Degeneracy loci, determinants, Pfaffians, and Schubert polynomials
by Dave Anderson (OSU)
A very old problem asks for the degree of a variety defined by rank conditions on matrices. A version of this problem was studied by Giambelli, and his solution is one of the cornerstones of Schubert calculus. The story of the modern approach begins in the 1970’s, when Kempf and Laksov proved that the cohomology class of a degeneracy locus for a map of vector bundles is given by a Giambelli-type determinant in their Chern classes.
Since then, many variations have been studied — for example, when the vector bundles are equipped with a symplectic or quadratic form, the formulas become Pfaffians. In this series of lectures, I will describe some recent extensions of these results — beyond determinants and Pfaffians, and beyond ordinary cohomology — including my joint work with W. Fulton, as well as work of several others.
These formulas are closely related to Schubert polynomials, which were introduced combinatorially by Lascoux and Schutzenberger in 1982, and I will discuss some of the combinatorics involved. There are many delicate combinatorial and algebraic questions connected with this subject, some of which remain unanswered.
Berkovich analytic spaces from the tropical perspective
by María Angélica Cueto (OSU)
Algebraic varietes over fields equipped with non-Archimedean valuations can be studied from the perspective of analytification (in the sense of Berkovich) and tropicalization. The purpose of this course is to provide a gentle introduction to Berkovich’s theory through the lens of tropical geometry. Special emphasis will be given on computationally effective tools to examine these complicated spaces of valuations through their (easier) tropical shadows.
A central theme of the course will be the notion of faithful tropicalization: when the fixed tropical variety best reflects the topology of the analytic space. The guiding questions will be: is a tropicalization faithful, and if not, can we fix it? We will illustrate the approach to tackle these questions via two examples with rich combinatorics: (hyperelliptic) curves and Grassmannians.
The first two talks will be devoted to a hands-on introduction to Berkovich spaces and its connection to tropicalization, following the work of Payne. In the last two talks we will focus on the concrete examples mentioned earlier and provide some open questions.
Torus Actions and Combinatorics
by Nathan Ilten (Simon Fraser University)
The well-known connection between toric varieties and convex polyhedral objects has led to a fairly complete understanding of the geometry of toric varieties in combinatorial terms. A little over a decade ago, Altmann and Hausen developed a theory generalizing this connection to so-called T-varieties: normal varieties X endowed with an effective action of an algebraic torus T. Examples of such varieties include spherical varieties and toric vector bundles, among other examples. The situation where the dimension of X equals that of T is exactly the toric case.
The price to be paid for this generalization is that the theory of T-varieties is no longer purely combinatorial. Instead, a T-variety X may be described in terms of a quotient variety Y, endowed with combinatorial information in the form of a “polyhedral divisor” (in the affine case) or a “divisorial fan” (in general). The dimension of the quotient variety is the codimension of T in X; in the toric case, the quotient collapses to a point, and only the combinatorial information remains. In situations where the geometry of the quotient variety Y is relatively accessible (e.g. Y is a curve or projective space), many results concerning toric varieties can be generalized.
In this series of lectures, I will give an introduction to the Altmann-Hausen theory of polyhedral divisors. After outlining the general theory, I will focus on the case of complexity-one actions with two applications in mind: proving that rank two toric vector bundles are Mori Dream Spaces, and studying the deformation theory of toric varieties.
Newton-Okounkov bodies, Khovanskii bases, and tropical geometry
by Kiumars Kaveh (Pittsburgh)
We start with a review of basic concepts and constructions in the theory of Newton-Okounkov bodies. I will then talk about some related results and touch on some applications e.g. constructing toric degenerations and applications to symplectic geometry, and multiplicities of ideals.
Toric degenerations constructed in this context are directly related to the notion of a Khovanskii basis. This in turn is related to the key question of finite generation of the value semigroup. This approach provides a natural setup to generalize the notion of a SAGBI basis for a subalgebra of polynomials to an arbitrary algebra equipped with a full rank valuation. We will talk about some basic results on Khovanskii bases and the subduction algorithm.
Finally, I will discuss some recent results (joint with Chris Manon) about a connection between full rank valuations (appearing in the theory of Newton-Okounkov bodies) and rank one valuations (appearing in tropical geometry). More precisely, let A be a finitely generated positively graded domain. Then we make a correspondence between prime cones in tropical varieties of different ideals presenting A, and certain class of full rank valuations on A (which we call “good” valuations). Roughly speaking, a valuation v is good if its values semigroup is finitely generated and the subduction algorithm for v terminates.
Many well-known examples such as Plucker coordinates and coordinate rings of Grassmannians, as well as the Gelfand-Zetlin bases and coordinate rings of flag varieties fit into this general picture.
Tropical geometry of algebraic curves and their moduli
by Sam Payne (Yale)
Combinatorics and algebraic geometry related to hyperplane arrangements
by David Speyer (Michigan)
In the first talk, I will explain the language of matroids for
those who haven’t seen it. This is a language which has been to develop to
describe and extract the combinatorial structure of hyperplane arrangements.
In the remaining talks, I will present several of the main geometric
constructions related to hyperplane arrangement complements and how they
have been combinatorially described in terms of matroids. Particularly, I
want to talk about the maximum likelihood degree, about the wonderful
compactification and about the Chow quotient of the Grassmannian.
I hope that these topics should be interesting for people thinking about
CSM classes, about June Huh’s recent work, about tropical linear spaces and
about Hacking-Keel-Tevelev and Alexeev’s compactifications of the moduli
space of hyperplane arrangements, and I will work to make them accessible
to people who have not heard of matroids before.
Combinatorial Algebraic Geometry 