Click here for the School Schedule
Venue: Room L4E48, located on floor E of Lab 4, on the OIST Campus (click for campus information and floor map)
Mini-courses by Osamu Iyama, Daniel Labardini-Fragoso, Charles Paquette, Hugh Thomas
(each mini-course consists of four 1-hour lectures)
Osamu Iyama: Silting theory and Grothendieck groups
This mini-course will discuss the following topics:
τ-tilting theory and silting theory (basics)
g-vector fans and polytopes in tilting theory
TF-equivalence and canonical decompositions
Lecture Notes: Complete Series of 4 Lectures
Daniel Labardini-Fragoso: Tame algebras arising from surfaces
The construction of various kinds of algebras from combinatorial or geometric information drawn on a compact differentiable two-dimensional real manifold with boundary, i.e., a surface, has been a very fruitful research stream for decades already. The algebras produced are predominantly tame. In this mini-course we will explore these constructions, as well as the striking ways in which the underlying surfaces provide geometric and combinatorial models that allow to visualize objects and prove phenomena from cluster algebras, tau-tilting theory, the geometry of representation varieties, derived categories and Fukaya categories.
Charles Paquette: Algebraic and geometric aspects of bricks, and different notions of tameness
A module over a finite dimensional algebra is called a brick if its endomorphism algebra is a division algebra. Understanding the bricks associated with a given algebra can provide significant insights into various aspects of the algebra, including its torsion theory, its τ-tilting theory, the moduli spaces of its representations, its invariant theory and its wide subcategories. In particular, bricks control some combinatorial, algebraic and geometric aspects of the representation theory of a finite dimensional algebra. This mini-course will explore the emergence of bricks in different contexts of the representation theory of an algebra, and how some of those are interconnected. By focusing on bricks and related concepts, one can consider many different notions of tameness for an algebra, including classical tameness, brick-tameness, g-tameness, E-tameness and τ-tilting tameness, just to name a few. After brick-finite algebras (i.e. those algebras having only finitely many bricks, up to isomorphism), those algebras that are tame with respect to one of these notions form natural classes of algebras to study. In the second part of the course, we will define all of these notions of tameness and consider their relationships and properties (especially the properties related to bricks). Many open problems will be presented.
Hugh Thomas: τ-tilting theory and the physics of scattering amplitudes
A new approach to the study of scattering amplitudes in quantum field theory has led to surprising connections to the representation theory of finite-dimensional algebras, and, more specifically, tau-tilting theory. The u-equations were introduced by Koba and Nielsen in 1969, and play a fundamental role in their definition of the string tree amplitude. I will explain how the u-equations of Koba and Nielsen are associated to a type A Dynkin diagram, and how it is possible to consider u-equations associated to other finite-dimensional algebras. I will focus in particular on three settings (i) finite representation type, (ii) gentle and locally gentle algebras, and (iii) arbitrary finite-dimensional algebras. This will also be an opportunity to discuss various features of tau-tilting theory as they become relevant. The applications in scattering amplitudes will be sketched in a way that does not require any background knowledge of physics. This minicourse will report primarily on a joint project with Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, and Giulio Salvatori, along with some earlier results, and necessary background from tau-tilting theory.
Exercise sessions & Discussion groups
You can also ask your questions through this online form!
You can see the questions already posed here!
Day 1 (Click for Some Exercises)
Day 2 (Click for Some Exercises)
In addition to the lectures, participants are encouraged to actively engage in other components of the school. In particular, after a couple of exercise sessions on the first two days, starting from day 3 of the school, participants will be divided into smaller groups and can collectively work on some research problems closer to their interests. The problems that will be treated during the discussion sessions will be primarily posed by the mini-course lecturers. Until the end of the conference week, each group can benefit from the discussion sessions and free slots to work on their project.
