My interests include:
- Algebro-geometric aspects of mirror symmetry (e.g., theta functions, coherent sheaves, polyvector fields)
- Tropical, log, and open Gromov-Witten theory
- Cluster algebras, both quantum and classical
- Quiver representations, DT theory, and Hall algebras
- The interplay of everything listed above.
More detail: Mirror symmetry is a phenomenon first observed by string theorists which relates algebro-gemetric structures (like vector bundles on subvarieties) for one space to symplectic features (like Lagrangian subspaces and holomorphic disks) for another. This geometric data for each space can be represented in terms of some combinatorial data like tropical curves. I am interested in using this tropical data to develop structures on the algebraic side that are mirror to various structures on the symplectic side.
Cluster varieties are a special type of space constructed by gluing together many copies of algebraic tori (ℂ*)n via certain birational maps called mutations. These spaces were motivated by structures arising in the study of representation theory and Teichmüller theory, and particularly by a desire to understand certain canonical bases and positivity properties arising in the study of quantum groups.
The tropical approach to mirror symmetry has been particularly fruitful when applied to cluster varieties. Here, certain generating functions for the tropical data can be used to build the canonical "theta" bases for cluster algebras. Ben Davison and I recently accomplished this for quantum cluster varieties, using ideas from the DT theory of quivers to prove the conjectured strong positivity properties.
According to mirror symmetry, these theta bases should be determined by certain Gromov-Witten numbers (i.e., counts of holomorphic curves) in the mirror/Langlands dual cluster variety. I have proved this result (the Frobenius structure conjecture) for the classical theta functions using new results (developed with Helge Ruddat) on the correspondence between counts of tropical curves and counts of holomorphic curves. One of my current goals is to prove a refined version of this, expressing the quantum theta functions in terms of certain open Gromov-Witten invariants.
- 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.
- Classification of rank 2 cluster varieties,
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 15:Paper 042, 32, (2019).
- Cluster algebras are Cox rings,
manuscripta mathematica, 160(1-2):153--171, 2019.
- Theta bases are atomic,
Compositio Mathematica, 153(6):1217--1219, 2017.
- Tropical Theta Functions and Log Calabi-Yau Surfaces,
Selecta Mathematica, 22(3):1289--1335, 2016.
Periods in Partial Words: An Algorithm, (with F. Blanchet-Sadri and Gautam Sisodia)
Journal of Discrete Algorithms, 16:113--128, 2012.
- Strong positivity for quantum theta bases of quantum cluster algebras, with Ben Davison.
- Fano mirror periods from the Frobenius structure conjecture.
- Theta bases and log Gromov-Witten invariants of cluster varieties.
- Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves, with Helge Ruddat.
- Donaldson-Thomas invariants from tropical disks, with Man-Wai Cheung.
- Scattering diagrams, theta functions, and refined tropical curve counts.
- Slides from a Zoom talk I gave on Tropical multiplicities from polyvector fields and QFT for the Sheffield Algebraic Geometry Seminar on April 21, 2020.
- Notes for my mini-course Log geometry, tropical geometry, and mirror symmetry for cluster varieties, presented at the conference Valuations and birational geometry in Lille, France, May 2019.
- Notes and videos from three lectures I gave at the KIAS scientific workshop Cluster Algebras and Log GW Invariants in GS program in 2017.
- Descendant log GW invariants are tropical curve counts.
- Broken lines and theta functions.
- Theta functions and log GW invariants.
- Slides from my talk Tropical curve counting and canonical bases at the 2015 AMS Summer Institute in Algebraic Geometry.
- Some incomplete notes on Mirror symmetry and cluster algebras from a course I taught at QGM (Fall, 2014).
- Notes from my talk "Gross-Hacking-Keel I" at the MIT-RTG Mirror Symmetry Workshop in 2013, explaining the main construction of the Gross-Hacking-Keel paper Mirror symmetry for log Calabi-Yau surfaces I.
- Some very short introductory notes on GIT from a talk I gave at UT Austin's student geometry seminary in 2013.
and accompanying slides
from a talk I gave on compass and straightedge constructions for Saturday Morning Math Group (SMMG), a UT Austin program where graduate students and faculty memebers give lectures to elementary, middle, and high-school students.
At University of Utah:
At University of Aarhus (QGM):
At University of Texas at Austin (teaching assistant and grader positions):