Colloquium: Titles and Abstracts

Q-Gorenstein deformations of toric singularities

by Klaus Altmann (FU Berlin)
Thursday, 10 October 2016 at 14:00 in Room 230, Fields Institute

We recall how toric deformations can be understood via the decomposition of certain polytopes into Minkowski summands. Together with a combinatorial description of the vector space of the infinitesimal deformations T1, this gives rise to a natural understanding of the Kodaira-Spencer map. Then, we focus on two-dimensional quotient singularities and compare Q-Gorenstein deformations in the global and in the infinitesimal level. We will see that the corresponding functor is obstructed and hence will replace it by gadgets we call qG-deformations. This talk is based on joint work with János Kollár.

Determinants, Pfaffians, and vexillary degeneracy loci

by Dave Anderson (Ohio)
Thursday, 27 October 2016 at 14:00 in Room 230, Fields Institute

The Giambelli–Thom–Porteous formula expresses the class of a degeneracy locus as a certain determinant in the Chern classes of the vector bundle involved. The conditions defining such loci can be encoded in what Lascoux and Schutzenberger called “vexillary” permutations. I will survey these formulas and their recent extensions to degeneracy loci of other types, where Pfaffians replace determinants, and certain signed permutations play the role of vexillary permutations. The focus will be on the interplay between geometry, algebra, and combinatorics.

Lefschetz Properties

by Mats Boij (KTH)
Thursday, 24 November 2016 at 14:00 in Room 230, Fields Institute

The study of Lefschetz properties of artinian algebras has its origin in the Hard Lefschetz Theorem and was encouraged by Stanley’s use of it in the proof of the g-Conjecture for simplicial polytopes. The Weak Lefschetz Property means that multiplication by a general linear form has maximal rank in each degree and the Strong Lefschetz Property that the same also holds for all powers of the form. Despite the fact that most algebras have these property it is surprisingly hard to prove general results in this direction. I will discuss some of the known results and some of the open problems in this area.

Tangency and discriminants in Combinatorial Algebraic Geometry

by Sandra Di Rocco (KTH)
Thursday, 1 December 2016 at 14:00 in Room 230, Fields Institute

The study of so called A-discriminants, i.e. projective duality of (possibly not normal) toric varieties, is a classical subject in Combinatorial Algebraic Geometry. An exciting theory, with deep connections to complex and real projective geometry, was developed during the last two decades.
I will recall some history, some recent generalisations and some classical and newer results, emphasising the interplay between convex and projective geometry.

Frobenius splittings of toric varieties

by Milena Hering (Edinburgh)
Thursday, 8 September 2016 at 14:00 in Room 230, Fields Institute

Varieties admitting Frobenius splittings exhibit very nice properties. For example, many nice properties of toric varieties can be deduced from the fact that they are Frobenius split. Varieties admitting a diagonal splitting exhibit even nicer properties. In this talk, I will give an overview over the consequences of the existence of such splittings and then discuss criteria for toric varieties to be diagonally split.

Bruhat atlases on stratified spaces

by Allen Knutson (Cornell)
Thursday, 3 November 2016 at 14:00 in Room 309, Fields Institute

The orbits of B on G/B, “Bruhat cells”, are simple as varieties (they’re just vector spaces) but have a very interesting and comprehensible stratifications by singular subvarieties, the “Kazhdan-Lusztig varieties”. I’ll define a “Bruhat atlas” on a stratified manifold as an atlas consisting of Bruhat cells, such that the chart maps correspond the stratifications. In particular, the poset is identified with an order ideal in a Bruhat order, which is already very restrictive.

The motivating example was the positroid stratification of the Grassmannian, whose strata we indexed by affine permutations in [Knutson-Lam-Speyer]; this was souped up to geometry in Snider’s 2010 thesis. I’ll explain how to put Bruhat atlases on general G/P (and more generally, Q-orbits on G/P) and on wonderful compactifications of groups. This work is joint with Jiang-Hua Lu and Xuhua He.

