Papers

Here's a list of my papers with some informal abstracts.
Also see my pages on arXiv, Google Scholar, and ResearchGate.

Publsihed:

• Stability scattering diagrams and quiver coverings, with Qiyue Chen and Fan Qin.
  Advances in Mathematics, 2024.
  • There is a useful technique in cluster theory called "folding" where you essentially take the quotient of a quiver by a finite group action and relate the corresponding cluster algebras. Other papers (including my other work with Qin) have examined the behavior of cluster scattering diagrams under folding. In this paper we instead focus on the behavior of Bridgeland's stability scattering diagrams and prove that these often behave as expected under folding. We apply this to cluster algebras from surfaces and find that the "missing wall" for the cluster scattering diagram of the once-punctured torus is in fact present in the stability scattering diagram.
• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves, with Helge Ruddat.
  International Mathematics Research Notices, 2023.
  • We set out to find a localized formula for computing the multiplicities of tropical curves. We develop a "tropical quantum field theory" to do this in general. Then in genus 0 we develop a simpler formula based on the Gerstenhaber algebra of polyvector fields.
• Strong positivity for quantum theta bases of quantum cluster algebras, with Ben Davison.
  Inventiones mathematicae, 2021.
  • Gross-Hacking-Keel-Kontsevich constructed canonical bases of theta functions for cluster algebras and proved that they satisfy certain positivity properties. By combining their techniques with Bridgeland's approach to scattering diagrams and Davison-Meinhardt's results on DT-invariants, we prove the analogous positivity properties for quantum theta function bases of quantum cluster algebras.
• Scattering diagrams, theta functions, and refined tropical curve counts,
  Journal of the London Mathematical Society,
  • Gross-Pandharipande-Siebert related two-dimensional scattering diagrams to log Gromov-Witten invariants (i.e., counts of holomorphic curves) by (1) relating the scattering function to certain counts of tropical curves and then (2) relating these tropical curve counts to the log Gromov-Witten invariants. We carry out (1) in higher dimensions and for more general Lie algebras, and also for the corresponding theta functions. As an application, we prove a quantum version of Carl-Pumperla-Siebert's result on consistency of theta functions. We also prove the non-degeneracy of the trace-pairing for the Frobenius structure conjecture of Gross-Hacking-Keel.
• Theta bases and log Gromov-Witten invariants of cluster varieties,
  Transactions of the American Mathematical Society, 374(8), 2021.
  • The Frobenius structure conjecture of Gross-Hacking-Keel predicts that, given a log Calabi-Yau variety Y with maximal boundary D, certain genus-0 descendant log Gromov-Witten invariants of (Y,D) will determine a mirror algebra with a canonical basis of theta functions. We prove this conjecture for cluster varieties and find that the invariants are in fact naive counts of certain holomorphic curves.
• Donaldson-Thomas invariants from tropical disks, with Man-Wai Cheung.
  Selecta Mathematica, 26(57), 2020.
  • The definition of a scattering diagram involves a choice of lattice-graded Lie algebra. E.g., classical scattering diagrams use a Poisson torus algebra or a module of log derivations, while quantum scattering diagrams use a quantum torus algebra. In this paper we examine what happens when the Lie algebra is the Hall algebra of a quiver with potential, applying the techniques of my work "Scattering diagrams, theta functions, and refined tropical curve counts" to this setting.
• Descendant log Gromov-Witten invariants for toric varieties and tropical curves, with Helge Ruddat.
  Transactions of the American Mathematical Society, 373(2):1109--1152, 2020.
  • Mikhalkin and Nishinou-Siebert have previously related certain counts of tropical curves to corresponding counts of algebraic curves. We prove that tropical curve counts with certain "higher-valency" conditions correspond to log Gromov-Witten invariants (counts of algebraic curves) with corresponding psi-class conditions that serve to fix the locations of marked points in the domain curve.
• Classification of rank 2 cluster varieties,
  Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 15:Paper 042, 32, (2019).
  • We classify rank 2 cluster varieties (those for which the span of the rows of the exchange matrix is 2-dimensional) according to the deformation type of a generic fiber of their corresponding cluster Poisson varieties (as defined by Fock-Goncharov). Equivalently, we classify Looijenga interiors U, as studied by Gross-Hacking-Keel, up to deformation. In particular, we find that U is positive (essentially meaning affine) and either finite-type or non-acyclic (in the usual cluster sense) if and only if the inverse-monodromy of the tropicalization of U is one of Kodaira's monodromies, and we prove that the tropicalization uniquely determines the deformation-type U in these cases.
• Cluster algebras are Cox rings,
  manuscripta mathematica, 160(1-2):153--171, 2019.
  • Gross-Hacking-Keel showed that a cluster algebra with special coefficients A can be identified with the Cox ring (i.e., the direct sum of the global sections of all isomorphism classes of line bundles) of a corresponding fiber U of the cluster Poisson variety. We extend this to partial compactifications of U by allowing A to have certain non-invertible frozen variables.
• Theta bases are atomic,
  Compositio Mathematica, 153(6):1217--1219, 2017.
  • Fock and Goncharov conjectured that the indecomposable universally positive (i.e., atomic) elements of a cluster algebra should form a basis for the algebra. This was shown to be false by Lee-Li-Zelevinsky. However, we find that the theta bases of Gross-Hacking-Keel-Kontsevich do satisfy this conjecture for a slightly modified definition of universal positivity in which one replaces the positive atlas consisting of the clusters by a larger collection of (formal) local coordinate systems that we call the scattering atlas.
• Tropical Theta Functions and Log Calabi-Yau Surfaces,
  Selecta Mathematica, 22(3):1289--1335, 2016.
  • We generalize the standard combinatorial techniques of toric geometry to the study of log Calabi-Yau surfaces. The character and cocharacter lattices are replaced by certain integral linear manifolds as in Gross-Hacking-Keel, and monomials on toric varieties are replaced with the canonical theta functions which GHK defined using ideas from mirror symmetry. We describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions.
• Periods in Partial Words: An Algorithm, (with F. Blanchet-Sadri and Gautam Sisodia)
  Journal of Discrete Algorithms, 16:113--128, 2012.
  • This article is based on an REU I participated in during Summer 2007. A theorem of Fine and Wilf says that if a word is p-periodic and q-periodic and has length at least L=p+q-gcd(p,q), then it must also be gcd(p,q)-periodic. If the word is allowed to have h holes (unknown letters where the periodicity condition does not apply), then the conclusion still holds for a larger value of L. We give an algorithm for computing this L.

