropical geometry both as a subject which is interesting and important unto itself, and also as a tool through which one may study moduli spaces and other topics in classical (nontropical) geometry.

  • Chow Rings of Heavy/Light Hassett Spaces via Tropical Geometry. With Siddarth Kannan and Shiyue Li. Journal of Combinatorial Theory, Series A 178 (2021) 105348.

    In this project, we compute the Chow ring of Heavy/Light Hassett spaces in genus zero. This builds on the work of Cavalieri-Hampe-Markwig-Ranganathan, who show that the moduli space of weighted stable tropical curves have the structure of a balanced fan in (and only in) the heavy/light case. We exploit the resulting Bergman fan of a graphic matroid. The resulting Chow ring presentation mirrors and generalizes Keel's presentation for the moduli space of stable marked curves of genus zero. This is an example of tropical geometry in service of classical geometry, and the study of moduli.

  • On the landscape of random tropical polynomials. Senior thesis of Chris Hoyt.

    In the real Euclidean plane, there is a taxonomy of non-degenerate conics including the ellipse, parabola, and the hyperbola. In the complex projective plane, there is a unique non-degenerate conic (up to projective transformation). The situation is more rich in the tropical plane.Tropical curves are stratified by topological type, roughly corresponding to lattice subdivisions of a simplex. Given this landscape, what is the expected tropical curve, of a fixed degree? Are some topological types more common, or are they homogeneously distributed? In his Senior Thesis, Chris Hoyt answered this question in the case of degree 2 polynomials in 2 variables (the one variable case had been studied in low degree). The results are surprising; some topological types are indeed more common than others.

Critical Education Theory

I'm interested in critical education theory in postsecondary mathematics. There is a rich universe of scholarship in critical pedagogy, anti-oppressive education, rehumanization, queer theory, and critical race theory, however scholarship in these areas tends to focus on K12 education. There is a need for further investigation in the postsecondary mathematics setting.

  • Fiber Bundles and Intersectional Feminism. Preprint. (Submitted.)

    In this paper, we introduce intersectionality for a mathematical audience. We use a fiber bundle as a model for social structure. This appears as a chapter in the book below.

  • Book A Conversation on Professional Norms. Mathilde Gerbelli-Gauthier, Pamela E Harris, Mike Hill, Dagan Karp, and Emily Riehl, Editors. (Submitted.)

    This book originated as a conference proceedings for a workshop organized by Emily Riehl in September, 2019. It has since grown to include contributions from mathematicians who did not present, and did not attend, the workshop. As the title eludes, the goal was to articulate, investigate, and interrogate the professional norms in mathematics. How does mathematics function as an institution? What are our norms and practices? This volume explores these questions through a wide ranging assortment of articles.

  • Standards Based Grading and Equity in Postsecondary Mathematics Education. With Jenny Lee and Laura Palucki Blake.

    One of the central aims in critical pedagogy is to deconstruct the student/teacher dichotomy and shift some of the locus of control from the instructor to the students. How can one do so in a technical, demanding undergraduate mathematics course? In this project, we explore this question in the context of Linear Algebra. We conduct an experiment between two sections of Math 40 at Mudd. In the control section, assessment consists of homework and two exams (a midterm and final). In the experimental section, homework remains identical, but the exams are replaced by short assessment instruments (essentially quizzes). Whereas exams are fixed in time and representative of the material covered, the experimental quizzes may be taken at any time, and retaken without penalty, and they exhaustively cover all major topics in the course. The control and experimental sections are identical in all other aspects, including the same instructor. We find that students report being less stressed out in the experimental section. Students also view the classtime environment more positively. Other analyses are given and discussed.

Toric Symmetry in Gromov-Witten Theory

Gromov-Witten theory is both the mathematical underpinning of type IIA string theory, and the language of contemporary enumerative geometry. The field has a very rich canon of scholarship and much has been accomplished. However computation of GW invariants remains, in general, quite difficult. We remain ignorant of the full GW theory of spaces long studied and of basic interest in the subject (e.g. the quintic threefold).

The central idea in this series of projects is quite simple. For any toric variety, a symmetry of the fan corresponds to an isomorphism of varieties. Gromov-Witten invariants are indeed invariant under symplectomorphism, so they are certainly invariant under algebraic isomorphism. So the idea is to identify and exploit toric symmetries which correspond to nontrivial symmetries at the level of GW invariants.

  • Cremona symmetry in Gromov-Witten theory. Pro Mathematica Volume 29, Num. 57(2016) 129-149. With Amin Gholampour and Sam Payne.

    We establish the existence of a symmetry within the Gromov-Witten theory of CP^n and its blowup along points. The nature of this symmetry is encoded in the Cremona transform and its resolution, which lives on the toric variety of the permutohedron. This symmetry expresses some difficult to compute invariants in terms of others less difficult to compute. We focus on enumerative implications; in particular this technique yields a one line proof of the uniqueness of the rational normal curve. Our method involves a study of the toric geometry of the permutohedron, and degeneration of Gromov-Witten invariants.

  • Gromov-Witten Theory of P^1xP^1xP^1. With Dhruv Ranganathan. Journal of Pure and Applied Algebra, Volume 220, Issue 8. (2016) 3000-3009.

