Speaker: Brundan, Jonathan (University of Oregon)
Title: Super Kac-Moody 2-categories
Abstract: I’ll talk about some recent joint work with Alex Ellis. We introduce some generalizations of Kac-Moody 2-categories in which the quiver Hecke algebras of Khovanov, Lauda and Rouquier are replaced by the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka.

Speaker: Cheng, Shun-Jen (Academia Sinica)
Title: CHARACTER FORMULA FOR QUEER LIE SUPERALGEBRAS AND CANONICAL BASES OF TYPE C
Abstract: For the BGG category of modules over the queer Lie superalgebra at half-integer weights we formulate a Kazhdan-Lusztig conjecture in terms of the canonical basis on the tensor power of the natural representation of the quantum group of type C, similar to Brundan’s formulation at integer weights in terms of quantum group of type B. We establish a character formula for the finite-dimensional irreducible of half-integer weights in terms of the type C-canonical basis of a q-deformed wedge space. We then formulate a Kazhdan-Lusztig conjecture for the BGG category of modules over the queer Lie superalgebra of arbitrary weights. This is a joint work with Jae-Hoon Kwon and Weiqiang Wang.

Speaker: Geiß, Christof (Universidad Nacional Autónoma de México)
Title: Geometric construction of simple Lie algebras with symmetrizable Cartan matrix
Abstract: This is jont work with B. Leclerc and J. Schröer. In arXiv:1410.1403 we constructed for a symmetrizable Cartan matrix C with symmetrizer D and an orientation Omega a 1-Iwanaga Gorenstein algebra H(C,D,Omega) over an arbitrary field. The construction is quite similar to the Dlab-Ringel construction of a species for this data, however, we replace field extensions by truncated polynomial rings. In many senses, modules of finite projective dimension over this algebra behave like representations of the corresponding species. We show in this talk how one can extend the geometric realization in terms of constructible functions, due to Schofield, Ringel and Lusztig, of the positive part of a symmetric Kac-Moody algebra at least to all simple Lie algebras by replacing quiver representations by representations of finite projective dimension over the algebras H(C,D,Omega). For now, this requires a case by case analysis but it is remarkably independent of the choice of symmetrizer.

Speaker: Hernandez, David (Univ. Paris Diderot-Paris 7)
Title: Asymptotical representations categorifying cluster algebras
Abstract: In his seminar work, Baxter established that the spectrum of the 6 (and 8)-vertex model can be described in terms of polynomials and of the Baxter's QT-relation. Frenkel-Reshetikhin conjectured that the there is an analog form for the spectrum of more general quantum integrable systems.
Our recent proof (with E. Frenkel) of this conjecture for arbitrary untwisted affine types is based on the study of the "prefundamental representations". We had previous constructed these prefundamental representations with M. Jimbo as asymptotical limits of the Kirillov-Reshetikhin modules over the quantum affine algebra. One of the crucial point for our proof of the conjecture is to establish generalized Baxter's as relations in the Grothendieck ring of a category O containing these prefundamental representations.
In a joint work  with B. Leclerc, we also use this category O and such asymptotical representations to obtain new monoidal categorification of (infinite rank) cluster algebras. In particular, the celebrated Baxter relations for the 6-vertex model get interpreted as Fomin-Zelevinsky mutation relations.
This project is supported by the European Research Council under the European Union's Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine.

Speaker: Kim, Myungho (IMJ-PRG(CNRS/UP7D/UPMC), Paris)
Title: Monoidal categorification of the quantum coordinate ring $A_q(n(w))$
Abstract: In this talk, I will explain our work on the monoidal categorification of the quantum coordinate ring $A_q(n(w))$ of the unipotnet subgroup associated with a symmetric Kac-Moody algebra $g$ and an element $w$ in the Weyl group. The ring $A_q(n(w))$ can be understood as the Grothendieck ring of a certain subcategory $C_w$ of the category of finite-dimensional graded modules over the quiver Hecke algebras corresponding to $g$. Moreover, the set of isomorphism classes of self-dual simple objects in $C_w$ is the upper global basis (dual canonical basis) of the algebra $A_q(n(w))$. These are consequences of the works of Khovanov-Lauda, Rouquier and Varagnolo-Vasserot. On the other hand, due to Geiss-Leclerc-Schröer, it is known that the algebra $A_q(n(w))$ admits a quantum cluster algebra structure. By considering both pictures together, we establish that the quantum cluster monomials in $A_q(n(w)$ belongs to the upper global basis, as conjectured by Geiss-Leclerc-Schröer. We achieve it by showing that for each quantum cluster variable, there is a simple module in $C_w$ whose isomorphism class is the same as the given quantum cluster variable.
This is a joint work with Seok-Jin Kang, Masaki Kashiwara, and Se-jin Oh. 

