Speaker: Cameron Ruether, Memorial University of Newfoundland
Time/Date:
Wednesday, March 8, 2023, 1 p.m.,
Wednesday, March 22, 2023, 1 p.m.
Room: HH-3017
Title: Cohomology and quadratic pairs on schemes - Parts 1 & 2

Abstract:
We begin by reviewing the setting of working with algebras over a scheme (or more generally over a ringed site) and reviewing Cech cohomology of abelian sheaves. We will then ˇ return to the topic of quadratic pairs, now over schemes, and discuss some cohomological obstructions which may appear only when the base scheme is not affine. Time permitting, we will sketch the construction of some explicit examples with non-trivial obstructions.

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Speaker: Cameron Ruether, Memorial University of Newfoundland
Time/Date:
Wednesday, February 15, 2023, 1 p.m.,
Wednesday, March 1, 2023, 1 p.m. 
Room: HH-3017
Title: Cohomological obstructions to quadratic pairs over schemes - Parts 1 & 2

Abstract:
The concept of quadratic pair was generalized by Calm ́es and Fasel to the setting of Azumaya algebras over an arbitrary base scheme, also with groups of type D in mind. We will review these definitions before discussing recent work with Philippe Gille and Erhard Neher. We define two cohomological obstructions attached to an Azumaya algebra with orthogonal involution. The weak obstruction prevents the existence of a quadratic pair, and the strong obstruction prevents potential quadratic pairs from being described as in the field/ring case. Interestingly, both these obstructions are trivial over affine schemes, and so quadratic pairs have noticeably different behaviour when working over arbitrary schemes. To demonstrate that this behaviour is possible, we will also present examples where one or both obstructions are non-trivial.

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Speaker: Cameron Ruether, Memorial University of Newfoundland
Time/Date: Wednesday, January 18, 2023, 1 p.m.
Room: HH-3017
Title: Quadratic pairs over fields and rings

Abstract:
Quadratic pairs on a central simple algebra over a field were introduced by Knus, Merkurjev, Rost, and Tignol in “The Book of Involutions” in order to work with semisimple linear algebraic groups of type D in characteristic 2. In this first talk we will review the definition, basic properties, and classification of quadratic pairs. Many of these results still hold when working over an arbitrary base ring instead of a field, so we will discuss this generalization as well - in preparation for the second talk.

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Speaker: Alberto Daza García, University of Zaragoza
Time/Date: Wednesday, November 9, 2022, 1 p.m.
Room: HH-3017
Title: Tensor categories and superalgebras
         (Joint work with Alberto Elduque and Umut Sayin)

Abstract:
In this talk we will focus on three tensor categories over on a field of characteristic 3: the category $\mathrm{Rep}\,C_3$ of finite dimensional representations of the cyclic group of order 3, the Verlinde category $\mathrm{Ver}_3$, and the category sVec of finite dimensional super vector spaces. In each of these categories, we can define the concept of an algebra. For example, an algebra in $\mathrm{Rep}\,C_3$ is equivalent to an ordinary algebra together with an automorphism of order 3, while an algebra in sVec is a superalgebra.

In characteristic 3, the category $\mathrm{Rep}\,C_3$ is not semisimple, but we can use the concept of semisimplification, introduced in 1999 by J.W. Barrett and B.W. Westbury, to define a new semisiple category ($\mathrm{Ver}_3$) together with a functor S: $\mathrm{Rep}\,C_3\to\mathrm{Ver}_3$ called the semisimplification functor. Moreover, $\mathrm{Ver}_3$ is equivalent to the category of super vector spaces, so we can use this to construct a superalgebra starting from an algebra. This is similar to the idea used by A. Kannan to construct exceptional simple Lie superalgebras.

The main purpose of this talk will be to introduce a recipe to construct superalgebras from algebras with an automorphism of order 3. We will illustrate this by constructing one of the two non unital composition superalgebras (with non trivial odd part) starting from the split Cayley algebra and an automorphism of order 3.

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Speaker: Pavel Holba, Silesian University, Czech Republic
Time/Date: Friday, November 4, 2022, 3 p.m.
Room: HH-3017
Title: Complete classification of local conservation laws for generalized CHKSequations

Abstract: In this talk we consider nonlinear multidimensional Cahn--Hilliard and Kuramoto--Sivashinsky equations that have many important applications in physics, chemistry, and biology, and a certain natural generalization of these equations. For an arbitrary number of spatial independent variables we present a complete list of cases when this PDE in $n+1$ independent variables $t, x_1, \ldots, x_n$ and one dependent variable $u$
\[u_t = a \Delta^2 u + b(u)\Delta u + f(u)\lvert \nabla u \rvert^2 + g(u),\]
namely generalized Cahn--Hilliard--Kuramoto--Sivashinsky equation, admits nontrivial local conservation laws of any order, and for each of those cases we give an explicit form of all the local conservation laws of all orders modulo trivial ones admitted by the equation under study. Here $b, f, g$ are arbitrary smooth functions of the dependent variable $u$, $a$ is a nonzero constant, $\Delta = \sum_{i=1}^n \partial^2/\partial x_i^2$ is the Laplace operator and $\lvert \nabla u \rvert^2 = \sum_{i=1}^n (\partial u/\partial x_i)^2$.

In particular, we show that the Kuramoto--Sivashinsky equation admits no nontrivial local conservation laws and list all of the nontrivial local conservation laws for the original Cahn--Hilliard equation.

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Speaker: Yuri Bahturin, Memorial University of Newfoundland
Time/Date:
Wednesday, October 19, 2022, 1 p.m.
Wednesday, November 23, 2022, 1 p.m.
Room: HH-3017
Title: Nilpotent algebras, implicit function theorem, and polynomial quasigroups I & II (Joint work with Alexander Olshanskii)

Abstract:
We study finite-dimensional nonassociative algebras. We prove the implicit function theorem for such algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of the classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras.

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Speaker: Alberto Daza García, University of Zaragoza
Time/Date: Wednesday, October 5, 2022, 1 p.m.
Room: HH-3017
Title: Structurable algebras, automorphisms and gradings

Abstract:
Structurable algebras are a class of algebras with involution. They are a generalization of Jordan algebras in the sense that they admit a Tits-Kantor-Koecher (TKK) construction of a Lie algebra.

In this talk we introduce structurable algebras and discuss one of their classes, namely,the class of central simple structurable algebras related to an hermitian form. We will see their relation to associative algebras with involution and how it can be used to calculate their automorphisms and gradings. We will finish by discussing a special algebra in this class which behaves somewhat differently from the others: the split quartic Cayley algebra.