Ben Wormleighton

My domain research interests are around the geometry of problems arising from theoretical physics and, increasingly, in applied topology. My education research centers on formative learning rhythms and assessment methods. I've also dabbled in genre theory.

Here are some projects that I've worked on or am currently working on along with relevant articles. For a more compact list of my publications, see my CV or a past research statement. I recently wrote this summary for an Oberwolfach workshop. You can also find me on LinkedIn.

Symplectic Embeddings & Algebraic Positivity

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?

Formative Assessment & Pedagogy

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 assessment in proof-based mathematics classes, with Nelson Carter, Devin Akman, Barry Henaku, Peyton Gozon & Rachel Wu (submitted)
A genre analysis of mathematical remarks, with Valentin Küchle (submitted)

Questions I'm interested in

How can we maintain the impact of formative assessment on learning and student experience while integrating it into courses with other, more familiar assessment practices?
How can we develop efficient and effective formative assessment practices at R1 institutions?
How can we most effectively overcome the 'novelty' of formative methods for students?
What do formative learning spaces look like for master's or PhD students?

McKay Correspondence & Wall-Crossing

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?

Geography of Varieties & Ehrhart Theory

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 Varieties & Mirror Symmetry

Cluster algebras and cluster varieties are widespread, highly symmetric objects that are related to important constructs in geo metry, 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?

Theses & Other Writings

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.

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