Toric Degenerations and Moduli
by Valery Alexeev (Georgia)
I will begin by explaining the fundamental construction of a convex toric degeneration which has been rediscovered by many people and which appears in different guises in many fields of mathematics. I will illustrate it by giving examples of toric degenerations as varieties, as pairs, as cycles, and as subschemes. I will also explain the extension of this construction to degenerations of abelian varieties, and of hyperplane arrangements. In the second talk, I will apply the fundamental construction to making various combinatorial moduli spaces: of stable toric pairs, toric Hilbert scheme, toric Chow varieties, stable semiabelic varieties, and stable hyperplane arrangements. Time permitting, I will go over applications of the theory of degenerations of abelian varieties to a problem of extending various Torelli maps to compactified moduli spaces: for Jacobians, Prym varieties, Prym-Tyurin varieties, and Intermediate jacobians.
Combinatorial Aspects of Moduli Space of Curves
by Angela Gibney (Georgia)
In algebraic geometry, families of curves are studied using moduli spaces. Besides providing deep insights about curves themselves, moduli spaces exhibit important behaviors which help shape our understanding of how geometric objects may be parametrized, giving valuable insight into the development of higher dimensional theory. Geometric aspects of moduli spaces of curves are reflected in their underlying combinatorial structures.
In these lectures, I will discuss combinatorial aspects of the moduli space of curves. In the first talk, I will give an introduction and overview, aimed at those with no prior background about moduli spaces or the moduli space of curves. In the second talk I will discuss recent developments and open problems in the area.
An Invitation to Newton–Okounkov Bodies
by Megumi Harada (McMaster)
In the theory of toric varieties, the combinatorics of a convex integral
polytope Delta is intimately linked with the geometry of an associated toric variety X(Delta); indeed, the polytope Delta fully encodes the geometry of X(Delta) in this case. In more general situations arising in geometry, one can often associate combinatorial data to a group action on a manifold, but usually the combinatorics doesn’t completely encode the original geometric data. The recent theory of Newton–Okounkov bodies initiated by Kaveh–Khovanskii and Lazarsfeld–Mustata can be viewed as a generalization of the theory of toric varieties to a much more general setting: given an arbitrary algebraic variety X, together with some auxiliary data, this theory produces a convex body Delta, and in many cases Delta is a rational polytope. This talk will be a basic introduction to this relatively recent theory, and some of the questions it raises.
by Milena Hering (Edinburgh)
In these two talks, I will explain what a toric variety is and briefly mention generalizations, such as spherical varieties and T-varieties. I will then continue to cover some recent developments in toric varieties and mention some open questions.
Combinatorics in Geometric Representation Theory
by Joel Kamnitzer (Toronto)
In geometric representation theory, we study algebraic varieties related to reductive groups. Of particular interest are a collection of varieties called Mirkovic–Vilonen cycles. These varieties can be profitably studied by means of their moment polytopes, called Mirkovic–Vilonen polytopes. These MV polytopes are fundamental objects in combinatorial representation theory. We will introduce all these objects, assuming no background knowledge in geometric representation theory.
by Allen Knutson (Cornell)
“The set of k-planes in n-space satisfying the following list of closed conditions” defines a cycle in the Grassmannian (of all k-planes), about which one can ask (to begin with) two kinds of questions: (1) what does this cycle look like as a variety, e.g. how singular is it? and (2) what is its homology class inside the Grassmannian.
It turns out that every cycle is (uniquely) a positive combination of
“Schubert classes”, the classes of “Schubert varieties”. In the first
lecture I’ll define these, and explain enough equivariant cohomology (from the ground up) to compute the Schubert classes and explain their connection to semistandard Young tableaux. (It will turn out, for inductive purposes, to be useful to go beyond Grassmannians to flag manifolds.) I’ll also give a big picture of the problems of Schubert “calculus”, which concerns the structure constants for the cohomology product, and admits many generalizations that are still open.
In the second lecture, we’ll look at the singularities of Schubert varieties, using Bott-Samelson manifolds to help define a Gröbner basis for local patches on Schubert varieties; the resulting combinatorics is that of”subword complexes”. Then I’ll talk about recent work of [Maulik–Okounkov], [C. Su], [Aluffi–Mihalcea], [Huh], and [me–Zinn–Justin] about D-modules/Chern–Schwarz–MacPherson classes of Schubert varieties.
Tropical Geometry of Curves and their Moduli
by Sam Payne (Yale)
Tropical geometry provides an array of combinatorial techniques for studying compactifications and degenerations of fundamental objects in algebraic geometry. The piecewise linear objects appearing in tropical geometry are shadows (or skeletons) of nonarchimedean analytic spaces, in the sense of Berkovich, and often capture enough essential information about those spaces to resolve interesting questions about classical algebraic varieties. I will give an overview of tropical geometry as it relates to the study of algebraic curves, touching on applications to moduli spaces of algebraic curves, and refined (or quantum) curve counting as time permits.
The Combinatorics of Hilbert Schemes
by Gregory G. Smith (Queen’s)
Hilbert schemes are the prototypical parameter spaces in algebraic geometry. By definition, there is a bijection between subschemes of a fixed projective space with prescribed Hilbert polynomial and the points in a Hilbert scheme. From a geometric perspective, these parameter spaces provide both an important source of higher-dimensional varieties and fundamental insights into families of projective subschemes. Given the inherent complexity of these objects, combinatorics plays an indispensable role in understanding their structure.
These two lectures examine the geometry of Hilbert schemes highlighting the underlying combinatorial features. Starting from the definition, we discuss the construction of Hilbert schemes and identify the nonempty ones. Interspersed with classic examples and open problems, we then examine various geometric properties including connectedness, irreducibility, and singularities.
Cluster Algebras and Varieties
by David Speyer (Michigan)
I will explain both the original definition of a cluster algebra, and the recent recasting by Gross, Hacking, Keel and Kontsevich in terms of a holomorphic symplectic variety with a compatible dense torus. I will then present several of the most important examples of cluster varieties and see how their cluster structure illuminates their geometry.
In the second lecture, I will present work in progress which attempts to describe the (de Rham) cohomology of cluster varieties and the mixed Hodge structure on it. Neither background in cluster varieteis (beyond the first talk) nor in Hodge theory will be assumed. Joint work with Thomas Lam.
Partition Functions and Threefolds
by Balázs Szendrői (Oxford)
The first lecture will be a general introduction to counting problems in Calabi–Yau threefold geometry, illustrated throughout by several manifestations of a single function of two variables, the partition function attached to the Calabi–Yau threefold geometry often called the local projective line. I’ll explain the Gromov–Witten, Donaldson–Thomas, Pandharipande–Thomas and noncommutative Donaldson–Thomas points of view, all of which include very explicit though highly non-trivial combinatorics.
This second lecture will continue the theme of threefold partition functions, explaining some coloured and quantized versions, arising from the McKay correspondence and motivic Donaldson–Thomas theory.