Wall crossing and quiver representations
by Mandy Cheung (IAS)
Tuesday, 2 August 2016 at 14:00 in Room 309, Fields Institute
Cluster algebras were introduced to understand canonical bases in a semisimple Lie group. Later, Gross–Hacking–Keel–Kontsevich related scattering diagrams from mirror symmetry to cluster algebras. They further proposed that the sets of theta functions give canonical bases for cluster algebras. Using this linkage, we will see how the theory in quiver representation, namely Auslander–Reiten theory, can be visualized on scattering diagrams. Furthermore, we will discuss how broken lines, which are used to constructed theta functions, give stratifications for quiver Grassmannians.
Boij-Söderberg theory for Grassmannians
by Jake Levinson (Michigan)
Wednesday, 23 November 2016 at 14:00 in Room 309, Fields Institute
Boij-Söderberg theory is a “structure theory” for classifying syzygies of graded modules and sheaves on projective space. I’ll describe some recent work on extending it to the setting of GLk-equivariant modules and sheaves on Grassmannians. The algebraic setting concerns modules over a polynomial ring in kn variables, thought of as the entries of a k-by-n matrix. The goal is to understand “equivariant Betti tables” of these modules.
I’ll focus on motivating and stating our main result, which is an equivariant analog of a key feature of graded Boij-Söderberg theory: the pairing between Betti tables and cohomology tables of sheaves. A key ingredient is the base case of square matrices, where Betti tables have an interesting connection to perfect matchings.
Time permitting, I’ll sketch out one other result, a method for detecting extremal Betti tables in the equivariant setting (via an analog of the “Herzog-Kühl equations”).
Waring problem and Cremona transformations
by Massimiliano Mella (Ferrara)
Friday, 5 August 2016 at 14:00 in Room 309, Fields Institute
Waring problem is about additive decompositions of homogeneous polynomials. I am interested in understanding when such decomposition is unique, this is called a canonical form. The problem of canonical forms is related to the existence of Cremona transformation associated to linear sytstem with assigned double points. I will report on the state of the art and give some new contribution that confirm the conjecture that there are very few canonical forms.
Milnor number, intersection multiplicity, and number of zeroes
by Pinaki Mondal (College of Bahamas)
Friday, 12 August 2016 at 15:20 in Room 309, Fields Institute
We talk about two of the original problems that shaped the theory of Newton polyhedra: the problem of computing the Milnor number of the singularity at the origin of a generic polynomial, and computing the number of zeroes of generic polynomials. The former was addressed by Kushnirenko, who gave a beautiful formula in terms of Newton diagrams in a special case. Bernstein (following work of Kushnirenko) solved completely the latter problem for the case of (C^*)^n. In the case of C^n the problem was partially solved following the work of Khovanskii, Huber-Sturmfels, and many others. We give complete solution to both these problems. A common theme in our solution to both problems is the computation of intersection multiplicity at the origin of the hypersurfaces determined by n generic polynomials.
Higher convexity for complements of tropical objects
by Frank Sottile (Texas A&M)
Wednesday, 23 November 2016 at 13:00 in Room 210, Fields Institute
Gromov generalized the notion of convexity for open subsets of R^n with hypesurface boundary, defining k-convexity, or higher convexity and Henriques applied the same notion to complements of amoebas. He conjectured that the complement of an amoeba of a variety of codimension k+1 is k-convex. I will discuss work with Mounir Nisse in which we study the higher convexity of complements of coamoebas and of tropical varieties, proving Henriques’ conjecture for coamoebas and establishing a form of Henriques’ conjecture for tropical varieties in some cases.
Irreducibility of random Hilbert schemes
by Andrew Staal (Queen’s)
Tuesday, 26 July 2016 at 15:00 in Room 210, Fields Institute
What are the geometric properties of a typical Hilbert scheme? In this talk, we consider Hilbert schemes parametrizing closed subschemes in some projective space with a fixed Hilbert polynomial. We first explain how to make the collection of all such Hilbert schemes into a discrete probability space. Exploiting the underlying combinatorial structure, we show why a random Hilbert scheme is smooth and irreducible with probability greater than 0.5.
Preprojective algebras and Nathan Reading’s shards
by Hugh Thomas (UQAM)
Friday, 26 August 2016 at 13:00 in Room 6183, Bahen Centre
I will recall the notion of stability conditions in the sense of geometric invariant theory (as carried into the theory of finite-dimensional algebras by King). In order to explain the structure of the semi-stable subcategories of finite type preprojective algebras, I will introduce Reading’s combinatorics of shards, defined based on the reflection arrangement of the corresponding type. This project is joint work with David Speyer, and also draws on a previous joint project with Nathan Reading, Idun Reiten, and Osamu Iyama.