University of Utah, Salt Lake City, UT

Reeder and Yu have introduced certain low positive depth supercuspidal representations of p-adic groups called epipelagic representations. These representations generalize the simple supercuspidal representations of Gross and Reeder, which have the lowest possible possible depth. Epipelagic representations 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.

In this presentation, we will expand on the notions of maximal green and reddening sequences for quivers associated to cluster algebras. The existence of these sequences has been studied for a variety of applications related to Fomin and Zelevinsky’s cluster algebras. Ahmad and Li considered a numerical measure of how close a quiver is to admitting a maximal green sequence called a red number. In this paper we generalized this notion to what we call unrestricted red numbers which are related to reddening sequences. In addition to establishing this more general framework we completely determine the red numbers and unrestricted red numbers for all finite mutation type quivers. Furthermore, we give conjectures on the possible values of red numbers and unrestricted red numbers in general.

Let be a complex, connected, reductive, algebraic group and g be it’s Lie algebra. We fix a cocharacter map χ: C*→G. This defines a Z-grading on g and G_0 be the centralizer of χ(C*). In this talk we would explore Fourier transformation on the graded pieces. Fourier-Sato transformation has been studied on the nilpotent cone in both characteristic 0 and positive. One of the importance on nilpotent cone is that as a Weyl group representation Fourier transformation of the springer sheaf corresponds to a sign change. Similar results can be proved for partial springer sheaf. Following these motivation and Lusztig’s work on graded Lie algebras in characteristic 0, I have tried to study this functor on the G_0-equivariant bounded derived category on g_n, D^b_{G_0}(g_n,k), when charateristic of k is positive. To use some already proven result in positive characteristic we make some assumptions on the field coefficient, which includes Mautner’s cleanness conjecture. Parity sheaves are some important and new object of D^b_{G_0}(g_n,k) in positive characteristic. One main result in positive characteristic is that Fourier transformation sends parity to parity.

Demazure modules are Borel modules whose crystals are truncations of highest weight irreducible crystals. Whereas tensor products of full crystals are well understood, tensor products of Demazure crystals are not. This is in part because Kashiwara’s tensor product rule does not generally preserve the Demazure crystal structure. In these two talks we will explain how Kashiwara’s tensor product rule fails and discuss how solutions to special cases can shed light on a new way to tensor Demazure crystals that does preserve the desired structure. Time permitting, we will also discuss how answers to these questions allow us to explore certain filtrations for the corresponding Demazure modules.

Demazure modules are Borel modules whose crystals are truncations of highest weight irreducible crystals. Whereas tensor products of full crystals are well understood, tensor products of Demazure crystals are not. This is in part because Kashiwara’s tensor product rule does not generally preserve the Demazure crystal structure. In these two talks we will explain how Kashiwara’s tensor product rule fails and discuss how solutions to special cases can shed light on a new way to tensor Demazure crystals that does preserve the desired structure. Time permitting, we will also discuss how answers to these questions allow us to explore certain filtrations for the corresponding Demazure modules.

It is known that the Gerstenhaber bracket on Hopf algebra cohomology of a quasi-triangular algebra is abelian. We first calculate the bracket structure is indeed zero on Hopf algebra cohomology of a Taft algebra T_p by computing the bracket on the Hochschild cohomology of T_p and finding the corresponding bracket on Hopf algebra cohomology of T_p. Later, we introduce a simpler technique by using Volkov’s homotopy lifting and obtain a same bracket structure on the Hopf algebra cohomology of T_p. The second technique allows us to bypass the work on Hochschild cohomology. By second technique, we also show that this Lie structure on Hopf algebra cohomology is abelian in positive degrees for all quantum elementary abelian groups.

Let H ⊆ B ⊆ G be Cartan and Borel subgroups in a connected reductive complex algebraic group. If G = G' x C*, H = H' x C*, and V is a representation of G corresponding to a dominant weight λ = (λ', 1), then there exists a cocharacter χ = (χ', r), with r > 0, such that all weights in V pair positively with the cocharacter. Grojnowski generalizes flag varieties and weighted projective space by considering the quotient of the G'-saturation of all extremal weight vectors in V by the action of the cocharacter. More generally, even if G is not a product G' x C*, we consider Z the complement of the zero section of a G-equivariant line bundle on a flag variety corresponding to λ, and assume that for every Weyl group element w ∈ W, the pairing of wλ with χ is positive. We define the weighted flag variety as X = S\Z, where S is the image of χ.

