The heart fan is a new convex-geometric invariant of an abelian category which captures interesting aspects of the related homological algebra. I will review the construction and some of its key properties, illustrating them through examples. In particular, I will explain how the heart fan can be viewed as a 'universal phase diagram' for Bridgeland stability conditions with the given heart.
Describing the conjugacy classes and/or irreducible characters of the unitriangular group over a finite field is known to be an impossibly difficult problem. Superclasses and supercharacters have been introduced (under the names of "basic varieties" and "basic characters") as an attempt to approximate conjugacy classes and irreducible characters using a cruder version of Kirillov's method of coadjoint orbits. In the past thirty years, these notions have been recognized in several areas (seemingly unrelated to representation theory): exponential sums in number theory, random walks in probability and statistics, association schemes in algebraic combinatorics... In this talk, we will describe and illustrate the main ideas and recent developments of the standard supercharacter theory of adjoint groups of radical rings. We will explore the close relation to Schur rings, and extend a well-known factorization of supercharacters of unitriangular groups which explains the alternative definition as basic characters.
Many results about schemes can be generalized to the non-commutative setting of stable ∞-categories. For bounded weighted categories, things are even better: one can formulate and prove a natural analogue of the theorem of Dundas-Goodwillie-McCarthy which is one of the fundamental tools in studying algebraic K-theory. However, in algebraic geometry bounded weighted categories do not show up very often: for instance, the ∞-category of perfect complexes over a scheme X only admits a reasonable weight structure when X is affine. We introduce a new notion of a c-category which is designed to cover a diverse class of geometric examples, including all quasi-compact quasi-separated schemes, yet allowing for all the weighted arguments to work out in this setting. Our main result shows that a c-category can be resolved in finitely many steps by categories of perfect complexes over connective ring spectra. This allows us to prove an analogue of the DGM theorem for c-categories, as well as the vanishing of their Hochschild homology below a certain degree. We also show that either of the following admits a structure of a c-category:
1. the derived category of any exact (∞-)category that has finite Ext-dimension;
2. the subcategory of compact objects in any weakly approximable stable ∞-category in the sense of Neeman.
In particular, using the computation of the Hochschild homology for c-categories mentioned before we obtain that the category of module-spectra over the ring C*(𝕊2) of cochains over the 2-sphere is not weakly approximable.
By an ultra classical result, the tensor product of a simple representation of 𝔤𝔩n(ℂ) and its defining representation decomposes as a direct sum of simple representations without multiplicities. This means that for each highest weight, the space of highest weight vectors is 1-dimensional. We will give an explicit construction of these highest weight vectors, and show that they arise from the action of certain elements in the enveloping algebra of 𝔤𝔩n(ℂ) + 𝔤𝔩n(ℂ) on the tensor product. These elements are independent of the simple representation we started with, and in fact produce highest weight vectors in several other contexts.
Bridgeland stability conditions were introduced about 20 years ago, with motivations from algebraic geometry, representation theory, and physics. One of the fundamental problems is that we currently lack methods to construct and study such stability conditions in full generality. In this talk, I will present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari, and Zhao. As an application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces, and we prove a conjecture by Kuznetsov and Shinder on quartic double solids.
Given a graded-commutative ring acting centrally on a triangulated category, the main result of this talk shows that if the cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.
Many mathematical and scientific problems concern systems of linear operators (A1,...,An). Spectral theory is expected to provide a mechanism for studying their properties, just like the case for an individual operator. However, defining a spectrum for non-commuting operator systems has been a difficult task. The challenge stems from an inherent problem in finite dimension: is there an analogue of eigenvalues in several variables? Or equivalently, is there a suitable notion of joint characteristic polynomial for multiple matrices A1,...,An? A positive answer to this question seems to have emerged in recent years.
Definition. Given square matrices A1,...,An of equal size, their characteristic polynomial is defined as
QA(z):=det(z0I + z1A1 + ⋯ + znAn), z=(z0,...,zn) ∈ ℂn+1.
Hence, a multivariable analogue of the set of eigenvalues is the eigensurface (or eigenvariety) Z(QA):={z ∈ ℂn+1 ∣ QA(z) = 0}. This talk will review some applications of this idea to problems involving projection matrices and finite-dimensional complex algebras. The talk is self-contained and friendly to graduate students.
Geiss, Keller, and Oppermann introduced n-angulated categories to capture the structure found in certain cluster tilting subcategories in quiver representation theory. Jasso and Muro investigated Toda brackets and Massey products in such cluster tilting subcategories by using the ambient triangulated category. In joint work with Sebastian Martensen and Marius Thaule, we introduce Toda brackets in n-angulated categories, generalizing Toda brackets in triangulated categories (the case n = 3). We will look at different constructions of the brackets, their properties, some examples, and some applications.
This talk will consist of two parts. In the first part, we will see how certain results (such as the Nakayama 'Conjecture') for the symmetric groups and Iwahori-Hecke algebras of type A can be generalised to Ariki-Koike algebras using the map from the set of multipartitions to that of (single) partitions first defined by Uglov. In the second part, we look at Fayers's core blocks, and see how these blocks may be classified using the notation of moving vectors first introduced by Yanbo Li and Xiangyu Qi. If time allows, we will discuss Scopes equivalences between these blocks arising as a consequence of this classification
We will discuss joint work with Victor Ginzburg that proves a conjecture of Nadler on the existence of a quantization, or non-commutative deformation, of the Knop-Ngô morphism, a morphism of group schemes used in particular by Ngô in his proof of the fundamental lemma in the Langlands programme. We will first explain the representation-theoretic background, give an extended example of this morphism for the group GLn(ℂ), and then present a precise statement of our theorem.
Time permitting, we will also discuss how the tools used to construct this quantization can also be used to prove conjectures of Ben-Zvi and Gunningham, which predict a relationship between the quantization of the Knop-Ngô morphism and the parabolic induction functor, as well as an "exactness" conjecture of Braverman and Kazhdan in the D-module setting.
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