University of South Alabama, Mobile, AL

Saturday Morning 

8:00 - 8:20 am  Tolulope Oke,  Lie bracket structure on Hochschild cohomology of some twisted tensor products using Homotopy liftings.  [slides]

8:30 - 8:50 am  Radmila Sazdanovic,  Bilinear pairings on two-dimensional cobordisms and generalizations of the Deligne category.

9:00 - 9:45 am  Mikhail Khovanov,  Introduction to foam evaluation.

Saturday Afternoon

1:30 - 2:15 pm  Lev Rozansky,  Link Homology and Symplectic Geometry.  [slides]

2:30 - 2:50 pm  Joshua Sussan,  p-DG structures in higher representation theory.  [slides]

3:00 - 3:20 pm  Elizabeth Jurisich,  A Lie group analog for the monster Lie algebra. 

3:30 - 3:50 pm  Break 

4:00 - 4:20 pm  Anton Zeitlin,  q-Opers and Applications. 

4:30 - 4:50 pm  Scott Joseph Larson,  Equivariant Cohomology of Weighted Flag Varieties.

5:00 - 5:20 pm  Tamanna Chatterjee,  Study of parity sheaves arising from graded Lie algebras. 

5:30 - 5:50 pm  Yiqiang Li,  Langlands reciprocity for affine q-Schur algebra associated with a reductive group.  [slides] 

Sunday Morning

9:00 - 9:20 am  Kyu-Hwan Lee,  Mutations of reflections and existence of pseudo-acyclic orderings for type A_n. 

9:30 - 9:50 am  Shifra Reif,  Grothendieck Rings of Strange Lie Superalgebras.  [slides]

10:00 - 10:20 am  Lee Andrew Jenkins,  On the geometry of the nilpotent cone for the Lie superalgebra gl(m|n).

10:30 - 10:50 am  David Galban,  First and Second Cohomology Groups for BBW Parabolics for Lie Superalgebras. 

11:00 - 11:20 am  Discussion 

Sunday Afternoon

1:30 - 1:50 pm  Kyungyong Lee,  A new approach to the two dimensional Jacobian conjecture.  [slides]

2:00 - 2:20 pm  Jason Gaddis,  Cancellation and skew cancellation of Poisson algebras.

2:30 - 2:50 pm  Alisina Azhang,  Rigid Connections on the Projective Line with Elliptic Toral Singularities.  [slides] 

3:00 - 3:20 pm  Garrett Johnson,  Quantized nilradicals of parabolic subalgebras of simple Lie algebras.

3:30 - 3:50 pm  Break

4:00 - 4:20 pm  Kurt Trey Trampel,  Root of unity quantum cluster algebras.

4:30 - 4:50 pm  Shengnan Huang,  Root of unity quantum cluster algebras and Cayley-Hamilton algebras. 

5:00 - 5:20 pm  Discussion 

We generalize two studies of rigid G-connections on P^1 which have an irregular singularity at origin and a regular singularity at infinity with unipotent monodromy: one is the work of Kamgarpour-Sage which classifies rigid homogeneous Coxeter G-connections with slope r/h, where h is the Coxeter number of G, and the other is the work of Chen, which proves the existence of rigid homogeneous elliptic regular G-connections with slope 1/m, where m is an elliptic number for G. In our work, similar to Chen, we look for rigid homogeneous elliptic regular G-connections, but we allow the slope to have a numerator greater than 1. However, for the present purpose, we essentially restrict to the case where G is either Sp(2n) or SO(2n+1). For Sp(2n), we show that Kamgarpour-Sage connections and Chen connections exhaust all the rigid homogeneous elliptic regular connections. For SO(2n+1)-connections, having introduced the notion of "generalized Chen connections," we classify all rigid connections of this type. We conjecture that any rigid homogeneous elliptic regular SO(2n+1)-connection is in this form.