The torus T = H/S acts on X, which enables the study of T-equivariant cohomology of X. In the case where X = G/P, Graham proved that the equivariant structure constants with respect to a Schubert basis satisfy positivity with respect to a system of simple roots. Abe–Matsumura study the special case of weighted Grassmannians, where G = GL(n) x C*, and prove that there exists a basis satisfying positivity with respect to certain mysterious parameters. We generalize all positivity results to any weighted flag variety, interpret the basis of H_T^*(X) as Poincaré dual to weighted Schubert varieties, and define the notion of weighted root to interpret geometrically the parameters in H_T^*(pt).

We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated with integrability, namely preservation of a symplectic form, and in the tropical case, the existence of a conserved quantity. We show in certain cases that the orbits are unbounded. The tropical dynamics are related to matrix mutation, from the theory of cluster algebras. We are able to show that in certain special cases, the tropical map is periodic. We also explain how our dynamics imply the asymptotic sign-coherence observed by Gekhtman and Nakanishi in the 2-dimensional situation.

Let G be a complex reductive group and P be a parabolic subgroup of G. In this talk the presenter will address questions involving the realization of the G-module of the global sections of the (twisted) cotangent bundle over the flag variety G/P via the cohomology of the small quantum group.

Our main results generalize the important computation of the cohomology ring for the small quantum group by Ginzburg and Kumar, and provides a generalization of well-known calculations by Kumar, Lauritzen, and Thomsen to the quantum case and the parabolic setting. As an application we answer the question (first posed by Friedlander and Parshall for Frobenius kernels) about the realization of coordinate rings of Richardson orbit closures for complex semisimple groups via quantum group cohomology. Formulas will be provided which relate the multiplicities of simple G-modules in the global sections with the dimensions of extension groups over the large quantum group.

This is a report on joint work with K.Coulembier and P.Etingof. We give an intrinsic characterization of symmetric tensor categories over fields of positive characteristic admitting a tensor functor to so called Verlinde category. This is a partial analogue of well known theorems by Deligne giving an intrinsic characterization of Tannakian and super Tannakian categories over fields of characteristic zero.

Using chord diagrams, we construct a diagrammatic basis for the space of invariant tensors of certain Type B representations. This basis carries the property that rotation of the chord diagrams intertwines with the natural action of the longest cycle of the symmetric group on the tensor powers. Our approach involves a generalization of Schützenberger's promotion operator on crystal graphs. As a consequence of this, we are able to show that fans of Dyck paths satisfy a cyclic sieving phenomenon. This is joint work with Stephan Pfannerer, Anne Schilling, and Mary Claire Simone.

There are several obstructions to extending the Zariski spectrum Spec as a functor from commutative rings to noncommutative rings. In an attempt to escape these limitations, we are forced to search for a category of "noncommutative sets" that is strictly larger then the ordinary category of sets. Restricting to cases where the maximal spectrum Max is functorial for commutative algebras, we argue that coalgebras can reasonably serve as generalized sets, and that the finite dual coalgebra is a suitable quantization of Max. We will discuss how the finite dual behaves under twisted tensor products and how it can be understood relative to the center of an affine noetherian PI algebra. If time permits, we will also discuss a conjectural path to quantizing the functor Spec itself.

This is the first half of a joint talk whose goal is to give an exposition of the theory of renormalized quantum link invariants due to Costantino, Geer, Patureau-Mirand, Turaev and others. The second half will be given by Jan-Luca Spellman. I define the unrolled quantum group \bar{U}_q^H(sl(2,C)) and identify a particular subcategory of the category of \bar{U}_q^H(sl(2,C))-representations. I explain why this subcategory is a non-semisimple ribbon category. Such a category is a suitable candidate for constructing a quantum link invariant using the Reshetikhin–Turaev approach. I show that, unfortunately, because the ribbon category is non-semisimple, the link invariant constructed is trivial. Based on a collaboration with Nathan Geer, Jan-Luca Spellman, and Matt Young.