Preprints:

• Valuative independence and cluster theta reciprocity, with Man-Wai Cheung, Timothy Magee, and Greg Muller.
  • We prove that theta functions constructed from positive scattering diagrams satisfy valuative independence: \(\operatorname{val}_v(\sum_u c_u \vartheta_u)=\min_{c_u\neq 0} \operatorname{val}_v(\vartheta_u)\). As applications, we prove linear independence of theta functions with specialized coefficients and characterize when theta functions for cluster varieties are unchanged by the unfreezing of an index. This yields a general gluing result for theta functions from moduli of local systems on marked surfaces. We then prove that theta functions for cluster varieties satisfy a symmetry property called theta reciprocity: briefly, \(\operatorname{val}_v(\vartheta_u)=\operatorname{val}_u(\vartheta_v)\). One may apply valuative independence and theta reciprocity together to identify theta function bases for global sections of line bundles on partial compactifications of cluster varieties.
• Bracelets bases are theta bases, with Fan Qin.
  • The (quantum) skein algebra of a marked surface has a canonical collection of elements called the "bracelets basis," constructed geometrically by Fock-Goncharov and combinatorially by Musiker-Schiffler-Williams. We prove that this coincides with the (quantum) theta function basis of Gross-Hacking-Keel-Kontsevich (quantized in my work with Davison). Similarly for the corresponding cluster Poisson varieties. D. Thurston's conjecture on strong positivity of quantum bracelets follows as a corollary.
• Fano mirror periods from the Frobenius structure conjecture.
  • The Fano classification program proposed by Coates-Corti-Galkin-Golyshev-Kasprzyk is based on the mirror symmetry prediction that the regularized quantum period of a Fano should be equivalent to the classical period of its mirror Landau-Ginzburg potential. We prove that this mirror equivalence follows from versions of the Frobenius structure conjecture of Gross-Hacking-Keel.

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