Last updated: March 13, 2020,
12:18 EDT

Conference Titles, Abstracts, and Slides

Abstract: The Cartan-Killing classification of simple complex Lie algebras is a principal result of the past century. With its four infinite families and five exceptions, it provides a classification both on the level of Lie groups and of algebraic groups. For the latter, this can be extended to a solid theory of group schemes over arbitrary commutative rings. This is useful in geometry and number theory, and achieved through the algebraic geometry developed by the school of Grothendieck.

Exceptional groups have, as natural representations, a number of algebras and other objects that are themselves in some sense exceptional, such as composition algebras, triality, and exceptional Jordan algebras. I will report on recent work, studying these groups and objects over rings from a geometric point of view, using homogeneous spaces (torsors) under exceptional groups. As it turns out, this not only provides a new framework to familiar constructions, but unveils irregularities and new structures that were not expected from the objects' behaviour in the classical setting.

Abstract: "This is joint work with Ted Chinburg and Alex Lubotzky. Let $d\ge 2$, let $F_d$ be a free discrete group of rank $d$, and let $\hat{F}_d$ be its profinite completion. Grunewald and Lubotzky developed a method to construct, under some technical condition, representations of finite index subgroups of $\mathrm{Aut}(F_d)$ that have as images certain large arithmetic groups. In this talk, I will show how we can apply their method to $\mathrm{Aut}(\hat{F}_d)$. In this case, we obtain a stronger result in which we can describe much more precisely the images of the constructed representations and without any assumption. I will also discuss an application of this result to Galois theory. This uses a result by Belyi who showed that there is a natural embedding of the absolute Galois group $G_{\mathbb{Q}}$ of $\overline{\mathbb{Q}}$ over $\mathbb{Q}$ into $\mathrm{Aut}(\hat{F}_2)$. In particular, I will show how the natural action of certain subgroups of $G_{\mathbb{Q}}$ on the Tate modules of generalized Jacobians of covers of $\mathbb{P}^1$ over $\overline{\mathbb{Q}}$ that are branched at 3 points can be extended, up to a finite index subgroup, to an action of a finite index subgroup of $\mathrm{Aut}(\hat{F}_2)$.

Ales M.

Abstract: We are concerned with relating derived categories of all modules of two dual Koszul algebras defined by a locally bounded quiver. We construct a Koszul functor associated with a Koszul algebra defined by a gradable quiver and we obtain a Koszul complex functor, that descends to an equivalence of a continuous family of pairs of triangulated subcategories of doubly unbounded complexes of the respective derived categories of all modules of the Koszul algebra and its Koszul dual. Under this special setting, this extends Beilinson, Ginzburg and Soegelâ€™s Koszul duality. In case the Koszul algebra is right or left locally bounded and its Koszul dual is left or right locally bounded respectively (for instance, the quiver has no right infinite path or no left infinite path), our Koszul duality restricts to an equivalence of the bounded derived categories of finitely supported modules, and an equivalence of the bounded derived categories of finite dimensional modules. This is a joint work with Shiping Liu and Min Huang.

If time permits, I will show that it is possible to relate the bounded derived category of the category of modules over any Koszul algebra with finite global dimension and the bounded derived category of the category of modules over its Koszul dual.

Abstract: In this talk we first address the problem of deciding whether a given quiver representation is semi-stable with respect to a fixed stability weight. We show that this problem can always be solved in deterministic polynomial time. Next, we discuss the membership problem for orbit semi-groups of representations of bound quiver algebras. For a faithful representation W of a tame quasi-tilted algebra, we show that checking whether a stability weight belongs to the orbit semi-group of W can be done in deterministic polynomial time. Finally, applications to Edmonds problem in algebraic complexity theory will be discussed. This talk is based on joint work with Harm Derksen and Dan Kline.

Eric Hanson

Abstract: TBA

Peter JĂ¸rgensen

Abstract: We introduce the green groupoid G of a 2-Calabi--Yau triangulated category C. It is an augmentation of the mutation graph of C, which is defined by means of silting theory. The green groupoid G has certain key properties:

- If C is the stable category of a Frobenius category E, then G acts on the derived categories of the endomorphism rings E(m,m) where m is a maximal rigid object.
- G can be obtained geometrically from the g-vector fan of C.
- If the g-vector fan of C is a hyperplane
arrangement H, then G specialises to the Deligne groupoid
of H, and G acts faithfully on the derived categories of
the endomorphism rings E(m,m).

Abstract: TBA

Ryan Kinser

Abstract: A classical result of Gabriel states that every finite-dimensional algebra over an algebraically closed field K is Morita equivalent to a quotient of a path algebra of a quiver. This can be viewed as a classification of algebras in the tensor category of finite-dimensional K-vector spaces. An algebra equipped with additional structure (e.g. an action of a finite group on A, or a grading of A by a finite group, or a twisting of the associativity of A) can be viewed as an algebra in a more interesting tensor category. I will give an introduction to the basic notions of tensor algebras in finite tensor categories, and state our classification result for them.