For inductive classification purposes, one wants to study not just the manifolds but their strata, which inherit “Kazhdan-Lusztig atlases”. The first nontrivial problem is to classify equivariant K-L atlases on smooth projective toric surfaces. These have been largely classified by Bal\’azs Elek in his thesis: there are 18 or 19 simply-laced atlases, and < 8000 non-simply-laced.

Algebraic geometry and convex geometry

by Askold Khovanskii (Toronto)
Thursday, 22 September 2016 at 14:00 in Room 230, Fields Institute

I will review some results which relate these areas of mathematics. Newton polyhedra theory connects algebraic geometry and the theory of singularities to the geometry of convex polyhedra with integral vertices in the framework of toric geometry. The theory of Newton-Okounkov bodies relates algebra, singularities and geometry of convex bodies outside of that framework. An intermediate version of these theories provides such a relation in the framework of spherical varieties. Those relations are useful in many directions and suggest new unexpected results.

Tropical Schemes

by Diane Maclagan (Warwick)
Thursday, 20 October 2016 at 14:00 in Room 230, Fields Institute

Tropical geometry allows varieties, and their compatifications and degenerations, to be studied using combinatorial and polyhedral techniques. While this idea has proved surprisingly effective over the last decade, it has so far been restricted to the study of varieties and algebraic cycles. I will discuss joint work with Felipe Rincon that gives a definition for of a subscheme of a tropical toric variety. This builds on work of Jeff and Noah Giansiracusa on tropicalizing subschemes, and uses the theory of valuated matroids.

Fans of Logarithmic Structures

by Steffen Marcus (TCNJ)
Thursday, 1 September 2016 at 14:00 in Room 230, Fields Institute

In this talk I will outline recent developments in logarithmic Gromov-Witten theory, a variant of relative Gromov-Witten theory first proposed by Siebert in 2002 and constructed in 2011 by Abramovich-Chen and Gross-Siebert. My focus will be on some of the combinatorial tools in logarithmic geometry this theory has helped illuminate, and how these tools have been useful in a broader context.

Counting curves in surfaces: the tropical and the Fock space approach

by Hannah Markwig (Tübingen)
Thursday, 29 September 2016 at 14:00 in Room 310, Fields Institute

Tropical geometry can be viewed as an efficient tool to organize degenerations. The techniques to count curves in surfaces via tropical geometry are related to the Fock space approach initiated by Cooper–Pandharipande, via floor diagrams, which can be viewed as the combinatorial essence of a tropical curve count (following Block–Goettsche). Our own contribution relates the tropical and the Fock space approach for descendant Gromov-Witten invariants. Joint work with Renzo Cavalieri, Paul Johnson and Dhruv Ranganathan.

Littlewood Richardson coefficients—moving between geometry and combinatorics

by David Speyer (Michigan)
Thursday, 15 September 2016 at 14:00 in Room 230, Fields Institute

Littlewood-Richardson coefficients in geometry count the number of points of a Grassmannian lying on certain Schubert varieties (and many other objects: components of Springer components, multiplicities of irreducible representations in GL_n tensor products…). Littlewood-Richardson coefficients in combinatorics count the number of Young tableaux obeying various conditions (and many other objects: hives, cartons, puzzles…). I’ll describe work of Eremenko and Gabrielov, of Mukhin, Tarasov and Varchenko, of Purbhoo, and of myself connecting the Schubert variety story to the combinatorics story. No previous knowledge of Schubert calculus or tableaux combinatorics is assumed.

Nearest Points on Toric Varieties

by Bernd Sturmfels (Berkeley)
Thursday, 25 August 2016 at 14:00 in Room 1180, Bahen Centre

We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the A-discriminant in terms of Euler obstructions. The motivation for this work is the development of reliable methods for computing the points on a real toric variety that are closest to a given data point. In this lecture we emphasise the role played by characteristic classes such as the Chern-Mather class. This is joint work with Martin Helmer.

Derived category of the moduli space of stable rational curves

by Jenia Tevelev (UMass Amherst)
Thursday, 10 November 2016 at 14:00 in Room 230, Fields Institute

I will discuss work in progress with Ana-Maria Castravet verifying a surprising conjecture of Orlov and Kuznetsov on the equivariant structure of the derived category (and even K-theory) of the moduli space of stable rational curves.