Department of Mathematics and Statistics

Department of Mathematics and Statistics
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Geometry and Representation Theory Seminar

Geometry & Representation - Souheila Hassoun (Sherbrooke)

Monday, November 30th, 2020

Time: 4:30p.m.  Place: Zoom

Speaker: Souheila Hassoun (Université de Sherbrooke)

Title: Jordan-Hölder exact categories

Abstract:  Exact categories goes back to the work of Yoneda and generalise the important and widely used notion of abelian categories. In a joint work with T. Brüstle and A. Tattar, we generalise the famous Jordan-Hölder theorem to the realm of Quillen exact categories.

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Geometry & Representation - Kaveh Mousavand (Queen's University)

Monday, October 26th, 2020

Time: 4:30p.m.  Place: Zoom

Speaker: Kaveh Mousavand (Queen's University)

Title: A Categorification of Biclosed Sets of Strings.

Abstract:  In joint work with A. Garver and T. McConville, for any gentle algebra of finite representation type we studied a closure operator on the set of strings. Shortly before, Palu, Pilaud, and Plamondon had proved that the collection of all biclosed sets of strings forms a lattice, and moreover, that this lattice is congruence-uniform. Many interesting examples of finite congruence-uniform lattices may be represented as the lattice of torsion classes of an associative algebra. To extend this result, we introduced a generalization-- the lattice of torsion shadows-- and proved that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows.

Furthermore, we introduced the analogous notion of wide shadows to extend wide subcategories, so we could find a new realization for another important lattice theoretical phenomenon. In fact, we showed the shard intersection order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows. Consequently, in our setting, we established a bijection between torsion shadows and wide shadows, analogous to those between functorially finite torsion classes and wide subcategories studied by Ingalls-Thomas and Marks-Stovicek.

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Geometry & Representation - Kaiming Zhao (Wilfrid Laurier)

Monday, October 5th, 2020

Time: 4:30p.m.  Place: Zoom

Speaker: Kaiming Zhao (Wilfrid Laurier University, Waterloo, Canada)

Title: Simple modules over Witt algebras Wd.

Abstract:  Let W_d be the Witt algebra, that is, the derivation Lie algebra of the Laurent polynomial algebra $A_d=C[x_1^{\pm1},x_2^{\pm1}, . . .,x_d^{\pm1}]$. Let H be the Cartan subalgebra of W_d. All simple W_d-modules that are U(H)-free modules of finite rank will be given.

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Geometry & Representation - Michael Perlman (Queen's University)

Monday, September 28th, 2020

Time: 4:30p.m.  Place: Zoom

Speaker: Michael Perlman (Queen's University)

Title: Equivariant D-modules on spaces with finitely many orbits.

Abstract:  Let X be a variety endowed with the action of a linear algebraic group G. Given an orbit, the local cohomology modules with support in its closure encode a great deal of information about its singularities and topology. We will discuss how techniques from representation theory and the theory of D-modules (and quivers!) may be used to compute these local cohomology modules in the case when G acts with finitely-many orbits.

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Geometry & Representation - Atabey Kaygun (Istanbul Technical University)

Monday, September 21st, 2020

Time: 4:30p.m.  Place:

Speaker: Atabey Kaygun (Istanbul Technical University)

Title: Generalized Weyl Algebras.

Abstract:  Generalized Weyl algebras (GWAs) form a class of algebras whose representation theory resembles those of Lie algebras. They are defined by Bavula and Hodges and found extensive use in noncommutative resolutions of Kleinian singularities. Apart from noncommutative resolutions of Kleinian singularities, the class is known to contain the ordinary Weyl algebras, the enveloping algebra U(sl2) and its primitive quotients, and the quantum enveloping algebra Uq(sl2). In this talk, I will define what GWAs are, and then show that the quantum monoid Oq(M2), the quantum groups Oq(GL2), Oq(SL2) and Oq(SU2), and Podles spheres Oq(S^2) are all examples of GWAs. Then, time permitting, I will give an account of their representation theory in terms of certain smash products and local isomorphisms.

