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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: TBD.
The April schedule is currently being cooked.
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