Speaker: Kimura, Yoshiyuki (Kobe University)
Title: Remarks on quantum unipotent subgroups and the dual canonical basis
Abstract: Quantum unipotent subgroup is the quantum coordinate ring of the unipotent subgroup associated with a finite subset that is defined by a Weyl group element of a symmetrizable Kac-Moody Lie algebra. By the works of Geiss-Leclerc-Schroer and Goodearl-Yakimov, it is known that it has a quantum cluster algebra structure. In this talk, I explain about the quantum coordinate ring of the pro-unipotent subgroup associated with a cofinite subset given by a Weyl group element and its compatibility with the dual canonical basis and the multiplicity-free property between the dual canonical basis element in the quantum unipotent subgroups and the one in the opposite. This talk is based on arXiv:1506.07912.

Speaker: Kleshchev, Alexander (University of Oregon)
Title: Stratifying KLR algebras
Abstract: TBA

Speaker: Lecouvey, Cédric (University of Tours)
Title: Random Littelmann paths and harmonic functions on graphs
Abstract: The purpose of this talk will be to describe some interactions between representation theory of Kac-Moody algebras and conditionings of random trajectories defined from the Littelmann path model. I will explain how representation theory can be used to study a wide class of random walks and, in the opposite direction, how asymptotic behaviors of tensor multiplicities can be deduced from probabilistic arguments. In the spirit of the works by Kerov and Vershik, this can also be related to the study of harmonic functions on certain graphs generalizing the Young graph of partitions. This yields natural problems I will try to evoke.

Speaker: Lee, Kyu-Hwan (University of Connecticut)
Title: Whittaker functions and Demazure characters
Abstract: In this talk, we will consider how to express an Iwahori-Whittaker vector through Demazure characters. Under some interesting combinatorial conditions, we will be able to obtain an inductive formula. It can be considered as a generalization of Casselman-Shalika formula. This is a joint work with Cristian Lenart and Dongwen Liu.

Speaker: Lenart, Cristian (State University of New York at Albany)
Title: Kirillov-Reshetikhin modules, Macdonald polynomials, and related topics
Abstract: In a series of papers with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, we developed a uniform combinatorial model for (tensor products of one-column) Kirillov-Reshetikhin (KR) modules of affine Lie algebras; we also showed that their graded characters coincide with the specialization of symmetric Macdonald polynomials at $t=0$. I will first present our latest work, on the extension of the above results corresponding to the non-symmetric Macdonald polynomials. I will then present a brief survey of related applications, which involve: various computations, properties of KR crystals, local and global Weyl modules, a categorification of Macdonald polynomials, and the quantum $K$-theory of flag varieties.

Speaker: Mathas, Andrew (University of Sydney)
Title: A KLR grading on the Hecke algebras of the alternating groups
Abstract: The Iwahori-Hecke algebra of the symmetric group has subalgebra that is a deformation of the group algebra of the alternating group. I will explain how this algebra inherits a Z-grading from the KLR grading of the Iwahori-Hecke algebra. This is joint work with Clinton Boys.

Speaker: McNamara, Peter J. (University of Queensland)
Title: Face functors for KLR algebras
Abstract: For each k-face of the Weyl polytope, there is an inclusion of semisimple Lie algebras. Using the categorification in terms of KLR algebras, we show how to categorify a quantum version of this inclusion. In finite types, this gives an equivalence of categories with a certain cuspidal subcategory of KLR-modules. We also discuss some applications of this functor in affine type.

Speaker: Morse, Jennifer (Drexel University)
Title: Applications of Macdonald polynomials to Schubert calculus
Abstract: We will discuss a long-standing open problem concerning the combinatorics of Macdonald polynomials and see how the study impacts problems in quantum, affine and equivariant Schubert calculus. This is joint work with Ryan Kaliszewski.

