Joint UOttawa/Carleton
Algebra Seminar
Winter 2025
Fridays, 2:30-3:30pm
UOttawa (STEM 664) / Carleton (HP 4369)
January 24 (Carleton)
Speaker: Kaveh Mousavand (Okinawa Institute of Science and Technology)
Title: Hom-orthogonal modules and geometry of representation varieties
Abstract: Originally motivated by some open conjectures in representation theory of finite dimensional algebras, we study the sets of pairwise Hom-orthogonal modules. In particular, we relate the study of arbitrary Hom-orthogonal modules to the distribution of bricks (a.k.a Schur representation) and consequently prove some new results on the geometry of representation varieties. For an algebra A, we use some algebraic and geometric tools to find necessary and sufficient conditions for the existence of an infinite family of Hom-orthogonal A-modules of the same dimension. This is based on joint work with Charles Paquette.
February 7 (UOttawa)
Speaker: Erhard Neher (Ottawa)
Title: Azumaya algebras with orthogonal involutions
Abstract: Azumaya algebras are the commutative ring version of central simple algebras over fields, for example matrix algebras. Orthogonal involutions are the analogue of the transpose involution of matrix algebras. For applications in the theory of algebraic groups, it is important to investigate so-called semi-traces associated with Azumaya algebras with orthogonal involutions. In this talk, I will describe their theory over rings, and if time sketch what is known over schemes. The talk is based on joint work with Philippe Gille and Cameron Ruether.
February 28 (UOttawa)
Speaker: Rita Fioresi (Universita di Bologna)
Title: Generalized Room Systems
Abstract: Generalized Root Systems present a unifying conceptual point of view on root systems in Lie Theory. They comprehend root systems for semisimple Lie algebras, contragredient superalgebras and also special types of hyperplane arrangements. In this talk, we will see their definition, main properties and a classification result in rank 2. This is a joint work with Dimitrov ”Generalized Root Systems”,TAMS 2024 and work in progress with Cuntz, Dimitrov, Mulherren.
March 14 (UOttawa)
Speaker: Erhard Neher (Ottawa)
Title: Local-global rings and applications
Abstract: Local-global rings R axiomatize an important property of semi-local rings: a multivariate polynomial over R has an invertible value in R if it has a non-zero value in every quotient field of R. In this talk, I will describe several nice features of local-global rings, like cancellation theorems and an application to the Brauer group of a local-global ring. The talk is based on joint work with Philippe Gille.
March 21 (Carleton)
Speaker: Steven Amelotte (Carleton)
Title: Moment-angle complexes and minimal free resolutions of Stanley-Reisner rings
Abstract: Toric topology assigns to each simplicial complex K a space with a torus action, called the moment-angle complex, whose equivariant topology neatly reflects homological properties of the Stanley-Reisner ring of K. A fundamental result in the subject identifies the cohomology ring of the moment-angle complex with the Tor-algebra of the Stanley-Reisner ring. In this talk, we describe how this can be extended to a topological interpretation of the entire minimal free resolution of the Stanley-Reisner ring, and consider the problem of reading off the homotopy types of these spaces from this data. We show that the Hurewicz image for any moment-angle complex contains the linear strand of its associated Stanley-Reisner ring. Combined with variants of a theorem of Eagon and Reiner well-known to commutative algebraists, we describe how this recovers results of various authors identifying the moment-angle complex up to homotopy as a wedge of spheres when the Stanley-Reisner ring satisfies certain Cohen-Macaulay or Koszulity properties. Going further, we introduce a large class of Gorenstein simplicial complexes which generalizes the homological behaviour of cyclic polytopes, stacked polytopes and neighbourly sphere triangulations. For these simplicial complexes, we show that the associated moment-angle manifolds are rationally homotopy equivalent to connected sums of sphere products. This is joint work with Ben Briggs.
Junior Algebra Seminar: May 6, 2:30 (UOttawa)
Speaker: Simon Larose (MSc student, UOttawa)
Title: Isometries of 2-adic Hermitian Lattices: the unramified case
Abstract: In basic linear algebra we learn about linear transformations which preserve inner products. One can then observe that these special linear transformations, called the isometries, are built out of simpler ones: hyperplane reflections. Like for so many other concepts from linear algebra, we can then ask: does a similar thing happen when a vector space is replaced by a module, and if so, over which ring? In this talk, we give the necessary background to explain a result of Brandhorst, Hoffman, and Manthe which extends the theory of the generation of the unitary group to Hermitian lattices over 2-adic rings. The talk is aimed at MSc. students, but undergraduate students are most welcome to attend. All that is required is rudimentary knowledge of rings and modules.
Algebra Seminar: June 26, 4:00pm (UOttawa)
Speaker: Raj Gandhi (PhD last year, Cornell University)
Title: Deforming the motivic Segre classes of Schubert cells in the Grassmannian
Abstract: A longstanding goal of Schubert calculus is to give a positive formula for the structure constants for the Schubert basis in the cohomology ring of the d-step flag variety. This goal can be generalized by replacing “cohomology ring” with “torus-equivariant cohomology ring”, “K-ring”, and “torus-equivariant K-ring”. It can also be generalized in an orthogonal direction by replacing “d-step flag variety” with “cotangent bundle of the d-step flag variety”.
Recently, Allen Knutson and Paul Zinn-Justin proved a positive formula in terms of Knutson-Tao puzzles for the structure constants in the basis of motivic Segre classes of Schubert cells in (a localization of) the torus-equivariant K-ring of the cotangent bundle of the Grassmannian. Their proof heavily uses the theory of quantum integrable systems.
In this talk, we will describe a one-parameter deformation of the motivic Segre classes of Schubert cells in the Grassmannian (using the method of “restriction to fixed points”), and we will prove a positive formula for the structure constants in the basis of deformed classes in terms of Knutson-Tao puzzles. We will also suggest where these deformed classes may come from geometrically.