Research
Publications
Multiplicative preprojective algebras are 2-Calabi-Yau (joint with Travis Schedler) in Algebra and Number Theory
Exceptional collections for mirrors of invertible polynomials (joint with David Favero and Tyler Kelly) in Mathematische Zeitschrift
Multiplicative preprojective algebras of Dynkin quivers Journal of Pure and Applied Algebra
A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles (joint with David Favero and Tyler Kelly) Forum of Mathematics, Sigma
Frobenius degenerations of preprojective algebras Journal of Noncommutative Geometry
Spectrum of the Laplacian on regular polyhedra (joint with Evan Greif, Robert Strichartz, and Samuel Weise) Communications on Pure and Applied Analysis
An endless pursuit, a review of Netflix's film "A Trip to Infinity" (joint with Michael Kaplan) Nature Physics, Books & Arts
Here is my Google Scholar profile, my ORCID profile, and my ResearchGate profile.
All of my papers are available in preprint form on ArXiv, here.
Feel free to email me with any questions or comments.
My recent preprint: Crepant resolutions of stratified spaces via gluing (joint with Travis Schedler) constructs new resolutions on e.g., symplectic singularities by building them in the neighborhood of a point and spreading them out first to an entire stratum and then to the entire space, if possible.
Oberwolfach talk as part of the Interactions between Algebraic Geometry and Noncommutative algebra
Noncommutative geometry
My research examines the role of non-commutative algebras in geometry and topology. Noncommutative algebras can arise in geometry by
quantizing the algebra of functions on a space,
considering the algebra of endomorphisms of a vector bundle (or sheaf) on a space, and
invariants such as the algebra of differential operators, or the Chekanov--Eliashberg dg-algebra of a link, or the algebra of differential operators
Noncommutative rings can be used in lieu of sheaf theory to study resolutions of singularities. And some spaces can be realized as a moduli space of representations for a noncommutative algebra. Groups, Lie algebras, and quivers can be studied using the group algebra, universal enveloping algebra, and path algebras respectively. Recasting a geometric problem in algebraic language often lends itself to combinatorial or computational approaches.
Videos
Three talks focusing on different aspects of my work with Travis Schedler on multiplicative preprojective algebras:
(1) The 2-CY property + Delgne-Simpson: Multiplicative preprojective algebras in geometry and topology (talk in the LAGOON seminar 2021) video
(2) The Diamond Lemma: The Diamond Lemma for multiplicative preprojective algebras (talk at Perimeter Institute 2019) video
(3) Formality + symplectic geometry: How I met the multiplicative preprojective algebra (talk at Yale 2020) video
A talk on my work with David Favero and Tyler Kelly on exceptional collections for mirrors of invertible polynomials:
Exceptional collections for invertible polynomials using VGIT (talk given in the Online Algebraic Geometry Seminar) video
A talk on my work in progress with Travis Schedler on local-to-global results for crepant resolutions of singularities:
Multiplicative quiver varieties and symplectic resolutions of singularities (talk at the Isaac Newton Institute conference -- Mathematical physics: algebraic cycles, strings, and amplitudes) video
Two talks on the construction and obstruction of isomorphisms between multiplicative preprojective algebras and preprojective algebras
(I) Multiplicative preprojective algebras of Dynkin quivers (talk given in the Paris Algebra Seminar) notes video
(II) MPAs of ADE Dynkin quivers (talk given in the Antwerp Algebra Seminar focusing on the D_4 example) video
Notes
The introduction of my thesis serves as an entry point to my work with preprojective algebras.
A poster on my work with symplectic resolutions of singularities
Koszul Duality: notes based on a talk by Yanki Lekili in the Topics in Geometry course for LSGNT
Deformation Theory of Associative and Frobenius algebras: my Early Stage Assessment at Imperial
(2, 1, 0)-TQFTs: notes for a talk at European Talbot, based on Chris-Schommer Preis' thesis
My undergraduate thesis at Northwestern concerning which finite groups can act freely on some n-sphere.
A one-paramter family of representations of a quiver, describing how five complex lines can sit inside two-dimensional space