University of South Alabama, Mobile, AL

4:00 - 4:20 pm  Zhongkai Mi

4:30 - 4:50 pm  Xin Tang

8:00 - 8:20 am  Aria Masoomi

8:30 - 8:50 am  Xin Jin

9:00 - 9:20 am  Anton Zeitlin

9:30 - 9:50 am  Coffee Break

10:00 - 10:20 am  Mikhail Khovanov

10:30 - 10:50 am  Joshua Sussan

11:00 - 12:00 pm  Cornelius Pillen  "Lifting to tilting: modular representations of algebraic groups and their Frobenius kernels,"  Shelby Hall,  Room 2119

12:00 pm  Partial solar eclipse,  beach,  USS Alabama,  Mobile downtown,  etc.  [free time]

4:30 - 4:50 pm  Prasad Senesi

5:00 - 5:20 pm  Jonas Hartwig

5:30 - 5:50 pm  Daniele Rosso

6:00 pm  Reception,  Mitchell Center,  Globe Grand Lobby  [hosted by USA's Department of Mathematics & Statistics]

7:30 pm  Party,  Cornelius Pillen's house  [Meet at Michell Center lobby  and  walk for 15 minutes]

8:00 - 8:20 am  Xingting Wang

8:30 - 8:50 am  Tianyuan Xu

9:00 - 9:20 am  Robert Muth

9:30 - 9:50 am  Nick Davidson

10:00 - 10:20 am  Coffee Break

10:30 - 10:50 am  Scott Larson

11:00 - 11:20 am  Matthew Hamil

11:30 - 11:50 am  Prerna Agarwal

Reeder-Yu introduced certain low positive depth supercuspidal representations of p-adic groups called epipelagic representations. These representations generalize the simple supercuspidal representations of Gross-Reeder, which have the lowest possible depth. Epipelagic representations also arise in recent work on the Langlands correspondence; for example, simple supercuspidals appear in the automorphic data corresponding to the Kloosterman l-adic sheaf. In this talk, we report on a project whose goal is the construction of "mesopelagic representation of Iwahori type", higher depth analogues of simple supercuspidal representations.

Webs are diagrams used to encode homomorphisms between certain representations of Lie (super)algebras. In this talk, I’ll discuss work which “deforms” web categories using an associative superalgebra. By specializing the superalgebra appropriately, we recover several existing constructions of web categories. The work also includes a new Howe duality and expected applications to the representation theory of the symmetric group.

In the study of cohomology of finite group schemes it is well-known that nilpotence theorems play a key role in determining the spectrum of the cohomology ring. Balmer recently showed that there is a more general notion of a nilpotence theorem for tensor triangulated categories through the use of homological residue fields and the connection with the homological spectrum. The homological spectrum can be viewed as a topological space that realizes the Balmer spectrum in a concrete way. 

Let g = g_0 ⊕ g_1 be a Type I classical Lie superalgebra with an ample detecting subalgebra. In this talk, the speaker will consider the tensor triangular geometry for the stable category of finite-dimensional Lie superalgebra representations: stab(F_{(g, g_0)}), The localizing subcategories for the detecting subalgebra f are classified which answers a question of Boe, Kujawa and Nakano. As a consequence of these results, we prove a nilpotence theorem and determine the homological spectrum for the stable module category of F_{(f, f_0)}. 

The orbit structure of the reductive group G_0 on g_1 where Lie(G_0) = g_0 along with the results for the detecting subalgebra is used to prove a nilpotence theorem for stab(F_{(g, g_0)}) and to determine the homological spectrum in this case. Now using work of Balmer and key assumptions in the work of Boe, Kujawa and Nakano, we provide a method to explicitly realize the Balmer spectrum of stab(F_{(g, g_0)}) for Type I classical Lie superalgebras (with an ample detecting subalgebra). Examples of such Lie superalgebras include the general linear Lie superalgebras gl(m|n) and the periplectic Lie superalgebra p(n). 

This is joint work with Daniel K. Nakano.

Gelfand-Tsetlin theory is concerned with chains of algebras where each inclusion satisfies a multiplicity-free branching rule. One example is the chain of Lie algebras gl(n) ⊃ ... ⊃ gl(1). Another is so(n) ⊃ ... ⊃ so(2). Explicit Gelfand-Tsetlin-Zhelobenko formulas for representations of gl(n) yield an embedding of U(gl(n)) in a skew group algebra (Futorny-Ovsienko 2010). This enables construction of irreducible Gelfand-Tsetlin modules. For so(n), only generic modules have been constructed (Mazorchuk 2001; Disch 2022). 

I will discuss a formula-free approach to this topic based on tensor products of Mickelsson algebras.

I’ll present recent work on mirror symmetry for the affine Toda systems, which can be viewed as a Betti Geometric Langlands correspondence (after Ben-Zvi—Nadler) in the wild setting. More explicitly, we realize the affine Toda system (associated to a complex semisimple group) as a moduli space of Higgs bundles on P^{1} with certain automorphic data, and the dual side is the group version of the universal centralizer (associated to the dual group), which is a wild character variety. We show that the wrapped Fukaya category of the former is equivalent to the category of coherent sheaves of the latter. The proof uses my previous result on the mirror symmetry for the (usual) Toda systems, also known as the universal centralizers. This is joint work with Zhiwei Yun.

I’ll explain the correspondence between Boolean one-dimensional TQFTs with defects and finite state automata. This talk is based on a recent joint work with P.Gustafson, M.S.Im, R.Kaldawy and Z.Lihn.

