My research interests are (1) algebraic geometry, including tropical
geometry, GromovWitten 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,
antioppressive 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. https://arxiv.org/abs/1910.10883
In this project, we compute the Chow ring of Heavy/Light Hassett
spaces in genus zero. This builds on the work of CavalieriHampeMarkwigRanganathan, 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.
https://scholarship.claremont.edu/hmc_theses/114/
In the real Euclidean plane, there is a taxonomy of nondegenerate
conics including the ellipse, parabola, and the hyperbola. In the
complex projective plane, there is a unique nondegenerate 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,
antioppressive 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 GerbelliGauthier, 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 GromovWitten Theory
GromovWitten 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. GromovWitten 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 GromovWitten theory. Pro Mathematica
Volume 29, Num. 57(2016) 129149. With Amin Gholampour and
Sam Payne. https://arxiv.org/abs/1412.1516
We establish the existence of a symmetry within the GromovWitten
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
GromovWitten invariants.
 GromovWitten Theory of P^1xP^1xP^1. With Dhruv
Ranganathan. Journal of Pure and Applied Algebra, Volume 220,
Issue 8. (2016) 30003009. http://arxiv.org/abs/1201.4414
We use elementary geometric techniques to exhibit an explicit
equivalence between certain sectors of the GromovWitten theories of
blowups of P^1xP^1xP^1 and P^3. In particular, we prove that the all
genus, virtual dimension zero GromovWitten theory of the blowup of
P^3 at points coincides with that of the blowup at points of
P^1xP^1xP^1, for nonexceptional classes. We observe a toric symmetry
of the GromovWitten 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) 12911314. arxiv.org/1109.5157.
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 GromovWitten theory and DonaldsonThomas theory. We
identify all nontrivial toric symmetries. The induced nontrivial
isomorphisms lift and provide new symmetries at the level of
GromovWitten Theory and DonaldsonThomas 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 GromovWitten invariants of configurations of rational
curves. With
ChiuChu Melissa Liu and Marcos Marino.
Geometry & Topology 10 (2006) 115168.
We compute the local GromovWitten invariants of certain
configurations of rational curves in a CalabiYau 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 GromovWitten 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
GromovWitten 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 GromovWitten invariants of certain
configurations of rational curves in a CalabiYau threefold. We first
transform this from a problem involving local GromovWitten 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
GromovWitten 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):529542, 2005. (pdf)
We compute the local GromovWitten invariants of the "closed
vertex", that is, a configuration of three rational curves meeting in
a single triple point in a CalabiYau threefold. The method is to
express the local invariants of the vertex in terms of ordinary
GromovWitten 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 RiemannRoch. Memorias de los
Grandes Maestros de la Matematica Columbiana. Vol. 2, 253259. (2018) With
Alfonso Castro.
El propósito de este trabajo es relacionar el Grado de Brouwer
con el Teorema de
RiemannRoch. 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 CalabiYau
Threefolds, Fields Institute Communications, Volume67. R. Laza,
M. Schutt, and N.Yui (Eds.). 2013. arxiv.org/1103.1892.
The largest group which occurs as the rotational symmetries of a
threedimensional reflexive polytope is the symmetric group on four
elements. There are three pairs of threedimensional reflexive
polytopes with this symmetry group, up to isomorphism. We identify a
natural oneparameter 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 PicardFuchs equation for the third Picard
rank 19 family by extending the GriffithsDwork technique for
computing PicardFuchs equations to the case of semiample
hypersurfaces in toric varieties. The holomorphic solutions to our
PicardFuchs 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.
