Combinatorial Algebraic Geometry Day

University of Bristol, 17 September 2018

Combinatorial Algebraic Geometry Day (ComAlgGeo) is a one-day meeting which is open to all mathematicians in the UK with interests in Combinatorics, Algebra, and Geometry.

The talks will be at the 4th Floor Seminar Room in Howard House.

Invited Speakers and Schedule

Short Talks:


Please register by emailing Fatemeh. Please indicate whether you are willing to have a 5 minutes presentation and if you would like to join us for dinner.

Titles and Abstracts:

Learning Bayesian Networks Using Generalized Permutohedra

Abstract: Graphical models (Bayesian networks) based on directed acyclic graphs (DAGs) are used to model complex cause-and-effect systems. A graphical model is a family of joint probability distributions over the nodes of a graph which encodes conditional independence relations via the Markov properties. One of the fundamental problems in causality is to learn an unknown graph based on a set of observed conditional independence relations. In this talk, I will describe a greedy algorithm for DAG model selection that operate via edge walks on so-called DAG associahedra. For an undirected graph the set of conditional independence relations are represented by a simple polytope known as the graph associahedron, which can be constructed as a Minkowski sum of standard simplices. For any regular Gaussian model, and its associated set of conditional independence relations we construct the analogous polytope DAG associahedon which can be defined using relative entropy. For DAGs we construct this polytope as a Minkowski sum of matroid polytopes corresponding to Bayes-ball paths in graph. This is a joint work with Caroline Uhler, Charles Wang, and Josephine Yu.

Polynomial interpolation in algebraic geometry

Abstract: I will give an introduction to polynomial interpolation problems in several variables and to their formulation in the algebraic as well as in the geometric setting. I will give an overview of conjectures and open problems arising from both settings and discuss some results in this direction. This is joint work with C. Brambilla and O. Dumitrescu.

Supports of Schubert polynomials

Abstract: Last year Monical, Tokcan and Yong conjectures about several families of combinatorial polynomials that their supports were the set of lattice points of some convex polytope, with no "internal" zeroes. We have proved this for the Schubert polynomials, well-loved for being the "best" lifts of the cohomology classes of Schubert varieties to the polynomial ring, as well as the key polynomials of Demazure. In both these cases the Newton polytope is in fact a generalised permutahedron. I'll explain these results and the technology involved, and if time permits discuss some progress towards the problem of identifying the Schubert polynomials which are exactly lattice point enumerators of a polytope. All work discussed is joint with Karola Mészáros and Avery St. Dizier.

The Hilbert function of the graph of the reciprocal plane

Abstract: Two different-looking ways to get the characteristic polynomial of a hyperplane arrangement by commutative algebra appear in the literature. One is from the Hilbert function of the reciprocal plane (Orlik-Terao); the other is from the multidegree of the graph over this plane (Adiprasito-Huh-Katz). After introducing these objects and constructions, I will give a Stanley-Reisner initial degeneration to an extension of the no broken circuit complex explaining why these two are not different after all. This is joint work with David Speyer and Alexander Woo.

Noncommutative resolutions and rings of differential operators of toric varieties

Abstract: Let R be the coordinate ring of an affine toric variety over a field k of arbitrary characteristic. The module M of p^e-th roots of R, where p and e are positive integers, is then the direct sum of so-called conic modules. In this talk we are interested in homological properties of the endomorphism ring End_R(M), in particular its global dimension.

With a combinatorial method we construct certain complexes of conic modules over R and explain how these yield projective resolutions of simple modules over End_R(M). Thus we obtain a bound on the global dimension of End_R(M), which shows that this endomorphism ring is a so-called noncommutative resolution of singularities (NCR) of R (or Spec(R)). If the characteristic of k is p>0, then this fact allows us to bound the global dimension of the ring of differential operators D(R). This is joint work with Greg Muller and Karen E. Smith.

Mirror symmetry for cominuscule homogeneous varieties

Abstract: In this talk reporting on joint work with K. Rietsch and L. Williams, I will explain a new version of the construction by Rietsch of a mirror for some varieties with a homogeneous Lie group action. The varieties we study include quadrics and Lagrangian Grassmannians (i.e., Grassmannians of Lagrangian vector subspaces of a symplectic vector space). The mirror takes the shape of a rational function, the superpotential, defined on a Langlands dual homogeneous variety. I will show that the mirror manifold has a particular combinatorial structure called a cluster structure, and that the superpotential is expressed in coordinates dual to the cohomology classes of the original variety.

I will also explain how these properties lead to new relations in the quantum cohomology, and a conjectural formula expressing solutions of the quantum differential equation in terms of the superpotential. If time allows, I will also explain how these results should extend to a larger family of homogeneous spaces called cominuscule homogeneous spaces.

Mirror symmetry for OGr(5,10)

Abstract: I will explain my attempt to understand mirror symmetry for the orthogonal Grassmannian X = OGr(5,10) by exhibiting a nice Laurent phenomenon for X which can be used to write X as both a log Calabi-Yau pair (X,D), for a nice anticanonical divisor D, and as a Landau-Ginzburg model mirror to (X,D). This is work in progress.

Short Talks:

  • Benjamin Smith (QMUL) Matching fields and tropical hyperplane arrangements

  • Ollie Clarke (Bristol) Toric degenerations of Schubert varieties

  • Charley Cummings (Bristol) Modules, categories and injective generation

  • Alessio D'Alì (Warwick) Translative group actions on simplicial posets