Fibers of resolutions of singularities have many connections to representation theory, for example, Springer’s resolution and Weyl groups. In the case of real reductive groups, we have resolutions of singularities of -orbit closures in flag varieties. It was shown by Larson–Romanov that cohomology of fibers are related to Kazhdan–Lusztig–Vogan polynomials. In joint work with Bill Graham and Minyoung Jeon, we analyze further the algebraic geometry of these resolutions. We prove that a family of small resolutions for U(p,q) have reduced scheme theoretic fibers. We furthermore prove that the corresponding Harish-Chandra modules for U(p,q) have irreducible characteristic cycles.

The scheme of point modules of an N-graded algebra plays an important role in noncommutative algebraic geometry after the works of Artin, Tate and Van den Bergh. A general treatment of this construction was given by Rogalski and Zhang. We will give a classification of the schemes of point modules of the flag varieties of all complex simple Lie groups. We will show that they are described as explicit unions of Richardson varieties classifying the points where the corresponding Poisson structure vanishes. We use methods from noncommutative algebra (normal separation), and algebraic and Poisson geometry of flag varieties.

Discriminant ideals for an algebra A module finite over a central subring C are indexed by positive integers. We study the lowest of them with nonempty zero set in Cayley Hamilton Hopf algebras whose identity fibers are basic algebras. Key results are obtained by considering actions of characters in the identity fiber on irreducible modules over maximal ideals of C and actions of winding automorphisms. We apply these results to examples in group algebras of central extensions of abelian groups, big quantum Borel subalgebras at roots of unity and quantum coordinate rings at roots of unity.

In studying compositions of induction and restriction functors on symmetric group modules, Khovanov defined the diagrammatic Frobenius Heisenberg category Heis_{-1}. More generally, for any Frobenius algebra A, Brundan, Savage and Webster have defined an ‘A-deformed’ analogue Heis_{-1}(A), and showed that it categorifies (via Grothendieck ring) the Frobenius Heisenberg algebra Heis_{-1}(A). On the other hand, for positively graded A, I will explain that the alternative trace decategorification of this category is isomorphic (as an algebra) to the Frobenius W-algebra W_{-1}(A). This proves a conjecture of Reeks and Savage, who previously established this result at the level of vector spaces. Of particular interest is the case when A is a zigzag algebra, in which case the category Heis_{-1}(A) was shown by Cautis and Licata to act on certain derived categories of coherent sheaves on Hilbert schemes of points, and W_{-1}(A) thus acts on the trace of this derived category.

Generalized Weyl algebras (GWAs) are an important class of algebras that are interesting for ring theory and representation theory. Even though they are not naturally Hopf algebras, the tensor product of a weight module for a GWA with a weight module for another GWA gives a weight module over a third GWA. As a consequence we get a ring structure on the sum of Grothendieck groups of the categories of weight modules over a tower of GWAs. In this work we describe this ring structure in the case of finite orbits (we had already described the infinite orbits case in previous work).

Quandles are algebraic structures that were developed, and are primarily used, in the context of knot theory. Their defining axioms arise from the Reidemeister moves on a knot diagram. In this talk, we introduce the representation theory of quandles, where a representation of a quandle Q is a quandle morphism from Q to the (quandle of) automorphisms of a finite-dimensional vector space. While the representation theories of groups, rings, and algebras are well known, representations of quandles remain unexplored in the literature. We will share recent representation-theoretic results for the families of dihedral quandles and cyclic quandles. This will include a complete decomposition of the regular representation of a dihedral quandle, and a partial classification of finite-dimensional representations of a cyclic quandle. We will also provide a complete classification of the cyclic quandles, extending previous work of Kamada, Tamaru, Wada, and Vendramin.

Topological quantum field theories coming from semisimple categories build upon interesting structures in representation theory and have important applications in low dimensional topology and physics. The construction of non-semisimple TQFTs is more recent and they shed new light on questions that seem to be inaccessible using their semisimple relatives. In order to have potential applications to physics, these non-semisimple categories and TQFTs should possess Hermitian structures. We will define these structures and give some applications. 

This is joint work with Nathan Geer, Aaron Lauda, and Bertrand Patureau-Mirand.

In this talk, we present a notion of twisting for graded Poisson algebras. We show that every graded Poisson algebra structure on a connected graded polynomial ring is a twist of a unimodular one. We define a rigidity in terms of graded twistings and calculate the rigidity for several families of graded Poisson algebras. We also compute the Poisson cohomology groups for several classes of Poisson algebras. This is joint work with Xingting Wang and James J. Zhang.

We compute the automorphism groups of unimodular Poisson algebras in dimension 3 given by a homogenous polynomial with an isolated singularity. We use the tool of Poisson valuation. This is joint work with Hongdi Huang, Xin Tang and James Zhang.

We introduce and study "2-roots", which are symmetrized tensor products of orthogonal roots of Kac–Moody algebras. We concentrate on the case where W is the Weyl group of a simply laced Y-shaped Dynkin diagram with three branches of arbitrary finite lengths a, b and c; special cases of this include types D_n, E_n (for arbitrary n ≥ 6), and affine E_6, E_7 and E_8. 

We construct a natural codimension-1 submodule M of the symmetric square of the reflection representation of W, as well as a canonical basis B of M that consists of 2-roots. The module M may be viewed as a specialization of a certain Kazhdan–Lusztig cell module of the Iwahori–Hecke algebra of W. We prove that with respect to B, every element of W is represented by a column sign-coherent matrix in the sense of cluster algebras. We also prove that if W is not of affine type, then the module M is completely reducible in characteristic zero and each of its nontrivial direct summands is spanned by a W-orbit of 2-roots. (This is joint work with Richard Green.)

I will talk about the concept of quantum/classical duality: the duality between quantum integrable systems solved by Bethe ansatz techniques and classical integrable systems of many-body type. That will include the interpretations of this phenomenon in various geometric settings, including enumerative geometry and the (q-)geometric Langlands correspondence.