Ottawa-Carleton joint algebra seminar (Fall 2015)

Generalized Hecke algebras. 11:30–12:30 in KED B015

Generalized Hecke algebras. 11:30–12:30 in KED B015

Abstract: In this talk I will review the algebraic definition of formal affine Hecke algebra, and its relation with equivariant oriented cohomology of Steinberg variety. Moreover, I will also briefly mention a parallel work, which shows that the elliptic affine Hecke algebra is isomorphic to the equivariant elliptic cohomology of Steinberg variety with some twist.
 

Heisenberg categorification and wreath product algebras

Heisenberg categorification and wreath product algebras

Abstract: The Heisenberg algebra plays a vital role in many areas of mathematics and physics. In this talk, we will discuss a general method for categorifying this algebra. That is, we introduce a family of categories, depending on a Frobenius superalgebra B, whose Grothendieck groups are isomorphic to the Heisenberg algebra. The categories are graphical in nature, consisting of planar diagrams modulo local relations, and they act naturally on the category of finitely generated projective modules over wreath product algebras corresponding to B. Appropriate specializations of B recover results of Khovanov and Cautis-Licata. This is joint work with Daniele Rosso.
 

Extensions For Current Algebras

Extensions For Current Algebras

Abstract: For this talk, a current algebra will be an infinite-dimensional Lie algebra formed as a tensor product of a finite-dimensional, simple Lie algebra over the complex numbers and the polynomial ring in one variable. Current algebras are closely related to affine Kac-Moody algebras. During this talk we will survey some recent results related to extensions between finite-dimensional, irreducible modules for current algebras and report on a work in progress, joint with B. Boe, C. Drupieski and D. Nakano.
 

Milnor-Witt K-theory is the generalized motivic cohomology of a field

Milnor-Witt K-theory is the generalized motivic cohomology of a field

Abstract: Together with Jean Fasel, we introduced generalized motivic cohomology groups of a smooth quasi-projective scheme over a field, and then we proved that in the simplest case, when this scheme is a field, its generalized motivic cohomology groups coincide with its Milnor-Witt K-theory groups (as defined by Morel). I will explain the proof of this isomorphism.
 

This talk is on Monday at 11:30 a.m. in Room FTX 361
Simple Lie conformal algebras

This talk is on Monday at 11:30 a.m. in Room FTX 361
Simple Lie conformal algebras

Abstract: The notion of a Lie conformal algebra (LCA) comes from physics, and is related to the operator product expansion. An LCA is a module over a ring of differential operators with constant coefficients, and with a bracket which may be seen as a deformation of a Lie bracket. LCA are related to linearly compact differential Lie algebras via the so-called annihilation functor. Using this observation and the Cartan's classification of linearly compact simple Lie algebras, Bakalov, D'Andrea and Kac classified finite simple LCA in 2000. I will define the notion of LCA over a ring R of differential operators with not necessarily constant coefficients, extending the known one for R=K[x]. I will explain why it is natural to study such an object and will suggest an approach for the classification of finite simple LCA over arbitrary differential fields.
 

Cohomological Invariants for stacks of algebraic curves

Cohomological Invariants for stacks of algebraic curves

Abstract: Cohomological invariants are arithmetic analogues of characteristic classes in topology, in which singular cohomology is replaced with Galois cohomology, and topological spaces with spectra of fields. Given an affine algebraic group G, a cohomological invariant for G is a way to functorially assign to each principal G-bundle over the spectrum a field k an element of the Galois cohomology of k. These invariants form a graded ring, which has been computed for many classes of algebraic groups by several authors, including Rost, Serre, Merkurjev and many others. In my talk I will show how to extend the classical theory to a theory of cohomological invariants for Deligne-Mumford stacks and in particular for the stacks of smooth genus g curves. The concept of general cohomological invariants turns out to be closely tied to the theory of unramified cohomology, which was introduced by Saltman, Ojanguren and Colliot-Thélène and is widely used to study rationality problems. I will also show how to compute the additive structure of the ring of cohomological invariants for the algebraic stacks of hyperelliptic curves of all even genera and genus three.
 

Optimal control on SO(3)

Optimal control on SO(3)

Abstract: Euler proved in 1776 that every rotation of a 3-dimensional body can be realized as a sequence of three rotations around two given axes. If we allow sequences of an arbitrary length, such a decomposition will not be unique. It is then natural to ask a question about decompositions that minimize the total angle of rotation. In the talk we present a solution to this problem. Orientation of Kepler space telescope is controlled with reaction wheels. In 2013, two of these wheels failed, and as a result Kepler may now be rotated only around two axes. Our theorem provides optimal algorithms for Kepler's attitude control. Other possible applications arise in quantum information theory, where transformations on a single qubit are described by the group SU(2), which is closely related to SO(3).
 

Elliptic affine Hecke algebras and their representations

Elliptic affine Hecke algebras and their representations

Abstract: This is based on a joint work with Changlong Zhong. In my talk, I will explain the basic notions of equivariant elliptic cohomology. I will prove that the convolution algebra of equivariant elliptic cohomology of Steinberg variety is isomorphic to the elliptic affine Hecke algebra constructed by Ginzburg-Kapranov-Vasserot. As an application, we study the Deligne-Langlands theory in the elliptic setting, and classify irreducible representations of the elliptic affine Hecke algebra. The irreducible representations are in one to one correspondence with certain nilpotent Higgs bundles on the elliptic curve. We also study representations at torsion points in type-A.