Algebra

This talk will not assume any knowledge of derived categories.
Two rings are "Morita equivalent" if they have equivalent module categories, and
if a property of rings depends only on the module category, then it is called
"Morita invariant". More generally, two rings are "derived equivalent" if they
have equivalent derived categories, and if a property of rings depends only on
the derived category, then it is called "derived invariant".
Fairly recently, Manuel Saorin asked me if I knew whether right coherence was a
derived invariant property. I didn't, but when my long term memory kicked in, I
realised that an example hidden in my 36 year old PhD thesis could be used to
give a counterexample.
Most of the talk will be an introduction to the background to this question and
variants, but it will finish with some fun examples of rings with strange
properties.

Counting the number of subgroups in a finite group has numerous applications, ranging from enumerating certain classes of finite graphs (up to isomorphism), to counting how many isomorphism classes of finite groups there are of a given order. In this talk, I will discuss the history behind the question; why it is important; and what we currently know.

A large class of algebras (those admitting quasi-Cartan subalgebras) can be reconstructed from naturally occurring inverse semigroups and groupoids. I will describe my work with collaborators comparing constructions of groupoids from inverse semigroups that supported this reconstruction. I will not assume any background on inverse semigroups or groupoids.

We all know that quandles are great, but there seems to have been resistance to their use as knot invariants because they're too hard to work with.  In this talk I will explain an older result of mine with Sheila Miller, giving rigorous mathematical substance to this sociological observation: we show that, in the framework of "Borel reducibility", the isomorphism problem for quandles is as complicated as it could possibly be.  The proof boils down to a very hands-on, combinatorial construction of a quandle from any given graph.

Exceptional sequences were first introduced in triangulated categories by the Moscow school of algebraic geometry. Later, Crawley-Boevey and Ringel studied exceptional sequences in the module categories of hereditary finite-dimensional algebras. Motivated by tau-tilting theory introduced by Adachi, Iyama, and Reiten, Jasso’s reduction for tau-tilting modules, and signed exceptional sequences introduced by Igusa and Todorov, Buan and Marsh developed the theory of (signed) tau-exceptional sequences – a natural generalization of (signed) exceptional sequences that behave well over arbitrary finite-dimensional algebras.

In this talk, we will study (signed) tau-exceptional sequences over the algebra Λ=RQ, where R is a finite-dimensional local commutative algebra over an algebraically closed field, and Q is an acyclic quiver. I will explain how (signed) tau-exceptional sequences over Λ can be fully understood in terms of (signed) exceptional sequences over kQ.

Over an algebraically closed field, Gabriel’s theorem states that the path algebra kQ of a connected quiver is representation-finite if and only if the underlying graph of Q is an ADE Dynkin diagram. Equivalently, kQ is representation-finite precisely when the preprojective algebra of Q is finite-dimensional.

d-Representation-finite (d-RF) algebras, introduced by Iyama and Oppermann, are a generalisation of representation-finite path algebras. Attached to each d-RF algebra is a (d+1)-preprojective algebra. Grant showed that a d-RF algebra is fractional Calabi-Yau if and only if the Nakayama automorphism of its (d+1)-preprojective algebra has finite order.

In this talk, we present a family of algebras which arise from the well-studied 3-preprojective algebras of type A by “taking orbifolds”. We show that a subset of these are themselves 3-preprojective algebras (of type D). Thus we provide new examples of 2-RF algebras, which we show are also fractional Calabi-Yau.

I shall explain Theorem (127) which determines the unique bijection BETWEEN the monomial basis called $(A_n)$- FC elements  AND the set of non-crossing diagrams of $n+1$ strings OF  our well-known Temperley-Lieb algebra, that respects the  diagrammatic multiplication by concatenation along with the two algorithms implementing this bijection and its inverse, in other terms : "Drawing" a basis element into a diagram and "writing" a diagram as member in the monomial basis.

Naturally  we shall find ourselves in CataLand, so I will try to give -as much as our time allows me- some consequences and open problems coming from the above holly marriage, that is to explain why did I commit this work (other than the obvious reasons). The talk is pretty simple & basically addressed to our Ph.D students, accessible to Master students and in which there is an introduction to my talk next week.

A gentle quiver is the data of a finite connected directed graph together with a collection of paths of length two satisfying additional conditions. A resolving subcategory of its representations is an additive subcategory that contains the projective objects and is closed by extensions and epimorphism kernels. In our framework, such a subcategory can be described combinatorially via a collection of indecomposable representations stable under some computational conditions.

In this algebraic context, a goal is to describe all resolving subcategories. To this end, we restrict ourselves to gentle trees (the directed graph is a tree) and use a geometric model to see indecomposable representations as curves on a disk. We then construct an algorithm that will enable us to compute them explicitly.

After reviewing all the essential notions and giving some motivations to understand the context, I will explain how we first describe the monogeneous resolving subcategories (generated by a single indecomposable nonprojective representation). Then, I will give some words on the design of the algorithm that allows the construction of all the resolving subcategories of any gentle tree. If time allows, I will share some expectations we can have following those results (link with tilting representations, generalization to gentle quivers, graduated cases, etc...) — all of this with combinatorial and geometrical perspectives.

This is a joint work in progress with Michael Schoonheere.

Stability conditions are an important tool in algebraic geometry for constructing moduli varieties. When applied to the varieties of modules over a finite-dimensional algebra, they give rise to the algebraic notion of semistable modules, which are closely linked to $tau$-tilting theory and cluster algebras. To find these semistable modules, one can compute a special class of regular functions known as determinantal semi-invariants. In this talk, we will revisit the relation of these semi-invariants to projective presentations and explore semistability for varieties of projective presentations. We will recall that determinantal semi-invariants give rise to two interesting types of subcategories, namely, wide subcategories of the module category and thick subcategories of the extriangulated category of projective presentations. Finally, we will introduce an extriangulated version of the correspondences among support $tau$-tilting objects, torsion classes, and wide subcategories. This correspondence extends classical results to the context of projective presentations.

Quinn's Finite Total Homotopy TQFT is a TQFT (topological quantum field theory) defined for any dimension, $n$, of space, and that depends on the choice of a homotopy finite space, $B$. For instance, $B$ can be the classifying space of a finite group or of a finite 2-group.
I will report on recent joint work with Tim Porter on once-extended versions of Quinn's Finite total homotopy TQFT, taking values in the (symmetric monoidal) bicategory of groupoids, linear profunctors, and natural transformations between linear profunctors. The construction works in all dimensions, thus in particular it yields (0,1,2)-, (1,2,3)- and (2,3,4)-extended TQFTs, any time we are given a homotopy finite space $B$. I will show how to compute these once-extended TQFTs for the case when $B$ is the classifying space of a homotopy 2-type, represented by a crossed module of groups.

References: Faria Martins J, Porter T : A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids. arXiv:2301.02491 [math.CT].