Algebraic capacities are analogues in algebraic geometry to symplectic capacities, which measure obstructions to embedding one symplectic manifold into another. Symplectic capacities date back to the Gromov width and the symplectic camel; more recently ECH capacities were developed by Hutchings for symplectic 4-manifolds, and have enjoyed much success in studying embeddings of ellipsoids and other toric manifolds. In the papers below we develop the theory of algebraic capacities and relate them to various symplectic capacities, which has exciting consequences for embeddings.

Papers

• Tropical Newton-Okounkov bodies and symplectic embeddings into cluster surfaces, with Julian Chaidez and Timothy Magee (in progress)

• Newton-Okounkov bodies and symplectic embeddings into non-toric rational surfaces, with Julian Chaidez (arXiv, appeared in JLMS)

• Lattice formulas for rational SFT capacities, with Julian Chaidez (arXiv)

• Towers of Looijenga pairs and asymptotics of ECH capacities (arXiv, appeared in manuscripta)

• ECH embedding obstructions for rational surfaces, with Julian Chaidez (arXiv)

• Algebraic capacities (arXiv, appeared in Selecta Mathematica)

• ECH capacities, Ehrhart theory, and toric varieties (arXiv, appeared in JSG)

Questions I'm interested in

• Can we use algebraic positivity to construct good embedding obstructions in higher dimensions?

• Can we use algebraic positivity to produce sharp embedding obstructions for non-toric rational surfaces?

• Can we largely or completely describe the extent to which the sub-leading asymptotics of ECH capacities converge?

I am interested in cultivating and more deeply understanding formative learning spaces that serve to shape and empower students in their process of intellectual maturation. See my page on this topic for more details.

Papers

• Mastery based grading as formative assessment, with Devin Akman, Barry Henaku, Peyton Gozon & Rachel Wu (in preparation)

• A genre analysis of mathematical remarks, with Valentin Küchle (submitted)

Questions I'm interested in

• What are effective metrics for assessing the impact of, and hence refining, formative learning spaces?

• What do formative spaces look like for master's or PhD students?

• What techniques of formative assessment are ultimately (un)fruitful?

• How to most effectively overcome the 'novelty factor' of formative methods for individual students?

The two-dimensional McKay correspondence supplies the geometry behind the ADE classification of finite subgroups of SL(2) via studying the associated quotient singularities and their resolutions. This has a beautiful extension to three dimensions, where the non-uniqueness of minimal resolutions makes the theory much richer. I have previously led the McKayCorr Research Group investigating various aspects of wall-crossing phenomena for minimal resolutions with a view to better understanding their birational geometry. I have also worked on derived aspects of the McKay correspondence.

Papers

• Crepant resolutions, mutations, and the space of potentials, with Mary Barker and Ben Standaert (arXiv, appeared in Exp. Math.)

• Wall-crossing for iterated Hilbert schemes (or 'Hilb of Hilb') (arXiv, appeared in ASPM)

• Walls for G-Hilb via Reid's recipe (arXiv, appeared in SIGMA)

Questions I'm interested in

• Is there a good symplectic or mirror symmetric approach to the Craw-Ishii conjecture?

• Is there a good mutation description for 3-fold flops over toric and non-toric singularities?

• How should Reid's recipe look for non-abelian groups? Or how can it be related iterated Hilbert schemes?

There are a host of interesting geography problems for (polarised) varieties: studying the range of values important invariants can take over a certain class of varieties. I like to explore several aspects of these problems including reconstruction problems (when does a variety exist with given invariants?), classification problems for Fano and toric varieties, and studying richer 'tropical' invariants. In equivariant contexts these invariants often have combinatorial avatars, such as lattice point counts from Ehrhart theory. There are many mysteries about such invariants, such as understanding periods of Ehrhart quasi-polynomials. My work also uses tools and intuition from geometry to investigate questions about combinatorial invariants.

Papers

• Picard bounds for toric Fano varieties with minimal rational curve constraints, with Roya Beheshti (arXiv, appeared in PAMS)

• Quasi-period collapse for duals to Fano polygons: an explanation from algebraic geometry, with Al Kasprzyk (arXiv, to appear in Proc. Nott. Alg. Geo. Sem.)

• Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series (arXiv, appeared in Commun. Alg.)

Questions I'm interested in

• Can we use geometric methods to classify or at least construct many rational polytopes with prescribed periods?

• Can we understand how generically quasi-period collapse occurs via studying moduli spaces of polytopes?

Cluster algebras and cluster varieties are widespread, highly symmetric objects that are related to important constructs in geometry, representation theory, combinatorics, and physics. I explore their associated geometry, such as their applications to the [homological] minimal model program (with Tom Ducat) and to mirror symmetry (with Jonathan Lai and Tim Magee).

Papers

• Tropical Newton-Okounkov bodies and symplectic embeddings into cluster surfaces, with Julian Chaidez and Timothy Magee (in progress)

• Crepant resolutions, mutations, and the space of potentials, with Mary Barker and Ben Standaert (arXiv, appeared in Exp. Math.)

• Reconciling mutations and potentials for Fano and cluster varieties, with Jonathan Lai and Tim Magee (pre-arXiv version)

• Wall-crossing for iterated Hilbert schemes (or 'Hilb of Hilb') (arXiv, appeared in ASPM)

• Towers of Looijenga pairs and asymptotics of ECH capacities (arXiv, appeared in manuscripta)

• Quasi-period collapse for duals to Fano polygons: an explanation from algebraic geometry, with Al Kasprzyk (arXiv, to appear in Proc. Nott. Alg. Geo. Sem.)

Questions I'm interested in

• How can we describe "class TG", the class of Fano varieties admitting a Q-Gorenstein toric degeneration?

• What do mirrors to non-toric 3-fold quotient singularities look like?

• Can we classify interesting classes of LCY varieties with a given tropicalisation?

• Numerics and stability for orbifolds, with applications to symplectic embeddings, UC Berkeley PhD thesis (link, slides)

• On the non-abelian McKay correspondence, University of Warwick master's thesis (link)

• I maintained a math & poetry blog for a while.

• Some fun with Pythagorean triples and cluster algebras with help from Alex Best.

Images: lattice paths computing ECH capacities (top left, ref.); final curves in G-Hilb for G = 1/35(1,3,31) (top center, ref.); global scattering tiles for P1 x P1 (top right, ref.); schematic of possible reduced baskets (bottom left, ref.); scissors congruence via mutation (bottom center, ref.); unlocking for G = 1/25(1,3,21) (bottom right, ref.). Produced by TikZ and CoCalc.