Last updated: March 13, 2020,
12:18 EDT
Conference Titles, Abstracts, and Slides
Seidon Alsaody, A Geometric
approach to exceptional groups and exceptional algebras.
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.
Frauke Bleher, Representations of automorphism
groups of free profinite groups.
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. Bouhada, Koszul duality in
the non-graded case.
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 Soegel’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.
Calin Chindris, Constructive aspects of
semi-stability of quiver representations.
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, A counterexample to the đťś‘-dimension conjecture.
Abstract: TBA
Souheila Hassoun, The lattice of exact structures.
Abstract: TBA
Peter Jørgensen, The green groupoid and its action
on derived categories.
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).
The situation in (3) occurs if \Sigma_{C }^2,
the square of the suspension functor, is the identity. It
recovers results by Donovan, Hirano, and Wemyss where E is the
category of maximal Cohen--Macaulay modules over a suitable
singularity.
Bernhard Keller, Categorifying cluster algebras
with invertible coefficients.
Abstract: TBA
Ryan Kinser, Tensor algebras in finite tensor
categories.
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.
Visu Makam, Weyl's polarization theorem in positive
characteristic.
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.
Weyl’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 Weyl’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.
Bach Nguyen, Global Weyl module of quantum affine
algebra and Yangian.
Abstract: Quantum affine algebra and Yangian are important
classes of quantum groups. Both were invented in the early
80’s by L. Faddeev, V. Drinfeld and M. Jimbo in order to
study the quantum inverse scattering problem and to solve
the quantum Yang–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.
Job Rock, Cluster theories and continuous mutation.
Abstract: TBA
Krystyna Serhiyenko, Maximal
green sequences for string algebras.
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.
Alexander Sistko, Ore extensions and
infinite triangularization.
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.
David Speyer, Real bricks for preprojective
algebras.
Abstract: I'll describe work in progress with Hugh Thomas,
describing real bricks for preprojective modules and their
stability domains.
Yadira Valdivieso DĂaz, Skew-gentle
algebras and orbifolds.
Abstract: Skew-gentle algebras, a generelisation of gentle
algebras, naturally appear in many different contexts such as
in the framework of cluster algebras where they arise as
Jacobian algebras of certain triangulations of surfaces with
punctures. In this talk we will give a geometric model of the
bounded derived category of a skew-gentle algebra in the terms
of graded curves in an generelised orbifold dissection with
orbifold points of order two with boundary and punctures. We
show that the geometric model of a skew-gentle algebras is
closed related to the model of the underlying gentle algebra
defined in joint work with Opper-Plamonodon-Schroll and which
by work of Haiden-Katzarkov-Kontsevich and Lekili-Polishchuk
is closely linked with the partially wrapped Fukaya category
of a surface with stops. This is a report on joint work in
progress with Sibylle Schroll and Daniel Labardini-Fragoso.
Kent Vashaw, Noncommutative
tensor triangular geometry.
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.
Jose Velez Marulanda,
String and band complexes for a class of Brauer graph
algebras.
Abstract: TBA
Nick Williams, Mutation in higher cluster categories.
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
Iyama’s higher Auslander–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.
Adam Wood, Module structure of the space of
holomorphic polydifferentials.
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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