    We use elementary geometric techniques to exhibit an explicit equivalence between certain sectors of the Gromov-Witten theories of blowups of P^1xP^1xP^1 and P^3. In particular, we prove that the all genus, virtual dimension zero Gromov-Witten theory of the blowup of P^3 at points coincides with that of the blowup at points of P^1xP^1xP^1, for non-exceptional classes. We observe a toric symmetry of the Gromov-Witten theory of P^1xP^1xP^1 analogous and intimately related to Cremona symmetry of P^3. Enumerative applications are given.

  • Toric Symmetry of CP^3. With Dhruv Ranganathan, Paul Riggins, and Ursula Whitcher. Adv. Theor. Math. Phys 10 (2012) 1291-1314.

    We exhaustively analyze the toric symmetries of CP^3 and its toric blowups. Our motivation is to study toric symmetry as a computational technique in Gromov-Witten theory and Donaldson-Thomas theory. We identify all nontrivial toric symmetries. The induced nontrivial isomorphisms lift and provide new symmetries at the level of Gromov-Witten Theory and Donaldson-Thomas Theory. The polytopes of the toric varieties in question include the permutohedron, the cyclohedron, the associahedron, and in fact all graph associahedra, among others.

  • The local Gromov-Witten invariants of configurations of rational curves. With Chiu-Chu Melissa Liu and Marcos Marino. Geometry & Topology 10 (2006) 115-168.

    We compute the local Gromov-Witten invariants of certain configurations of rational curves in a Calabi-Yau threefold. These configurations are connected subcurves of the `minimal trivalent configuration', which is a particular tree of P^1's with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary Gromov-Witten invariants of a blowup of P^3 at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal Gromov--Witten invariants using the mathematical and physical theories of the topological vertex. In particular, we provide further evidence equating the vertex amplitudes derived from physical and mathematical theories of the topological vertex.

  • PhD Thesis, University of British Columbia, 2005. Under Jim Bryan. (pdf)

    We compute the local Gromov-Witten invariants of certain configurations of rational curves in a Calabi-Yau threefold. We first transform this from a problem involving local Gromov-Witten invariants to one involving global or ordinary invariants. We do so by expressing the local invariants of a configuration of curves in terms of ordinary Gromov-Witten invariants of a blowup of CP3 at points. The Gromov- Witten invariants of a blowup of CP3 along points have a symmetry, which arises from the geometry of the Cremona transformation, and transforms some difficult to compute invariants into others that are less difficult or already known. This symmetry is then used to compute the global invariants.

  • The closed topological vertex via the Cremona transform. With Jim Bryan. Journal of Algebraic Geometry , 14(3):529-542, 2005. (pdf)

    We compute the local Gromov-Witten invariants of the "closed vertex", that is, a configuration of three rational curves meeting in a single triple point in a Calabi-Yau threefold. The method is to express the local invariants of the vertex in terms of ordinary Gromov-Witten invariants of a certain blowup of CP^3 and then to compute those invariants via the geometry of the Cremona transformation.

Additional Projects
  • El Grado de Brower y el Teorema de Riemann-Roch. Memorias de los Grandes Maestros de la Matematica Columbiana. Vol. 2, 253-259. (2018) With Alfonso Castro.

    El propósito de este trabajo es relacionar el Grado de Brouwer con el Teorema de Riemann-Roch. El grado de Brouwer es una herramienta básica en el estudio de la solubilidad de ecuaciones no lineales mientras que el Teorema de Riemann Roch relaciona dimensión del conjunto de funciones meromorfas definidas en una superficie de Riemann con la topología de la misma y es el origen de lo que conoce como las teorías de índice. La principal observación es que los polos simples de funciones meromorfas dan lugar a puntos de índice los local −1 mientras que los ceros simples dan lugar a puntos de índice los local +1.

  • On a family of K3 surfaces with S_4 symmetry. With Jacob Lewis, Daniel Moore, Dmitri Skjorshammer and Ursula Whitcher. Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds, Fields Institute Communications, Volume67. R. Laza, M. Schutt, and N.Yui (Eds.). 2013.

    The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is the symmetric group on four elements. There are three pairs of three-dimensional reflexive polytopes with this symmetry group, up to isomorphism. We identify a natural one-parameter family of K3 surfaces corresponding to each of these pairs, show that the symmetric group on four elements acts symplectically on members of these families, and show that a general K3 surface in each family has Picard rank 19. The properties of two of these families have been analyzed in the literature using other methods. We compute the Picard-Fuchs equation for the third Picard rank 19 family by extending the Griffiths-Dwork technique for computing Picard-Fuchs equations to the case of semi-ample hypersurfaces in toric varieties. The holomorphic solutions to our Picard-Fuchs equation exhibit modularity properties known as "Mirror Moonshine"; we relate these properties to the geometric structure of our family.

  • An extension of a criterion for unimodality. With Jenny Alvarez, Miguel Amadis, George Boros, Victor H.Moll, and Leobardo Rosales. The Electronic Journal of Combinatorics , volume 8. (pdf)

    We prove that if P(x) is a polynomial with nonnegative nondecreasing coefficients and n is a positive integer, then P(x+n) is unimodal. Applications and open problems are presented.