Geometry & Representation - Andrew Harder (Lehigh University)

Monday, March 9th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 102

Speaker: Andrew Harder (Lehigh University)

Title: Log symplectic pairs and mixed Hodge structures.

Abstract:  A log symplectic pair is a pair (X,Y) consisting of a smooth projective variety X and a divisor Y in X so that there is a non-degenerate log 2-form on X with poles along Y. I will discuss the relationship between log symplectic pairs and degenerations of hyperkaehler varieties, and how this naturally leads to a class of log symplectic pairs called log symplectic pairs of pure weight. I will give examples of families log symplectic pairs of pure weight; one coming from elliptic curves, and one coming from a hybrid toric/cluster construction. Finally, I will explain that if Y is a simple normal crossings divisor, the cohomology of a log symplectic pair (X,Y) is incredibly restricted. In particular, if there are dim(X) components of Y meeting in a point, the cohomology ring of (X,Y) has the "curious hard Lefschetz" property of Hausel and Rodriguez-Villegas.

Geometry & Representation - Veronique Bazier-Matte (UQAM)

Monday, December 2nd, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Véronique Bazier-Matte (Université du Québec à Montréal)

Title: Quasi-cluster algebras.

Abstract:  In 2015, Dupont and Palesi defined quasi-cluster algebra from non-orientable surfaces. The goal of this talk is to compare cluster algebras and quasi-cluster algebras and to explain some conjectures about them.

Geometry & Representation - Gregory G. Smith

Monday, November 18th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Gregory G. Smith (Queen's University)

Title: Smooth Hilbert schemes.

Abstract:  In algebraic geometry, Hilbert schemes are the prototypical parameter spaces: their points correspond to closed subschemes in a projective space with a fixed Hilbert polynomial. After surveying some of their known features, we will present new numerical conditions on the polynomial that completely characterize when the associated Hilbert scheme is smooth. In this smooth situation, our explicit description of the subschemes being parametrized also provides new insights into the global geometry of Hilbert schemes. This talk is based on joint work with Roy Skjelnes (KTH).

Geometry & Representation - Chris Brav (Higher School of Economics, Moscow)

Monday, November 11th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Chris Brav (Higher School of Economics, Moscow)

Title: Cartan's magic without the formulas.

Abstract:  The Cartan calculus concerns vector fields on a smooth variety X acting on differential forms via Lie derivative and contraction, with Cartan's magic formula expressing the relation between the two actions: Lie derivative is the graded commutator of the de Rham differential with contraction. On a smooth variety, the magic formula can be checked in local coordinates, while for singular schemes (and more general prestacks) it is necessary to work with the tangent complex, where it is no longer feasible to give explicit local formulas. Interpreting the magic formula as giving Griffiths transversality for the Gauss-Manin connection of the universal infinitesimal deformation of X, we are able to construct a formula-free, chain level Cartan calculus using the tangent complex of a singular scheme, and to establish the compatibility of this calculus with the noncommutative calculus of Hochschild cochains acting on Hochschild chains. This is joint work with Nick Rozenblyum.

Geometry & Representation - Anne Dranowski (University of Toronto)

Monday, November 4th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Anne Dranowski (University of Toronto)

Title: Generalized orbital varieties and MV modules.

Abstract:  Let O be the conjugacy class of a nilpotent matrix, and let C be its closure. By work of Joseph and Spaltenstein, the irreducible components of the subvariety of uppertriangular matrices in C, (aka orbital varieties,) can be labeled by standard Young tableaux. We explain how this labeling generalizes to the intersection of C and and a Slodowy slice, S. This question is motivated by the fact (due to Mirkovic-Vybornov) that such intersections are related to the Mirkovic-Vilonen (MV) construction of a cohomological crystal basis of GL(m). By D., the Mirkovic-Vybornov isomorphism maps the generalized orbital varieties to the MV cycles such that the crystal structure on tableaux matches the crystal structure on MV cycles. Our labeling enables us to determine equations of MV cycles and therefore compare the MV basis to another basis in bijection with tableaux - Lusztig's dual semicanonical basis - under the magnifying glass of the Duistermaat-Heckman measure.