Let G be a complex, connected, reductive, algebraic group, and χ: C* → G be a fixed cocharacter that defines a grading on g, the Lie algebra of G. Let G_0 be the centralizer of χ(C*). Here I will talk about G_0-equivariant parity sheaves on the n-graded piece, g_n. I will define parabolic induction and restriction in graded setting. We will dive into the results of Lusztig in characteristic 0 in the graded setting. Under some assumptions on the field k and the group G we will recover some results of Lusztig in characteristic 0. These assumption together with Mautner’s cleanness conjecture will play a vital role. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair. Lusztig’s work on Z-graded Lie algebras is related to representations of affine Hecke algebras, so a long term goal of this work will be to interpret parity sheaves in the context of affine Hecke algebras.

In its algebraic form, the Zariski cancellation problem asks whether an isomorphism of algebras A[x] is isomorphic to B[x] implies A is isomorphic to B. In this talk, I will discuss this problem in the context of Poisson algebras. Our results show that quadratic Poisson algebras in three variables are cancellative. I will also discuss skew Poisson cancellation for Poisson Ore extensions and invariants related to this problem, including the divisor Poisson subalgebra and Poisson stratiform length. This is joint work with Xingting Wang and Dan Yee.

For semisimple Lie algebras, a well-known theorem of Kostant computes the cohomology groups of parabolic subalgebras, but it is unknown whether an analog of Kostant’s theorem exists for Lie superalgebras. Seeking to provide the first calculations in this direction, in this talk, I will describe the cohomology groups for the subalgebra n^+ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu. These classical Lie superalgebras have a triangular decomposition g = n^- ⊕ f ⊕ n^+, where f is a detecting subalgebra as introduced by Boe, Kujawa and Nakano. I will show that there exists a Hochschild-Serre spectral sequence that collapses for all infinite families of classical simple Lie superalgebras. Using this, I will provide examples of computation of the first and second cohomologies for some of the simpler cases.

The structures of maximal orders and Cayley-Hamilton algebras exist in many algebras within the framework of root of unity quantum cluster algebras. In this talk, I will show that the root of unity upper quantum cluster algebra is a maximal order, and the pair of it and its central subalgebra is a Cayley-Hamilton algebra. I will also discuss that all monomial subalgebras of root of unity quantum tori and any intersections of them over subsets of seeds are Cayley-Hamilton algebras.

Many aspects of the representation theory of a Lie algebra and its associated algebraic group are governed by the geometry of their nilpotent cone. An analogue of the nilpotent cone N for Lie superalgebras has been introduced by the author and Nakano, and it has been shown that for a simple classical Lie superalgebra the number of nilpotent orbits is finite. Furthermore, the finiteness result for N extends and generalizes the work of Duflo and Serganova on the commuting variety X. However, much is still unknown about the geometry and representation theory of the nilpotent orbits for Lie superalgebras. In this talk, we discuss more recent geometric results on N for the general linear Lie superalgebra gl(m|n). In particular, we compute the dimensions of N and the centralizers of the nilpotent orbits, determine the irreducible components of N, and show that N is a complete intersection. 

We describe the automorphism groups and some other properties of certain quantum Schubert cells, namely the quantized nilradicals of parabolic subalgebras of complex simple Lie algebras. This is joint work with Hayk Melikyan.

Let m be the monster Lie algebra. We construct a group G(m) associated to m via generators and relations. The generators used in this construction are chosen to correspond to a specific decomposition m = u^- ⊕ gl_2 ⊕ u^+ where u^{±} are free subalgebras. Furthermore we investigate constructing a pro-unipotent group of automorphisms of a completion of m.

In this talk we will review foam evaluation and its relation to link homology and the universal construction.

The nilpotent cone of a finite dimensional representation V of a reductive algebraic group G plays a key role in many areas of current interest; e.g., geometric invariant theory, infinite dimensional representations of real reductive groups, and representations of Weyl groups. Let χ be a cocharacter of G such that the image S acts with all positive weights. If V is irreducible and G is connected one may consider the orbit Z through extremal weight vectors. Following Ian Grojnowski and first published by Corti-Reid (2002), we consider the weighted flag variety X = S\Z, which generalizes weighted projective space and all G/P. 