Speaker: Oh, Se-jin (University of Oregon)
Title: Special family of reduced expressions of the longest element and categorification of PBW bases associated with the family
Abstract: It is well-known that (dual) PBW bases for quantum group of finite type are associated by reduced expressions of the longest element of Weyl group. The categorifications of (dual) PBW bases were studied by Kato, MaNamara, Brundan-Kleshchev-McNamara via modules over KLR-algebras. On the other hand, dual PBW bases associated by special family of reduced expressions are also categorified via modules over quantum affine algebras due to the work of Hernandez-Leclerc and Kang-Kashiwara-Kim-Oh. In this talk, I will explain the structure of the PBW bases associate the special family more explicity by introducing new notion on the sequence of positive roots.

Speaker: Park, Euiyong (University of Seoul)
Title: Affinizations and $R$-matrices for quiver Hecke algebras
Abstract: In this talk, I will explain our works on affinizations and R-matrices for arbitrary quiver Hekcke algebras, which generalizes affinizations and R-matrices for symmetric quiver Hecke algbras. It is shown that they enjoy similar properties to those for symmetric quiver Hecke algebras. I will also talk about a duality datum $mathcal{D}$ and the corresponding duality functor $mathfrak{F}^{mathcal{D}}$ between graded module categories of two quiver Hecke algebras $R$ and $R^{mathcal{D}}$ arising from $mathcal{D}$. The functor $mathfrak{F}^{mathcal{D}}$ sends finite-dimensional modules to finite-dimensional modules, and is exact when $R^{mathcal{D}}$ is of finite type. It is proved that affinizations of real simple modules and their $R$-matrices give a duality datum. This is a joint work with Masaki Kashiwara (arXiv:1505:03241).

Speaker: Schilling, Anne (University of California, Davis)
Title: Braid Moves in Commutation Classes of the Symmetric Group
Abstract: We prove that the expected number of braid moves in the commutation class of the word $(s_1 s_2 cdots s_{n-1})(s_1 s_2 cdots s_{n-2}) cdots (s_1 s_2)(s_1)$ for the long element in the symmetric group $mathfrak{S}_n$ is one. This is a variant of a similar result by V.~Reiner, who proved that the expected number of braid moves in a random reduced word for the long element is one. The proof uses X.~Viennot's theory of heaps and variants of the promotion operator. In addition, we provide a refinement of this result on orbits under the action of even and odd promotion operators. This gives an example of a homomesy for a nonabelian (dihedral) group that is not induced by an abelian subgroup. We extend this further to more general posets. This is joint work with Nicolas Thiery, Graham White and Nathan Williams.

Speaker: Tanisaki, Toshiyuki (Osaka City University)
Title: Characters of integrable highest weight modules over a quantum group
Abstract: We show that the Weyl-Kac type character formula holds for the integrable highest weight modules over the quantized enveloping algebra of any symmetrizable Kac-Moody Lie algebra, when the parameter $q$ is not a root of unity.

Speaker: Tsuchioka, Shunsuke (The University of Tokyo)
Title: On a general Schur's partition identity
Abstract: We will talk on a generalization of Schur's partition identity which is a kind of Rogers-Ramanujan type identity. Our identity comes from K"ulshammer-Olsson-Robinson theory of generalized blocks and the Fock space representations of quantum affine algebras due to Kashiwara-Miwa-Petersen-Yung. This is a joint work with Masaki Watanabe (University of Tokyo).

Speaker: Wang, Weiqiang (University of Virginia)
Title: Quantum symmetric pairs, Hecke algebras, and flag varieties of affine type
Abstract: Generalizing the work of Beilinson-Lusztig-MacPherson of finite type A, Bao-Kujawa-Li-Wang constructed the quantum algebras arising from partial flag varieties of finite type C, altogether with their canonical bases. They also provided a geometric realization of a Schur-type duality between these algebras and the Hecke algebras of type C acting on a tensor space. These quantum algebras are coideal subalgebras of quantum gl(n), which form quantum symmetric pairs with quantum gl(n). The above can be redeveloped within the framework of Hecke algebras without geometry. In this talk I will explain the affinization of above in both (Hecke algebraic and geometric) approaches, which requires substantial new work. This is joint work with Zhaobing Fan, Chun-Ju Lai (UVA), Yiqiang Li, Li Luo.