One advantage of this approach is that "categorical Morita equivalence" for tensor categories can be used to translate solutions between apparently quite different problems (e.g. classification of fin. dim. algebras with an action of a given finite group G is essentially the same as classification of fin. dim. algebras graded by G, up to an appropriate notion of equivalence). Finally, I will illustrate everything with a very concrete classification example coming from the 8-dimensional Kac-Paljutkin Hopf algebra. This talk is based on joint work with Pavel Etingof and Chelsea Walton.

Abstract: Let V be an n-dimensional representation of a group G (over an algebraically closed field of char 0). Starting from a set of generators for the invariant polynomials for the action of G on V^n, one can construct (via polarization) a generating set for the invariant polynomials for the action of G on V^m for any m, no matter how large. This is a fundamental result in classical invariant theory and was proven by Hermann Weyl.

Weylâ€™s result however fails in positive characteristic. For a large class of representations that includes quiver representations, and a modest lower bound on characteristic, we show that Weylâ€™s result still continues to hold. Key ideas are Hashimoto's results on good filtrations, Seshadri's work on geometric reductivity over arbitrary base, and new combinatorial result on Schur modules in positive characteristic. This is joint work with Harm Derksen.

Abstract: Quantum affine algebra and Yangian are important classes of quantum groups. Both were invented in the early 80â€™s by L. Faddeev, V. Drinfeld and M. Jimbo in order to study the quantum inverse scattering problem and to solve the quantum Yangâ€“Baxter equation. Since then, their investigation has been a very active research area with numerous important applications in mathematical physic, representation theory, and recently in the theory of cluster algebra with the work of D. Hernandez and B. Leclerc. Finite-dimensional representations of Yangian were studied by many people such as A. Molev, V. Chari, A. Pressley, and N. Guay. However, many aspects of the infinite-dimensional representation theory of Yangian are not fully developed. In particular, the global Weyl module for Yangian has not yet been defined in the current literature. In this talk, we will give a construction for the global Weyl module of Yangian associated to any complex simple Lie algebra and describe its properties. This is a joint work with M. Lee and P. Senesi.

Abstract: TBA

Abstract: Maximal green sequences are
certain transformations of quivers that were first introduced
by Keller in the context of cluster algebras. Later they were
generalized to the setting of finite dimensional algebras,
where a maximal green sequence is a finite maximal chain in
the lattice of torsion classes. More recently, it was
shown that these sequences are in bijection with forward
hom-orthogonal sequences of bricks in the module
category. We use the latter approach to study
existence and number of maximal green sequences for string
algebras. This is joint work with A. Garver.

Abstract: We introduce a class of algebras, the Ore-solvable algebras, and discuss recent results on their (strict) upper-triangularizability. Ore-solvable algebras include many examples as particular cases, such as group algebras of polycyclic groups or finite solvable groups, enveloping algebras of solvable Lie algebras, quantum planes and quantum matrices. The triangularization theorems we discuss can be seen to generalize and extend classical simultaneous triangularization results, such as the Lie and Engel theorems for Lie algebras. We also discuss connections between strict triangularization and nil/nilpotent algebras, including a result for algebras defined via a recursive ``Ore'' procedure on commutative, finite-dimensional and nilpotent algebras.

Abstract: I'll describe work in progress with Hugh Thomas, describing real bricks for preprojective modules and their stability domains.

Abstract: We describe a general theory of the prime spectrum of non-braided monoidal triangulated categories. These notions are a noncommutative analogue to Paul Balmer's prime spectra of symmetric tensor-triangulated categories. Noncommutative monoidal triangulated categories appear naturally as stable module categories for non-quasitriangular Hopf algebras and as derived categories of bimodules of noncommutative algebras. In stable module

categories of Hopf algebras, the support theory of the category, as described by Benson-Iyengar-Krause, is linked to the Balmer spectrum, which is shown to be the final support datum. We will describe how this connection can be used to compute Balmer spectra in general, and we will compute the Balmer spectra for stable module categories of the small quantum group of a Borel subalgebra at a root of unity, and the stable module categories for smash coproduct Hopf algebras of group algebras with coordinate rings of groups. This is joint work with Daniel Nakano and Milen Yakimov.

Abstract: TBA

Nick Williams

Abstract: Unlike for classical tilting modules, in the cluster-tilting of Buan, Marsh, Reineke, Reiten, and Todorov, one may always mutate every summand of a cluster-tilting object. Cluster categories were generalised in the context of Iyamaâ€™s higher Auslanderâ€“Reiten theory by Oppermann and Thomas. In these higher cluster categories, it is no longer possible to mutate a cluster-tilting object at every summand. We give a new criterion for which summands may be mutated.

Abstract: Let X be a smooth projective curve over an algebraically closed field and let G be a finite group acting on X. The space of holomorphic m-polydifferentials is defined to be the space of global sections of the m-fold tensor product of the sheaf of relative differentials with itself. This space provides a representation of G and a classical problem is to determine the decomposition of the space of holomorphic polydifferentials as a direct sum of indecomposable representations. We discuss work on this problem in the case when k has prime characteristic p and G has cyclic Sylow p-subgroups. The results have implications for cusp forms on modular curves as well as for the dimension of the tangent space of the deformation functor of curves.

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