• Symplectic Geometry and Representation Theory
• Workshop: Categorification, representation theory and symplectic geometry

• Schedule
• Participants

I will explain the definition of the affine VW supercategory, some properties, the motivation for studying it, and links to representation theory of the periplectic Lie superalgebra.

Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk, I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution. This result builds upon work of Crawley–Boevey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler.

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In the talk I will discuss how to obtain functoriality for coloured Khovanov–Rozansky homology. This uses a foam category constructed via a foam evaluation by Robert–Wagner and deformation technics. This is joint work with Daniel Tubbenhauer and Paul Wedrich.

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Khovanov and Seidel gave in 2000 an action of the classical braid group on a category of algebraic nature that categorifies the Burau representation. They proved the faithfulness of this action through the study of curves in a punctured disk (while Burau representation is not faithful for braids with five strands or more). In a recent article with Anne-Laure Thiel and Emmanuel Wagner, we extended this result to the braid group of the cylinder. The work of Khovanov and Seidel also had a symplectic aspect that we now generalize. In this talk, I will explain the strategy and tools to get a symplectic monodromy in our case and prove its faithfulness. If time permits, I will explain how this action lifts to a symplectic categorical representation on a Fukaya category that should be related to the algebraic categorical representation. This is a joint work in progress with Anne-Laure Thiel and Emmanuel Wagner.

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Ladder diagrams were invented by Cautis, Kamnitzer and Morrison, and form a monoidal category that is equivalent to the monoidal category generated by exterior powers of the standard representation of Uq(gl(m)). In this talk, I will talk about how ladders can be generalised to the superalgebras Uq(gl(m|n)) for all m,n ≥ 0. The new ladders involve relations that are significantly more complicated than those for Uq(gl(m)). I will also discuss how this affects problems relating to categorification of these representation categories, and their corresponding knot polynomials.

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I will describe a class of algebras which generalize the zigzag algebras of trees studied by Huerfano and Khovanov. An important class of examples comes from Iyama’s iterative construction of higher Auslander algebras. These higher zigzag algebras have spherical projective modules, and I will discuss relations in groups generated by the associated spherical twists.

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Ingalls and Thomas have shown that the lattice of non-crossing partitions of a regular polygon with n+1 vertices is isomorphic to the lattice of thick subcategories in the bounded derived category of representations of a Dynkin quiver of type A with n vertices. In joint work with Greg Stevenson we provide an infinite version of this result by showing that the lattice of non-crossing partitions of the infinitygon with a point at infinity is isomorphic to the lattice of thick subcategories in the bounded derived category of graded modules over the dual numbers.

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I will first recall the correspondence between the simple transitive 2-represen- tations of Uq(sl2)-mod, for q an even root of unity, and those of dihedral Soergel bimodules. Both have ADE classifications, which generalize the Mackey correspondence for finite subgroups of SL(2, C) and are related by the so called algebraic Satake correspondence, due to Elias. This is based on joint papers with Kildetoft, Mazorchuk, Miemietz, Tubbenhauer and Zimmermann. Recall that the Hecke algebras of the finite dihedral groups are quotients of the Hecke algebra of the affine A₁ Weyl group. In the second part of my talk, I will sketch how this naturally lead us to consider certain finite-dimensional subquotients of the Hecke algebra of the affine A₂ Weyl group. These subquotients are categorified by certain subquotients of the monoidal category of affine A₂ Soergel bimodules, which by the algebraic Satake correspondence are related to Uq(sl3)-mod- mod, for q an even root of unity (also due to Elias). Mathematical physicists have studied the simple transitive 2-representations of Uq(sl3)-mod, for q an even root of unity, although there is no complete classification yet. However, enough is known to show that our new subquotients of affine A₂ Soergel bimodules have a very interesting 2-representation theory, as I will sketch. This is joint work in progress with Elias, Mazorchuk, Miemietz and Tubbenhauer.

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Geometric extension algebras are convolution algebras in Borel–Moore homology, or equivalently sheaf-theoretic Ext algebras. Interesting examples include KLR algebras, algebras related to Schur algebras, category O and Webster algebras. We discuss how geometric parity vanishing properties are equivalent to representation-theoretic properties of these algebras. Some applications to the theory of KLR algebras will be discussed if time permits.

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I will talk about joint work with Robert Laugwitz on 2-representation theory of p-dg 2- categories, and how it relates to categorification of quantum groups at roots of unity by Khovanov, Qi, and Elias.

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Poisson manifolds are a generalization of symplectic manifolds, so one can ask in what sense Floer theory and the Fukaya category generalize to them. While the direct path to generalization is blocked by a lack of Gromov compactness, one can sidestep the difficulty by associating a certain symplectic manifold (an integration) to a Poisson manifold. The Fukaya category of the integration is monoidal, and it interacts in a rich way with the Poisson geometry of the original manifold. In this talk I will survey some applications of these ideas in the simplest examples.

