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Teams Meeting ID: 39675733658195
1pm-2pm, Tuesday, MB-503
27/1/2026: Abhiram Natarajan (Warwick)
Title: Semi-Pfaffian geometry - tools, and applications
References: [1] Larry Guth, Polynomial partitioning for a set of varieties, Mathematical Proceedings of the Cambridge
Philosophical Society, vol. 159, Cambridge University Press, 2015, pp. 459–469.
[2] Larry Guth and Nets Hawk Katz, On the Erdős distinct distances problem in the plane, Annals of
mathematics (2015), 155–190.
3/2/2026: Federico Ardila-Mantilla (QMUL)
Title: Tree metrics and log-concavity for matroids
Abstract: We show that a set function ν satisfies the gross substitutes property if and only if its homogeneous generating polynomial Zq,ν is a Lorentzian polynomial for all positive q≤1, answering a question of Eur-Huh. We achieve this by giving a rank 1 upper bound for the distance matrix of an ultrametric tree, refining a classical result of Graham-Pollak. This characterization enables us to resolve two open problems that strengthen Mason's log-concavity conjectures for the number of independent sets of a matroid: one posed by Giansiracusa-Rincón-Schleis-Ulirsch for valuated matroids, and two posed by Dowling in 1980 and Zhao in 1985 for ordinary matroids.
Special event: We enjoyed a pub trip in Rusty Bike!
10/2/2026: Abhiram Natarajan (Warwick)
Title: Gröbner Bases Native to Term-ordered Commutative Algebras, with Application to the Hodge Algebra of Minors
Abstract: Standard Gröbner basis methods are often too inefficient to handle even small cases arising in areas such as computational complexity theory—for instance, the orbit closure of the 3 × 3 determinant in geometric complexity theory. Motivated by this, and better understanding the bideterminant (=product of minors) basis on the polynomial ring in $n \times m$ variables, we develop theory \& algorithms for Gröbner bases in not only algebras with straightening law (ASLs or Hodge algebras), but in any commutative algebra over a field that comes equipped with a notion of ``monomial'' (generalizing the standard monomials of ASLs) and a suitable term order. Rather than treating such an algebra $A$ as a quotient of a polynomial ring and then ``lifting'' ideals from $A$ to ideals in the polynomial ring, the theory we develop is entirely ``native'' to $A$ and its given notion of monomial.
When applied to the case of bideterminants, this enables us to package several standard results on bideterminants in a clean way that enables new results. In particular, once the theory is set up, it lets us give an almost-trivial proof of a universal Gröbner basis (in our sense) for the ideal of $t$-minors for any $t$. We note that here it was crucial that theory be native to $A$ and its given monomial structure, as in the standard monomial structure given by bideterminants each $t$-minor is a single variable rather than a sum of $t!$ many terms (in the ``ordinary monomial'' structure).
17/2/2026: Sara Veneziale (Imperial)
Title: Machine learning detects terminal singularities
Abstract: In this talk, I will describe recent work in the application of machine learning to explore questions in algebraic geometry, specifically in the context of the study of Q-Fano varieties. These are Q-factorial terminal Fano varieties, and they are the key players in the Minimal Model Program. In this work, we ask and answer if machine learning can determine if a toric Fano variety has terminal singularities. We build a high-accuracy neural network that detects this, which has two consequences. Firstly, it inspires the formulation and proof of a new global, combinatorial criterion to determine if a toric variety of Picard rank two has terminal singularities. Secondly, the machine learning model is used directly to give the first sketch of the landscape of Q-Fano varieties in dimension eight. This is joint work with Tom Coates and Al Kasprzyk.
24/2/2026: Alejandro Vargas (Warwick)
Title: Arithmetic matroids and elliptic arrangements of complex multiplication type
Abstract: We recall how the matroid of a hyperplane arrangement in A^n encodes key geometric and topological invariants of the arrangement and its complement. This framework extends to arrangements of abelian subvarieties, where torsion and disconnected intersections introduce arithmetic phenomena not seen by ordinary matroids, prompting arithmetic refinements. We report on joint work with Moci, Pagaria, and Pismataro, where we define elliptic arrangements in powers of an elliptic curve with complex multiplication, associate to them a natural arithmetic matroid, and study how invariants such as the intersection lattice and the Euler characteristic of the complement are reflected in this enriched combinatorial structure. We also highlight how matroids realized via this construction are different from those in the toric case.
3/3/2026: Siao Chi Mok (Cambridge)
Title: Logarithmic Fulton–MacPherson configuration spaces
Abstract:The Fulton–MacPherson (FM) configuration space is a well-known compactification of the ordered configuration space of a projective variety. Given a semistable degeneration of X, we construct a log smooth degeneration of the FM space of X. This degeneration satisfies a “degeneration formula” — each irreducible component of its special fibre can be described as a proper birational modification of a product of log FM configuration spaces. These log FM spaces parametrise point configurations on certain target degenerations arising from both tropical/logarithmic geometry and the original Fulton–MacPherson construction.
Special event: Patience's leaving drink!
10/3/2026: Linxuan Li (QMUL)
Title: Tropical minimal model program.
Abstract: Regular tropical fans form an important class of tropical varieties, characterized by the property that their tropical intersection number is equal to 1. Classifying such tropical varieties is both natural and significant: for example, Alex Fink and Esterov–Gusev obtained classifications in the cases of Bergman fans and tropical complete intersections of hypersurfaces, respectively. Motivated by this classification problem, the tropical minimal model program was introduced, yielding statements that are formally very close to those of the classical minimal model program. In this talk, I will explain the tropical minimal model program and present its known results and examples in the cases where it has been established.
17/3/2026: Mark Gross (Cambridge)
Title: Vector fields on intrinsic mirrors.
Abstract: Siebert and I gave a general construction of mirror partners to log
Calabi-Yau pairs, we called these mirror partners "intrinsic mirrors". This talk
is about a small part of a larger project with Pomerleano and Siebert aimed
at understanding this construction at a deeper level. I will explain how to
construct vector fields on the mirror using enumerative geometry of the original
log Calabi-Yau pair.
24/3/2026: Alexander Esterov (London Institute for Mathematical Sciences)
Title: Tropical complete intersections
Abstract: The geometry of a toric complete intersection is encoded in the Newton polytopes of its equations. This story can be generalized to include many other special varieties, such as hyperplane arrangements, discriminant and incidence varieties in enumerative geometry and polynomial optimization, generalized Calabi-Yau complete intersections, general sections of toric vector bundles etc. The role of Newton polytopes is then played by more general objects -- tropical complete intersections, encoding the geometry of all of the above.
07/4/2026: Lilybelle Cowland Kellock (Manchester)
Title: Reduction types of genus 2 curves.
Abstract: Reduction types are used to recover lots of important arithmetic information attached to curves over local fields, for example Tamagawa numbers which show up in the Birch and Swinnerton-Dyer conjecture for Jacobians of curves. I will discuss a joint work in progress with E. Aylward, V. Dokchitser and E. Lupoian that classifies the (more than 100!) possible reduction types of genus 2 curves in terms of a combinatorial object known as the `cluster picture’.
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