Writing
Book:
I am currently writing a book, together with Dusty Ross, called Algebra and Geometry. It is intended as a textbook for a first course in algebraic geometry geared specifically at students who have been exposed to but not yet completely mastered such abstract algebra concepts as rings, fields, and quotients. The philosophy of the book is that by narrowing one's focus to the specific setting of polynomial rings, algebraic geometry can be used to reinforce comprehension of abstract algebra and to motivate the introduction of new algebraic notions, rather than assuming familiarity with those notions at the outset.
A preview of the book is available at the link above and will be updated periodically as we write. Please let us know if you use the book for a course or independent study, especially if you have feedback or spot any mistakes.
Publications and Preprints:
On the geometry and combinatorics of curves with cyclic action:
Multimatroids and rational curves with cyclic action, with Chiara Damiolini, Chris Eur, Daoji Huang, and Shiyue Li.
Wonderful compactifications and rational curves with cyclic action, with Chiara Damiolini, Shiyue Li, and Rohini Ramadas. To appear in Forum of Mathematics, Sigma.
Permutohedral complexes and rational curves with cyclic action, with Chiara Damiolini, Daoji Huang, Shiyue Li, and Rohini Ramadas. To appear in Manuscripta Mathematica.
On the moduli space of curves and its tautological ring:
Holomorphic forms and non-tautological cycles on moduli spaces of curves, with Veronica Arena, Samir Canning, Richard Haburcak, Amy Q. Li, Siao Chi Mok, and Carolina Tamborini.
Boundary complexes of moduli spaces of curves in higher genus, with Dante Luber and Kyla Quillin. In Proceedings of the American Mathematical Society, 2022. (This paper represents the combined work of Dante Luber and Kyla Quillin in their Master's theses.)
Topological recursion relations via Pixton's formula, with Felix Janda, Xin Wang, and Dmitry Zakharov. To appear in Michigan Mathematical Journal.
Powers of the theta divisor and vanishing in the tautological ring, with Samuel Grushevsky, Felix Janda, and Dmitry Zakharov. In International Mathematics Research Notices, 2017.
Pixton's double ramification cycle relations, with Felix Janda. In Geometry & Topology, 2018.
Relations on Mbar_{g,n} via orbifold stable maps. In Proceedings of the American Mathematical Society, 2017.
On Gromov-Witten theory, Landau-Ginzburg theory, and wall-crossing in both settings:
Wall-crossing in genus-zero hybrid theory, with Dustin Ross. In Advances in Geometry, 2021.
Higher-genus wall-crossing in the gauged linear sigma model, with Felix Janda and Yongbin Ruan. In Duke Mathematical Journal, 2021.
Higher-genus quasimap wall-crossing via localization, with Felix Janda and Yongbin Ruan. To appear in Algebraic Geometry.
Sigma models and phase transitions for complete intersections, with Dustin Ross. In International Mathematics Research Notices, 2017.
Landau-Ginzburg/Calabi-Yau correspondence for the complete intersections X_{3,3} and X_{2,2,2,2}. In Advances in Mathematics, 2017. (A more detailed version of this work appears in my Ph.D. Thesis.)
On r-spin theory for curves with boundary:
Open r-spin theory III: A prediction for higher genus, with Alexandr Buryak and Ran Tessler. To appear in Journal of Geometry and Physics.
Open r-spin theory II: The analogue of Witten's conjecture for r-spin disks, with Alexandr Buryak and Ran Tessler. To appear in Journal of Differential Geometry.
Open r-spin theory I: Foundations, with Alexandr Buryak and Ran Tessler. In International Mathematics Research Notices, 2021.
Closed extended r-spin theory and the Gelfand-Dickey wave function, with Alexandr Buryak and Ran Tessler. In Journal of Geometry and Physics, 2018.
Expository writing on Gromov-Witten theory and related topics:
Introduction to the gauged linear sigma model. To appear in Singularities, Mirror Symmetry, and the Gauged Linear Sigma Model, 2021.
Gromov-Witten Theory: From curve counts to string theory. In Surveys on Recent Developments in Algebraic Geometry, 2017.
Mirror Symmetry Constructions, with Yongbin Ruan. In B-Model Gromov-Witten Theory, 2018.
Geometric Quantization with Applications to Gromov-Witten Theory, with Nathan Priddis and Mark Shoemaker. In B-Model Gromov-Witten Theory, 2018.
Writing for a broader audience:
Why Twelve Tones? The Mathematics of Musical Tuning. In The Mathematical Intelligencer, 2018. (Also to appear on The Mathematical Intelligencer's Scientific American blog.)
What If? Mathematics, Creative Writing, and Play. In Journal of Humanistic Mathematics, 2016.
Infinite recess, with Daniel Reck. In 826 National's STEM to Story: Enthralling and Effective Lesson Plans for Grades 5 - 8, 2015.
Did chaos cause mayhem in Jurassic Park?. In Plus Magazine, 2014.
Other things:
Inverse limits of finite topological spaces (undergraduate thesis completed under the guidance of Robert Lipshitz). In Homotopy, Homology, and Applications, 2009. (These results were used, and significantly extended, in the Ph.D. thesis of Matthew Thibault at the University of Chicago. He also pointed out and corrected a mistake in the proof of Lemma 3.1; see Remark 2.4.17 of his work.)
Lacunarity of certain partition-theoretic generating functions (REU project completed under the guidance of Ken Ono), with Y. Kemper and M. Wage. In Proceedings of the American Mathematical Society, 2009.
Less formal notes:
The following are lectures notes from several talks I have given, as well as notes I compiled on a few other topics.
Wednesday Lecture Series, ETH:
Orbifolds and their cohomology. Notes from part one of a three-part lecture series at ETH in Fall 2014. This lecture covers the basics of orbifolds and Chen-Ruan cohomology, and is based mainly on Adem, Leida, and Ruan's Orbifolds and Stringy Topology.
Introduction to the Landau-Ginzburg Model. Part two of the series, covering the definition of FJRW theory.
The Landau-Ginzburg/Calabi-Yau correspondence. Part three of the series, on the idea of the LG/CY correspondence for hypersurfaces and its generalization to complete intersections.
The secondary fan. Lecture notes for a guest lecture in Yongbin Ruan's course on Mirror Symmetry in Fall 2011, covering the definition of the secondary fan and its relationship to Batyrev-Borisov's toric mirror symmetry. The main references are Cox, Little, and Schenck's Toric Varieties and Cox and Katz's Mirror Symmetry.
Mini-course on moduli spaces. Lecture notes for a four-session mini-course for graduate students taught in Summer 2011, based mainly on Kock and Vainsencher's An Invitation to Quantum Cohomology, with help from Renzo Cavalieri's notes for his VIGRE mini-course Introduction to the moduli space of curves.
Notes on localization. Notes I wrote for myself while learning, based mainly on Graber and Pandharipande's Localization of virtual cycles, with some help from Hori et al's Mirror Symmetry, Lotte Hollands's master's thesis Counting curves in topological string theory, and other sources.
Notes on Chen-Ruan cohomology. More notes I wrote for myself while learning, based on the material in Adem, Leida, and Ruan's Orbifolds and Stringy Topology.
Hilbert polynomials and the degree of a projective variety. Final paper for the Algebraic Geometry course taught by Bill Fulton at the University of Michigan in Fall 2010.