My research generally lies in the realm of algebraic geometry, but aspects of my work also touch on ideas from algebraic combinatorics, representation theory, symplectic geometry, and mathematical physics.
I am especially interested in tools coming from mirror symmetry, a phenomenon first observed by string theorists which relates the algebro-geometric structure of one space to the symplectic structure of a "mirror" space. The Gross-Siebert program uses tropical geometry (a sort of piecewise-linear skeleton of geometry) to create a combinatorial framework for understanding the structures arising in mirror symmetry. Much of my work is based on applying these tools from the Gross-Siebert program towards problems concerning cluster algebras.
Cluster algebras are a class of combinatorially defined commutative algebras which admit many different local coordinate systems, called clusters, related to each other via certain birational maps called mutations. They were defined by Fomin-Zelevinsky with the goal of better understanding certain canonical bases and positivity phenomena observed by Lusztig in the context of quantum groups and representation theory. Work of Gross-Hacking-Keel-Kontsevich applied the Gross-Siebert program to construct canonical "theta bases" for cluster algebras, thus settling many important conjectures in cluster theory. Most of my research is centered around better understanding every aspect of these theta bases, incluing their general properties, quantization, tropicalization, and connections to Gromov-Witten theory (counting holomorphic curves) and quiver DT-theory (e.g., Euler characteristics of certain spaces of quiver representations).
Also see my pages on arXiv
and Google Scholar
- Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves (with Helge Ruddat)
International Mathematics Research Notices, 2021.
- Strong positivity for quantum theta bases of quantum cluster algebras (with Ben Davison)
Inventiones mathematicae, 2021.
- Scattering diagrams, theta functions, and refined tropical curve counts,
Journal of the London Mathematical Society, (to appear)
- Theta bases and log Gromov-Witten invariants of cluster varieties,
Transactions of the American Mathematical Society, 374(8), 2021.
- Donaldson-Thomas invariants from tropical disks, (with Man-Wai Cheung)
Selecta Mathematica, 26(57), 2020.
- 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.
- Slides from my Zoom talk Bracelet bases are theta bases for the Workshop on Cluster Algebras and Related Topics, hosted by the Morningside Center of Mathematics, CAS, August 2-6, 2021.
- Slides and video from my Zoom talk Quantum theta bases for quantum cluster algebras for University of Nottingham's Online Algebraic Geometry Seminar on May 5, 2021.
- Slides and video from my Zoom talk Quantum theta bases for Cluster Algebras 2020.
- 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 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.
Current teaching (Spring 2022):
Differential and Integral Calculus III [MATH 2934-004 and 007 (honors)]
Past courses at University of Oklahoma:
Fall 2021: Topics in Algebra: Toric varieties and related topics (MATH-6393-001). Lecture notes on ``Introduction to toric varieties and algebraic geometry'' available here.
Spring 2021: Calculus and Analytic Geometry IV (Math 2443 003,004)
Fall 2020: Calculus and Analytic Geometry II (Math 2423 001,003)
Here is a list of courses I have taught in the past at other universities:
At University of Utah:
Spring 2018: Calculus III (Math 2210-001).
Fall 2017: Foundations of Analysis II (Math 3220-001). Syllabus. Notes
Spring 2017: Calculus III (2210-006).
Fall 2016: Foundations of Analysis I (3210-001).
Spring 2016: Calculus III (2210-003).
Fall 2015: Calculus II (1220-004).
At University of Aarhus (QGM):
Fall 2014: Graduate course on mirror symmetry and cluster algebras. Here are some (incomplete) notes meant to accompany that course.
At University of Texas at Austin (teaching assistant and grader positions):
Spring 2014: Teaching assistant for Integral Calculus for Science (M 408S).
Fall 2013: Grading for Algebraic Structures (M 373K, 57500 and
57510) and for Curves and Surfaces (M 365G, 57475).
Summer 2012: Teaching assistant for Integral Calculus (M 408L).
Summer 2010: Grader for Linear Algebra and for Real Analysis.
Spring 2010 and Fall 2009: Supplemental instructor for Differential Calculus (M 408K).
Spring 2009 and Fall 2008: Teaching Assistant for Differential and Integral Calculus (M 408C).