Time: Tuesday 4:30 - 5:30pm.
Room: Hodson 216.
Contact: Matthew Hamil and Haihan Wu.
Faculty contact: Mee Seong Im (Krieger 419).
The visitor talks are at the intersection of topology, algebra, geometry, category theory, mathematical physics, and related areas. Everyone interested is welcome to attend.
Weekly Social Events (everyone is welcome)
Department Afternoon Tea: Mondays, Tuesdays, and Thursdays at 4pm in Krieger 413
Department Wine and Cheese: Wednesdays at 4pm in Krieger 413
Title: The Delannoy category and its diagrammatics
Abstract: N.Harman and A.Snowden discovered a semisimple monoidal pivotal category, which they called the Delannoy category, where composition of morphisms is given by computing the compact Euler characteristic of subspaces of the Euclidean space described by inequalities on the coordinates. In the talk we will explain a diagrammatic description of their category, following a joint work with N.Snyder. The number of simple objects in the Delannoy category grows exponentially, but a suitable monoidal subcategory has the Grothendieck ring isomorphic to the ring of integer-valued one-variable polynomials. That subcategory can be viewed as a categorification of the latter ring.
Title: Springer Fibres, Parabolic Induction & Stacking Maps
Abstract: Fibres coming from the Springer resolution on the nilpotent cone are incredibly rich algebraic varieties that have many applications in algebraic geometry, representation theory and combinatorics. In this talk, I will describe how we can use the combinatorics of (bi)tableaux to describe their geometry in low dimensions, in particular, giving a description of their irreducible components. I will also describe a process of parabolic induction coming the nilpotent cone of a Levi subalgebra. This is joint work with Lewis Topley, and separately with Mee Seong Im and Arik Wilbert.
Title: Semisimplifying categorical Heisenberg actions, diagrammatics, and periodic equivalences
Abstract: Semisimplification functors on tensor categories underlie many powerful constructions in representation theory, including the Frobenius functor in modular representation theory and the Duflo-Serganova functor from Lie superalgebras. I will introduce a systematic approach to applying semisimplification functors to categories defined over positive characteristic, with the most important examples being representation categories and degenerate categorical Heisenberg actions. In the latter case, the functors will always define (non-exact!) morphisms of categorical actions, and in particular cases categorify a certain element of the mod-p centre of affine sl_p. These functors admit a convenient diagrammatic description which clarifies their properties. Finally, I will explain how these functors naturally globalize known equivalences of subcategories of representations of S_n, originally due to Henke-Koenig.
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Title: On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities
Abstract: Kleinian singularities are quotients of C^2 by finite subgroups of SL_2(C). They are in bijection with the ADE Dynkin diagrams via the McKay correspondence. In this talk, I will introduce certain singular Lagrangian subvarieties in the minimal resolutions of Kleinian singularities that are related to the geometric classification of certain unipotent Harish-Chandra (g,K)-modules. The irreducible components of these singular Lagrangian subvarieties are P^1's and A^1's. I will describe how they intersect with each other through the realization of Kleinian singularities as Nakajima quiver varieties. Time permitting, we will also discuss the deformations of these singular Lagrangian subvarieties.
Title: Graphs embedded on surfaces and their delta-matroids
Abstract: By “cutting out” the edges and vertices of a graph cellularly embedded in a surface, we obtain a ribbon graph. A partial Petrial is obtained from a ribbon graph by selecting a subset of the ribbon edges and adding a half-twist to each. The resulting ribbon graph corresponds to a cellularly embedded graph in a surface obtained by "filling in" the missing faces. Delta-matroids generalize ribbon graphs similarly to the way that matroids generalize graphs. The ribbon graph partial Petrial is the analogue of a more general delta-matroid operation. In this talk, we characterize the set of delta-matroids that is closed under this delta-matroid operation by a set of minimal obstructions. We also define delta-matroids and matroids.
Title: Profinite transfers in K(n)-local homotopy theory
Abstract: After K(1)-localization, the classical J-homomorphism can be interpreted as a profinite transfer map. More precisely, it is a transfer map \Sigma^{-1}KO^\wedge_2 \to \mathbb{S}_{K(1)} from the C_2-homotopy fixed points (with a twist) to the \mathbb{Z}_2^\times-homotopy fixed points of the 2-complete complex topological K-theory. In joint work in progress with Guchuan Li, we extend this idea to define and study profinite transfers between homotopy fixed points of the Morava E-theory by closed subgroups of the Morava stabilizer group.
We introduce two definitions of profinite transfer maps. The first defines them as duals to the profinite restriction maps in the appropriate category. At large primes, we show that the image of the transfer map \Sigma^{-n^2}E_n \to \mathbb{S}_{K(n)} on homotopy groups is the n^2-th filtration in the homotopy fixed point spectral sequence. A second definition is based on the 6-functor formalism for smooth representations of p-adic Lie groups after Heyer--Mann. We prove that the two definitions of profinite transfers are equivalent for homotopy fixed points of the Morava E-theory.
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Have a great break!
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