Research
Science Database
My research is centered around enumerative geometry, Gromov-Witten theory, and related aspects of mirror symmetry.
Recorded Talks
Recorded Transmissions
- (General mathematics audience) What is Mirror Symmetry?, CSU "Who is...?" Seminar, Dec. 2020.
- Towards a Mirror theorem for GLSMs, Columbia University, Oct. 2021.
- Genus-zero Gromov-Witten theory under extremal transitions, Harvard CMSA, Mar. 2020.
- Integral transforms and quantum correspondences, BIRS, Nov. 2020 and The Fields Institute, Sep. 2019.
- Matrix factorizations in Gromov-Witten theory, MSRI, Mar. 2018.
Papers
Research Archives
submitted.
We construct a Chern character from the Grothendieck group of the category of matrix factorizations of w to the critical cohomology of w, and show that it factors through a certain topological K-theory group. We prove a Grothendieck--Riemann--Roch theorem with respect to this Chern character, and verify several functorial properties.
With Yefeng Shen, submitted.
For a smooth projective variety X, the quantum spectrum of X is the set of eigenvalues of quantum multiplication by c_1(X). This depends on a given specialization of the Novikov parameter q of quantum multiplication. We show that for X a projective bundle, blow-up, or flip, the quantum spectrum, when restricted to exceptional curve classes, is compatible with a natural semi-orthogonal decomposition of the derived category of X.
With Nathan Priddis and Yaoxiong Wen, SIGMA (2025).
We prove the crepant transformation conjecture for relative Grassmann flops over a smooth base B. We show that the I-functions of the respective GIT quotients are related by analytic continuation and a symplectic transformation. We verify that the symplectic transformation is compatible with Iritani's integral structure, that is, that it is induced by a Fourier-Mukai transform in K-theory.
With Nathan Priddis and Yaoxiong Wen, submitted.
We study the genus-zero Gromov-Witten theory of two natural resolutions of determinantal varieties, termed the PAX and PAXY models. We realize each resolution as lying in a quiver bundle, and show that the respective quiver bundles are related by a quiver mutation. We prove that generating functions of genus-zero Gromov-Witten invariants for the two resolutions are related by a specific cluster change of variables. Along the way, we obtain a quantum Thom-Porteous formula for determinantal varieties and prove a Seiberg-like duality statement for certain quiver bundles.
Accepted to KIAS Springer Series in Mathematics: Categorical and Enumerative Aspects of Mirror Symmetry.
Kleiman's criterion states that, for X a projective scheme, a divisor D is ample if and only if it pairs positively with every element of the closure of the cone of curves. In other words, the cone of ample divisors in N^1(X) is the interior of the nef cone. In this paper we present an analogous statement for a variety X acted on by a reductive group G, together with a choice of G-linearization L -> X. In this new context, the ample cone of X is replaced by a cell in the variation of GIT decomposition of the G-ample cone, and curves in X are replaced by quasimaps to [X/G].
Accepted to Compositio Mathematica.
A gauged linear sigma model (GLSM) consists roughly of a group G, a vector space V on which G acts, a regular function on V which is invariant under the group action, and a choice of character theta of G. These data together define a GIT quotient Y = [V//G] and a function on Y. GLSMs provide a broad setting in which it is possible to define an enumerative curve counting theory, and simultaneously generalize many other such theories. In this paper we provide some of the first explicit computations of GLSM invariants. We show that generating functions of GLSM invariants arise as derivatives of generating functions of Gromov-Witten invariants on Y.
With Levi Heath, European Journal of Mathematics (2022).
Let Z be a smooth subvariety of a projective variety X, defined by the vanishing of a section of a vector bundle E -> X. Quantum Serre duality refers to a relationship between the Gromov-Witten theory of Z and of the vector bundle E^v. In this paper we show that both the statement and proof of this correspondence can be dramatically simplified if one considers quasimap invariants rather than Gromov-Witten invariants. This approach yields new instances of the correspondence, including for non-convex orbifolds.
With Rongxiao Mi, Mathematische Annalen (2022).
