Aberdeen Algebra Seminar

The seminar take place at 16:00-17:00 on Monday.

Monday 10.9.18: No Seminar. The Scottish Topology Seminar will meet in Aberdeen.

Monday 17.9.18: Double Seminar.
15:00-16:00 : Theo Raedschelders, University of Glasgow. Room: Meston 009
Title: Rings of differential operators and finite F-representation type
Abstract: For many nice classes of commutative rings, the corresponding differential operators form a simple ring. After explaining some of the history of this problem, I will specialize to the setting of invariant theory, where an old conjecture due to Levasseur and Stafford predicts that rings of invariants for reductive groups in characteristic zero form such a class. The conjecture is known to be true in quite a few examples, but the proofs typically require a very broad range of techniques. Smith and Van den Bergh first considered the corresponding problem in positive characteristic, which appears easier to handle, and showed how this might also lead to a systematic approach in characteristic zero. To do this they introduced the notion of finite F-representation type, and in joint work with Špela Špenko and Michel Van den Bergh, we give the first examples of invariant rings having finite F-representation type for a group which is not linearly reductive.

16:15-17:15 : Bernd Sturmfels, UC Berkley and MPI Leipzig
Abstract: This lecture addresses the role of algebraic geometry in data
science. We report on recent work with Paul Breiding, Sara Kalisnik and
Madeline Weinstein. We seek to determine a real algebraic variety from a
fixed finite subset of points. Existing methods are studied and new methods
are developed. Our focus lies on topological and algebraic features, such
as dimension and defining polynomials. All algorithms are tested on a
range of datasets and made available in a Julia package.

Monday 24.9.18: Angela Tabiri, University of Glasgow
Title: Plane Curves which are Quantum homogeneous spaces.
Abstract: Motivated by classical results for homogeneous spaces, for any plane curve of the form f(y)=g(x), we construct the Hopf algebra A(g,f) which is free over the coordinate ring C of the curve with C being a right coideal subalgebra of A(g,f). This partly answers the conjecture that all plane curves are quantum homogeneous spaces. We use auxiliary Hopf algebras A(x,a,g) and A(y,a,f) to construct A(g,f). Condition are given for when these Hopf algebras have properties such as  being domains, noetherian, having finite Gelfand-Kirillov dimension …

Monday 1.10.18: Lukas Müller, Heriot Watt University, Edinburgh
Title: A TFT perspective on twisted Drinfeld doubles and equivariant extensions.
Abstract: The representation category of the Drinfeld double of a finite group is closely related to low dimensional topology, more precisely to 3-dimensional topological field theories (TFTs).
This leads to a rich and interesting interaction between algebra and topology. In this talk we will discus an equivariant generalization of this relation. We discuss a general construction for G-modular categories from group homomorphisms.
In particular this gives a simple construction for the representation category of the twisted Drinfeld double of a finite group.
Furthermore, we explain how this construction is related to a result of Etingof, Nikshych, Ostrik and Meir applied to the representation category of twisted Drinfeld doubles. The talk is based on joint work with Richard J. Szabo and Lukas Woike.

Monday 8.10.18: No seminar

Monday 15.10.18: Lewis Topley, University of Kent
Title: The maximal dimensions of simple modules over restricted Lie algebras.
Abstract: Restricted Lie algebras were introduced by Jacobson in the 1940’s and ever since the first investigations into their representation theory, it has been understood that the simple modules of a given such algebra have bounded dimensions. In 1971 Kac and Weisfeiler made a striking conjecture (KW1) giving a precise formula for the maximal dimension M(g) of a restricted Lie algebra g.
In this talk I will give a general overview of this theory, and then I will describe a joint work with Ben Martin and David Stewart in which we apply the Leftschetz principle, along with many classical techniques from Lie theory, to prove the KW1 conjecture for all restricted Lie subalgebras of the general linear algebra gl_n, provided the characteristic of the field is much larger than n.

Monday 22.10.18: Jim Belk, University of St. Andrews
Title: Rational Embeddings of Hyperbolic Groups.
Abstract: The rational group R defined by Grigorchuk, Nekrashevych, and Sushchanskii is the group of all homeomorphisms of the Cantor set determined by asynchronous, finite-state automata. Subgroups of R are known as automata groups, and include Grigorchuk’s group and Thompson’s groups. In this talk I will discuss the theory of asynchronous automata groups and sketch a proof that all hyperbolic groups can be represented using finite-state automata. This is joint work with Collin Bleak and Francesco Matucci.

Monday 29.10.18: Ehud Meir, University of Aberdeen
Title: Symmetric monoidal categories, invariant theory and Hopf algerbas.
Abstract: In this talk I will explain a general approach for the study of algebraic structures using symmetric monoidal categories and geometric invariant theory. I will explain a construction of a symmetric monoidal category which is related to generic central simple algebras on the one hand, and to results of Deligne on the other hand. I will then concentrate especially on algebraic structures such as Hopf algebras and Hopf-Galois objects, Hopf orders, and finite groups schemes, and explain what new results can be proven for these structures using the symmetric monoidal categories approach.

Monday 5.11.18: Alexander Shapiro, University of Edinburgh
Title: Cluster structure on quantum Toda chain.
Abstract: Quantum open Toda chain is a quantum integrable system on the Coxeter double Bruhat cell. The latter is known to have a structure of a cluster variety. We will use this structure to write the Baxter operator as a sequence of quantum cluster mutations and obtain a Givental type formula for q-Whittaker functions, the eigenvectors of the quantum Toda Hamiltonians. This is a joint work with Gus Schrader.

Monday 12.11.18: Ben Martin, University of Aberdeen
Title: Reductive pairs
Abstract: Let K be a reductive algebraic group over an algebraically closed field k, and let G be a reductive subgroup of K. We call (K,G) a reductive pair if the Lie algebra Lie(G) has a G-stable complement inside Lie(K). This condition holds automatically if char(k)=0 because then every representation of a reductive group is completely reducible, but it can fail when char(k)>0. R.W. Richardson used reductive pairs to show that a connected reductive group over a field of good characteristic has only finitely many conjugacy classes of unipotent elements. Reductive pairs have also proved useful in the theory of G-complete reducibility. I will discuss some recent joint work with Oliver Goodbourn, giving criteria for (K,G) to be a reductive pair (or not).