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I will discuss a deformation of the annular Khovanov homology that carries an action of the quantum sl(2). The first step is to construct an isomorphism between undeformed annular Khovanov homology and Hochschild hyperhomology of the Chen-Khovanov invariant of tangles, conjectured by Auroux, Grigsby, and Wehrli. Rephrasing this in the language of categorical traces allows us to recover the sl(2) action due to Grigsby, Licata, and Wehrli, and then deform it. The new invariant has many interesting properties. For instance, the quantized colored Khovanov com- plex is homotopy equivalent to the Cooper–Krushal complex, and both are finite dimensional. This is a joint project with Anna Beliakova, Stephan Wehrli, and Matthew Hogancamp.

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What is the ‘mirror dual’ object to a Grassmannian X? In joint work with R. Marsh we wrote down a rational function on a Langlands dual Grassmannian, the ‘superpotential’ of X, and showed how it can be used to describe Gromov–Witten invariants of the original Grassmannian via a Dubrovin/Givental style of mirror symmetry construction. In this talk I will briefly report on these results, and then talk about joint work with L. Williams which makes use of the same superpotential, but in a very different way, to construct a class of Newton–Okounkov convex bodies of X.

The central characters of this talk are a restricted class of foams (which looks like tubes) and monomial polynomials. The aim to this talk is to present an evaluation which will provide a categorification of the symmetric MOY calculus using a restricted universal construction. The symmetric MOY calculus is a diagrammatical description of the sub-category of representations of the quantum group Uq(sln) monoidally generated by symmetric powers of the natural representation.

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Log schemes are an enlargement of the category of schemes that was introduced by Deligne, Faltings, Illlusie and Kato, and has applications to moduli theory and deformation problems. Log schemes play a central role in the Gross–Siebert program in mirror symmetry. In this talk I will introduce log schemes and then explain recent work joint with D. Carchedi, S. Scherotzke, and M. Talpo on various aspects of their geometry. I will discuss a comparison result between two different ways of associating to a log scheme its e ́tale homotopy type, respectively via root stacks and the Kato-Nakayama space. Our main result is a new categorified excision result for parabolic sheaves, which relies on the technology of root stacks.

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There exists a braid group action on the homotopy category of modules for a zigzag algebra associated to a linear quiver. We will explain several variations of this well known fact arising from various areas of categorical representation theory.

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The central characters of this talk are foams (which are certain kind of 2 dimensional CW-complexes) and Schur polynomials. The aim of this talk is to present a closed formula for the evaluation of closed foams in terms of the combinatorics of symmetric polynomials. We will then explain how this formula provides a categorification of the exterior MOY calculus using a procedure know as the universal construction. The exterior MOY calculus is a diagrammatical description of the sub-category of representations of the quantum group Uq(sln) monoidally generated by exterior powers of the natural representation.

Physicists have long been arguing that gauge theories at large rank are related to topolog- ical string theories. As a concrete example, I will describe a correspondence between the colored HOMFLY–PT polynomials of knots and the motivic DT invariants of certain symmetric quivers, which was recently proposed by Kucharski–Reineke–Stosic–Sulkowski. I will outline a proof of this correspondence for rational knots and then speculate about how much of the HOMFLY–PT skein theory might carry over to the realm of DT quiver invariants. This is joint work with Marko Stosic.

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We explain how to construct an explicit topological model for every two-block Springer fiber of type C and D. These so-called topological Springer fibers are homeomorphic to their corresponding algebro-geometric Springer fiber. They are defined combinatorially using cup diagrams which appear in the context of finding closed formulas for parabolic Kazhdan-Lusztig polynomials of type D with respect to a maximal parabolic of type A. As an application it is discussed how the topological Springer fibers can be used to reconstruct the famous Springer representation in an elementary and combinatorial way.

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I will give an overview of a program to study quantizations of slices in the affine Grassmannian of a semisimple group G. The slices are interesting Poisson varieties that arise in geometric representation theory, and, more recently, symplectic duality. Their quantizations have a beautiful representation theory, and in particular can be used to categorify interesting representations of the Langlands dual Lie algebra gL. Originally we constructed the quantizations using subquotients of Yangians. Recently, using work of Braverman–Finkelberg–Nakajima, these quantizations were shown to arise as quantum Coulomb branches associated to quiver gauge theories. This new perspective led Webster to define cylindrical KLR-type algebras, which we prove are Morita equivalent to the quantized slices. Using this we can prove many outstanding conjectures about the highest weight theory of the quantized slices. This project is joint with various subsets of {J. Kamnitzer, P. Tingley, B. Webster, A. Weekes}.

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