Abe-Matsumura (2014) studied the T-equivariant cohomology of X for G = GL(n, C) such that V corresponds to a fundamental weight, and Azam-Nazir-Qureshi (2018) more generally for the same group and any V . We study all G and V . A key fact we use is that X is rationally smooth, which enables us to explicitly describe equivariant Schubert classes via Poincare duality. We obtain a presentation of T-equivariant cohomology which generalizes the classical equivariant Borel presentation of T-equivariant cohomology of G/P. 

In a recent paper by K.-H. Lee, K. Lee and M. Mills, a mutation of reflections in the universal Coxeter group is defined in association with a mutation of a quiver. A matrix representation of these reflections is determined by a linear ordering on the set of vertices of the quiver. It was conjectured that there exists an ordering (called a pseudo-acyclic ordering) such that whenever two mutation sequences of a quiver lead to the same labeled seed, the representations of the associated reflections also coincide. In this talk, we prove this conjecture for every quiver mutation-equivalent to an orientation of a type A_n Dynkin diagram by decomposing a mutation sequence into a product of elementary swaps and checking relations studied by Barot and Marsh.

The Jacobian conjecture concerns the polynomial automorphisms. Motivated by cluster algebras, we consider the joint divisibility conditions associated with the Jacobian matrix whose determinant is a constant. By doing this, we get a substantially better understanding on the two dimensional case.

Langlands reciprocity for an affine Hecke algebra plays a key role in the resolution of the Deligne-Langlands conjecture. The reciprocity relates the constructible geometry on an affine flag variety on the one side and the coherent geometry on a Steinberg variety of the Langlands dual group on the other side. The extension of this reciprocity to quantum groups was done by Ginzburg-Vasserot for affine type A. In this talk, we should present a further generalization of this reciprocity.

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov’s homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. These results illustrate the utility of homotopy liftings for theoretical purposes.

The Grothendieck ring of the general linear Lie algebra gl(n) is isomorphic to the ring of symmetric Laurent polynomials C[x_1^{±1}, ... ,x_n^{±1}]. The strange series are two generalizations of gl(n) called the periplectic and queer Lie superalgebras. In the periplectic case, we describe the Grothendieck ring using the ring of supercharacters. We prove that it consists of all symmetric Laurent polynomials for which the evaluation x_1 = x_2^{-1} = t is independent of t. In the queer case, we describe the Grothendieck ring using the ring of characters. We prove that it consists of all symmetric Laurent polynomials for which the evaluation x_1 = -x_2 = t is independent of t. 

The first part is a joint work with Mee Seong Im and Vera Serganova.

This is a review of my joint work with A. Oblomkov. Quantum field theory considerations suggest that one can associate a 2-category to a symplectic variety. This 2-category is particularly well understood when the variety is a Hamiltonian reduction of a cotangent bundle. We consider a special object "FL" in the 2-category associated with the Hilbert scheme of points in C^2 and construct a homomorphism from the braid group to the monoidal category End(FL) which is a particular category of matrix factorizations. This homomorphism leads to yet another construction of the HOMFLY-PT link homology. The 'microscopic' view of the structure of the homomorphism is very similar to the pictures used by Khovanov-Lauda and others in categorifying Lie algebras, but at the next category level.

The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. It is semisimple for generic values of the parameter t while producing categories of representations of the symmetric group when modded out by the ideal of negligible morphisms when t is a non-negative integer. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. The Deligne category and its semisimple quotients admit similar interpretations. This viewpoint coupled to the universal construction of two-dimensional topological theories leads to multi-parameter monoidal generalizations of the partition and the Deligne categories, one for each rational function in one variable.

We review the machinery of p-DG theory and use it to categorify tensor products of representations of quantum sl(2) at prime roots of unity.

We will describe the setup for root of unity quantum cluster algebras and give fundamental results such as a version of the Laurent phenomenon and a theorem on specialization of generic quantum cluster algebras. We will review the concept of the discriminant and discuss recent progress to its use in understanding the representation theory of these quantum algebras.

I will talk about the finite-difference analog of oper connections, known as q-opers, and their applications. That includes various aspects of the theory of integrable systems and enumerative geometry, such as quantum/classical duality and 3d mirror symmetry.