We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams one compiles are hyperbolic. Furthermore, the diagrams can be arranged to have additional nice properties, such as being alternating with minimal crossing number. Various complexity-theoretic hardness results regarding the calculation of quantum invariants of knots follow as corollaries. In particular, we argue that the hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.

Brauer introduced the Brauer algebra, and established the double centralizer property between it and the orthogonal group or symplectic group. Wenzl defined a q-deformation of the Brauer algebra which contains the type A Hecke algebra as a natural subalgebra. It is well known that Jimbo-Schur duality relates Hecke algebras and quantum groups of type A. In recent years, Bao and Wang have formulated a q-Schur duality between a type B Hecke algebra and an iquantum group arising from quantum symmetric pairs. In this talk we focus on iquantum groups which specialize to the orthogonal or symplectic Lie algebra at q=1 and introduce an iSchur duality between them and the q-Brauer algebra. We also develop a natural bar involution and construct a Kazhdan-Lusztig type canonical basis on the q-Brauer algebra. This is joint work with Weideng Cui.

Donaldson-Thomas (DT) theory is an enumerative theory which produces a virtual count of stable coherent sheaves on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory, originally introduced by Kontsevich-Soibelman, is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will explain the role of d-critical locus structure in the definition of motivic DT invariant, following the definition by Bussi-Joyce-Meinhardt. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric Calabi-Yau threefolds. This is based on joint works with Sheldon Katz. The results have substantial overlap with recent work by Ricolfi-Savvas, but techniques used here are different.

This is the second half of a joint talk with Adam Robertson. The goal of the talk is to give an exposition of the theory of renormalized quantum link invariants due to Costantino, Geer, Patureau-Mirand, Turaev et al. I will explain how to modify the usual Reshetikhin-Turaev invariants associated to a subcategory of representations of the restricted unrolled quantum group \bar{U}_q^H(sl(2,C)) to give new non-trivial link invariants. Their characteristics will be exhibited with the help of several examples.

We construct an sl(3)-skein algebra on punctured surfaces generated by webs and formal invertible elements related to Goncharov-Shen potentials at punctures. We would like to embed the quantum Fock-Gonchaov higher Teichmuller space for PGL(3) into this skein algebra. This is based on joint work-in-progress with Linhui Shen and Zhe Sun.

In the area of cluster algebras, there are two general constructions: the Gekhtman-Shapiro-Vainshtein Poisson structures on cluster algebras and the associated root of unity quantum cluster algebras. We will prove that the spectrum of each of the former algebras has an explicit Zariski open torus orbit of symplectic leaves, which is a far reaching generalization of the Richardson divisor of Schubert cells in flag varieties. We will use these results, Poisson orders and Cayley-Hamilton algebras in the sense of Procesi, to describe explicitly the fully Azumaya loci of all (strict) root of unity quantum cluster algebras. This classifies their irreducible representations of maximal dimension.

In this talk, I will explain various relative modular structures on the category of representations of the quantum group of gl(1|1). These structures should be seen as non-semisimple analogues of the modular tensor categories associated to the quantum representation theory of a simple Lie algebra at a root of unity. I will also explain how these relative modular categories give a precise mathematical construction of U(1|1) Chern--Simons theory. Based on joint work with Nathan Geer.

Introduced by Lusztig in the 1990s, the braid group symmetries constitute an essential part in the theory of quantum groups. The iquantum groups are coideal subalgebras of quantum groups, which arise from quantum symmetric pairs. In recent years, many results for quantum groups have been generalized to iquantum groups.

In this talk, I will present our construction of relative braid group symmetries, associated to the relative Weyl group of the underlying symmetric pair, on iquantum groups. These new symmetries inherit most properties of Lusztig’s symmetries, including compatible relative braid group actions on modules. This is joint work with Weiqiang Wang.