Schedule for: 23w5047 - Silting in Representation Theory, Singularities, and Noncommutative Geometry

This is part of a joint project with Daniel Labardini-Fragoso and Jon Wilson. Let Σ be a marked surface with non-empty boundary together with a tagged triangulation T. Let A(T) be the corresponding Jacobian algebra. We show that in this situation we have an isomorphism of partial KRS-monoids Π_T: DecIrr^τ(A(T)) --> Lam(Σ), which intertwines generic g-vectors with (dual) shear coordinates. Such a result was previously only known for surfaces without punctures. In any case, it completes the known bijection between the basic support τ-tilting modules of A(T) and tagged triangulations of Σ.

During this talk we discuss the classification of spherical objects in various geometric setting including minimal resolution of surface ADE singularities and threefold flopping contractions with at worst Gorenstein terminal singularities. Namely, the main theorem classifies (a certain generalisation of) spherical objects in the associated null category. The technique in the proof comes from the analogy with the derived category of silting discrete algebras. The same technology can be applied to the classification of bounded t-structures of the null category. This talk is based on the joint work with Michael Wemyss.

Almost gentle algebras in the sense of Green and Schroll generalize gentle algebras by allowing more arrows at each vertex. For a class of rings including such algebras, the classification problem of silting complexes can be reduced to the study of rigid modules in a certain hereditary category. This reduction is based on the gluing technique by Burban and Drozd and Simson's approach to matrix problems.

Please submit a title for a 5 minute talk and send it to the organizers by Tuesday 13:00.

The Donovan-Wemyss Conjecture predicts that the isomorphism type of an isolated compound Du Val singularity R that admits a crepant resolution is completely determined by the derived-equivalence class of any of its contraction algebras. Crucial results of August, Hua-Keller and Wemyss reduced the DW conjecture to a problem closely related the question of uniqueness of enhancements of the singularity category of R. I will explain, based on an observation by Bernhard Keller, how the DW conjecture follows from a recent theorem of Fernando Muro and myself that we call the Derived Auslander-Iyama Correspondence.

In this talk we will explain how a simple tilt of any intermediate t-structure in the bounded derived category of an Artin algebra corresponds to an irreducible mutation of a cosilting set in the unbounded derived category. The talk will be based on joint work with Lidia Angeleri Hügel and Francesco Sentieri.

Cluster varieties are divided into two types: A-varieties (also known as cluster K2 varieties) and X-varietes (also known as cluster Poisson varieties). Each cluster variety has an associated cluster complex. The g-fans arising in the theory of cluster algebras provide a fan realization of the A-cluster complexes. In this talk I will motivate the study of the cluster complexes associated to cluster Poisson varieties. Then, in the acyclic case, I will provide a description of these complexes using c- and g-vectors. This is based on a joint project in preparation with Carolina Melo.

I will explain the concept of lax additivity which is an (infty,2)-categorical analog of the familiar 1-categorical notion of additivity. In this context, direct sums get replaced by lax sums leading to lax variants of matrices along with rules for how to multiply them. We will illustrate the resulting methods by investigating periodicitiy phenomena for mutations of semiorthogonal decompositions. Based on joint work with Christ-Walde and Kapranov-Schechtman.

A long-standing expectation in the study of total positivity, recently confirmed by Galashin and Lam, is that the cells in the positroid stratification of the Grassmannian have cluster algebra structures. The eventual construction actually provides two different such structures, but a conjecture by Muller and Speyer asserts that they should have a close relationship known as quasi-coincidence. In this talk, I will outline a proof of the conjecture. Despite the statement being about combinatorial geometry, the proof works by translating it into the language of representation theory, and then applying techniques from homological algebra: ultimately, the quasi-coincidence of the cluster structures follows from a statement about the derived category of a certain non-commutative Gorenstein ring.

Braid varieties are a class of algebraic varieties associated to elements in the positive braid monoid. They generalize Richardson varieties in the flag variety and double Bott-Samelson cells, among others. We construct a cluster algebra structure on the coordinate ring of a braid variety, using the combinatorics of algebraic weaves and cycles on them. This cluster structure turns out to be very nice, for example, it is locally acyclic and admits a reddening sequence. In particular, this gives a cluster structure on Richardson varieties. This is joint work with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le and Linhui Shen.

This talk is centered around the behavior of bricks over finite dimensional algebras. Following a conjecture that I first posed in 2019, we are primarily interested in those finite dimensional algebras $A$ over algebraically closed field $k$ which admit infinitely many isoclasses of bricks (i.e, finitely generated left $A$-module with $End_A(M)\simeq k$). Due to the conceptual analogy to the classical 2nd Brauer-Thrall conjecture (now theorem), we call the new statement the 2nd brick-Brauer-Thrall (2nd bBT) conjecture: If $A$ is a brick-infinite algebra, there is a positive integer $d$ for which $A$ admits infinitely many non-isomorphic bricks of length $d$. The 2nd bBT conjecture remains open in full generality. In this talk we look at some recent progress on this problem and show some direct consequences of the 2nd bBT conjecture. That is, we provide further evidence for the correctness of the new conjecture. Furthermore, I will highlight some significant contributions that the 2nd bBT conjecture provides to the study of geometry of moduli spaces of representations. Part of this talk is based on my joint work with Charles Paquette.

From the work of Adachi, Iyama and Reiten we know that there is a bijection between functorially finite torsion classes, support tau-tilting pairs, and 2-term silting complexes. In this talk we investigate to which extent this can be generalized to higher Auslander—Reiten theory. In particular, we explain how any functorially finite d-torsion class gives rise to a (d+1)-term silting complex. This is joint work with Jenny August, Johanne Haugland, Karin M. Jacobsen, Yann Palu, and Hipolito Treffinger.

I will first describe the space of stability conditions on a 3-Calabi-Yau triangulated category associated to a quiver with potential. I will then describe how, in a large class of examples, this space is related to the Teichmüller space of a surface. I conjecture that this is a special case of a general relationship between spaces of stability conditions and the positive real loci of cluster varieties.

The notion of an exceptional non-commutative curve was introduced by Lenzing in 1998. These are non-commutative hereditary projective curves, whose category of coherent sheaves has a tilting object. Basic examples of such curves are weighted projective lines of Geigle and Lenzing. However, the class of exceptional curves is not exhausted by weighted projective lines when the ground field is not algebraically closed. In my talk I shall discuss a specific example of an exceptional curve over a finite field, which is half ruled regular in the sense of a non-published work of Artin and de Jong. I shall prove that the composition Hall algebra of the corresponding category of coherent sheaves can be identified with the Drinfeld realization of the quantum affine algebra of type A_2^2. This is a joint work in progress with Heike Herr.

One of the well-known theorems of Maurice Auslander about artin algebras describes the correspondence between {algebras A of finite representation type} and {algebras B, with gl.dim.B ≤ 2 ≤ dom.dim.B} which are now called Auslander algebras. Higher Auslander algebras were introduced by O. Iyama as {algebras C, with gl.dim.C ≤ k ≤ dom.dim.C} and it was also shown that there is correspondence between {higher representation finite algebras} and {higher Auslander algebras} (with certain equivalences - there is more precise statement). I would talk about recent work of Emre Sen who had created several approaches for constructing new families of higher Auslander algebras and as a consequence he obtained new higher representation finite algebras. In addition to that Emre, Shijie Zhu and I, have a joint paper in which we have another method of constructing higher Auslander algebras in the family of Nakayama algebras. Higher Auslander algebras, higher representation finite algebras and higher cluster tilting modules are still not well understood, even in the families of well known classes of algebras, so creating families of such will be useful.