Given a singular variety Z_sing, there are two ways in which one could recover a smooth variety. One could smooth Z_sing to obtain Z_sm, or one could resolve Z_sing to obtain Z_res. Although the two smooth varieties Z_sm and Z_res are different (even topologically), one might hope to find connections between them. In this paper we show that in many cases, their Gromov-Witten invariants are closely related.
With Ionut Ciocan-Fontanine, David Favero, Jeremy Guere, and Bumsig Kim, Memoirs of the American Mathematical Society (2023).
We define enumerative invariants for a class of Gauged Linear Sigma Models via the derived category of factorizations, simultaneously generalizing Gromov-Witten invariants of complete intersections and FJRW invariants of singularities. We show that these invariants agree with previously defined invariants in these special cases.
In: Singularities, Mirror Symmetry, and the Gauged Linear Sigma Model. Contemporary Mathematics (2021).
This is an expository article based on a series of 5 lectures given at the Conference on Crossing the Walls in Enumerative Geometry at Snowbird, Utah in 2018. The notes give an introduction to the notion of a virtual fundamental class, and explain some of the main ideas behind the paper "Fundamental Factorization of a GLSM" above.
Advances in Mathematics (2020).
We show that a variety of correspondences in genus zero Gromov-Witten theory are compatible with integral transforms between appropriate derived categories. As a corollary we prove a strong form of the Landau-Ginzburg/Calabi-Yau correspondence.
Ann. Inst. Fourier (2021).
Given a non-compact manifold Y, we define the "narrow cohomology" of Y as a natural subspace of the usual cohomology. This subspace has a well-defined and non-degenerate Poincare pairing, which allows us to define a so-called "narrow quantum D-module." This results in a new formulation of quantum Serre duality. We show that the genus zero Gromov-Witten theory of a projective hypersurface is isomorphic in a certain sense to the narrow Gromov-Witten theory of the total space of an associated vector bundle.
With Pedro Acosta, Int. Math. Res. Not. (2019).
We study the relationship between the Gromov-Witten invariants of birational spaces. Given two toric orbifolds related by variation of GIT, we prove that specific generating functions of their genus zero Gromov-Witten invariants are identified after asymptotic expansion. This extends the crepant transformation conjecture to general toric birational transformations.
With Pedro Acosta, Algebraic Geometry (2018).
We establish a genus zero correspondence between the equivariant Gromov-Witten theory of an affine quotient and its blowup at the origin. The relationship generalizes the crepant transformation conjecture of Coates-Iritani-Tseng and Coates-Ruan to the discrepant (non-crepant) setting. We apply this result to prove LG/Fano and LG/general type correspondences for hypersurfaces.
With Y.-P. Lee and Nathan Priddis, Ann. Scient. Ec. Norm. Sup. (2016).
We establish a new relationship between twisted FJRW theory and the local Gromov-Witten theory of affine quotients. As a consequence we show that the Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence is implied by the crepant transformation conjecture for Fermat type polynomials. We use this to then prove the LG/CY correspondence in these cases.
With Nathan Priddis, Ann. Inst. Fourier (2016).
We give a correspondence between the orbifold Gromov-Witten theory of the mirror quintic and the FJRW theory of the corresponding Landau-Ginzburg theory. Mirror symmetry plays an important role in the proof.
With Emily Clader and Nathan Priddis, chapter in B-Model Gromov-Witten Theory (2018).
An expository article on quantization as it relates to Gromov-Witten theories, with an emphasis on computing explicit formulas.
Comm. Math. Phys. (2014).
Given a Calabi-Yau hypersurface in a quotient of weighted projective space, we prove that different mirrors which arise from BHK mirror symmetry are birational. This answers a question of Chiodo-Ruan.
With Y.-P. Lee, Geom. Topol. (2014).
We show that mirror symmetry for the quintic three-fold is in fact symmetric by demonstrating a correspondence between the B-model of the quintic and the A-model of the mirror quintic. This involves calculating orbifold Gromov-Witten invariants of the mirror quintic.
Other Writing
Archival Documents
- Enumerative Geometry Notes. These expository notes were written for a two week summer workshop at the University of Costa Rica in 2019.