Abstract. Cluster Algebras and their quantum counterparts play an important role in representation theory, combinatorics and topology. In relation to noncommutative algebra there are several open problems on the existence of cluster algebra structures on certain families of quantized coordinate rings. We will describe a result that proves the existence of quantum cluster algebra structures on a very general, axiomatically defined class of quantum nilpotent algebras. This has a broad range of applications, among which are a proof of the Berenstein-Zelevinsky conjecture for quantum double Bruhat cells, construction of quantum cluster algebra structures on quantum unipotent groups in full generality, and others.

Abstract. Let $\mathfrak g$ be a complex simple Lie algebra and $\mathfrak h$ its Cartan subalgebra. The Clifford algebra $C(\mathfrak g)$ of $\mathfrak g$ admits a Harish-Chandra map, which turns out to map primitive $\mathfrak g$-invariants in $C(\mathfrak g)$ to $\mathfrak h$. I will discuss a conjecture of Kostant which says that the image of a certain alternating invariant of degree $2m+1$ under this map is the zero weight vector of the simple $(2m+1)$-dimensional module of the principal $\mathfrak{sl}_2$-triple in the Langlands dual of $\mathfrak g$. My original proof of this conjecture was found to be incomplete if $\mathfrak g$ is not of type $A$. A complete proof was subsequently given by Joseph and Alekseev-Moreau.

Abstract. I this talk I wil briefly recall occurrences of parabolic Kazhdan-Lusztig polynomials of hermitian symmetric cases and related them to the representation theory of Brauer and walled Brauer algebras. In particular we will show that these algebras are Koszul.

Abstract. In this talk I will introduce the geometry of quiver Hecke algebras and their generalizations to $q$-Schur algebras. As a result we show that decomposition numbers of cyclotomic Schur algebras are governed by Fock space.

Abstract. We will prove that the category $\mathscr O$ of Cherednik algebras of cyclotomic type is equivalent, as a highest weight category, to the parabolic affine category $\mathscr O$ of type $A$ at a negative level. This implies several conjectures concerning the category $\mathscr O$ of Cherednik algebras.

Abstract. Group algebras and path algebras of quivers are two important types of finite-dimensional algebras. Many homological and combinatorial tools have been developed to study their representations. In this talk I will describe representations of some algebras with local group structures or local group actions. Examples includes category algebras of finite EI categories and skew group algebras. In particular, I will introduce a generalized Koszul theory for locally finite graded algebras with non-semisimple degree 0 parts. This generalized theory preserves many classical results such as the Koszul duality, and has a close relation to the classical theory.

Abstract. A finite EI category is a small category with finitely many morphisms such that every endomorphism is an isomorphism. They includes finite groups, posets, quivers as examples, and are studies in Topology (transformation groups), group representations (transporter categories, orbit categories, fusion systems), homological algebra (support varieties). In this talk I will describe representations of finite EI categories, characterize finite EI categories with hereditary category algebras, and discuss their representation types (if time allows).

Abstract. This is an introductory talk about a class of associative algebras constructed from a base ring $R$, a set of ring automorphisms of $R$, and a set of distinguished elements of $R$. This class contains many interesting algebras (skew group algebras with free abelian group, (quantized) Weyl algebras, (quantized) enveloping algebra of $\mathfrak{sl}_2$, certain Mickelsson-Zhelobenko step algebras, etc.). I will review some historical background which led to their definition, some structural results (consistency relations, Cartan matrices) and their representation theory (simple weight modules).

Abstract. In recent joint work with Vera Serganova we found that acyclic Dynkin diagrams with loops parametrize (equivalence classes of) certain embeddings of twisted generalized Weyl algebras (TGWAs) into the n:th Weyl algebra. This construction gives new solutions to the consistency relations for TGWAs. Examples include primitive quotients of enveloping algebras related to completely pointed simple weight modules in types $A$ and $C. We also have some partial results in the super algebra case.

Abstract. Almost split sequences are primary objects of study in Auslander-Reiten theory. In this talk, I will give the definition of an almost sequence and discuss various properties these sequences enjoy.

Abstract. Auslander-Reiten theory is a central topic of algebraic representation theory. Following the talk given by Jacob West, we continue to explore more aspects of this theory. In this talk I will introduce irreducible morphisms and Auslander-Reiten quivers. They have been proved to be very useful and have a lot of important applications. Topological and combinatorial structures of AR quivers, explored by Riedtmann, Gabriel and others, will be described. Some current developments (AR-triangles, interaction with tilting theory, etc.) will be mentioned as well.

Abstract. Quiver mutation is an elementary operation on quivers which appeared in physics in Seiberg duality in the 1990s and in mathematics in Fomin-Zelevinsky's definition of cluster algebras in 2002. In this talk, I will show how, by comparing sequences of quiver mutations, one can construct identities between products of quantum dilogarithm series. These identities generalize Faddeev-Kashaev-Volkov's classical pentagon identity and the identities obtained by Reineke. Morally, the new identities follow from Kontsevich-Soibelman's theory of Donaldson-Thomas invariants. They can be proved rigorously using the theory linking cluster algebras to quiver representations.

Abstract. In cluster algebras, after making several mutations of sends, you may sometimes end up with the initial seed. That is the periodicity phenomenon in cluster algebras. Periodicity is a rare event, but once you have it, you can also get the associated dilogarithm identity, plus its quantum version, for free!

There are two basic questions for periodicity: How to find it and how to prove it? The answer to the second question is given by the tropicalization method, which I explain in this talk by several examples.

The first question is more difficult, and I do not know the answer. However, we are lucky to have several (infinitely many) conjectured periodicities from the Bethe ansatz method in 90's, even before the birth of cluster algebras, and they are recently proved by the tropicalization method. There is always some root system behind the scene.

The talk is based on the work with R. Inoue, O. Iyama, B. Keller, and A. Kuniba.

Abstract. We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system. (This is joint work with Michael Barot.)

Abstract. We will discuss a realization of the crystal $B(\infty)$ adapted for the action of an admissible diagram automorphism, and our progress towards a combinatorial description of a natural subcrystal for folded Cartan datum.

Abstract. The study of the structure of irreducible representations of a quantum affine algebra can be reduced to the so called prime representations, those which cannot be written as a tensor product of two non-trivial simple representations. In their recent paper, Prime Representations from a Homological perspective, V. Chari, A. Moura and C. Young work to understand these prime representations via self extensions. Namely, they conjecture that an irreducible finite dimensional representation $V$ is prime if and only if the space of self extensions has dimension 1. I will be presenting some of the results of this paper that prove the conjecture for the $\mathfrak{sl}_2$ case, and give partial evidence in the case for general $\mathfrak g$.

Abstract. B. Kostant introduced a class of modules for finite dimensional complex semisimple Lie algebras. He called them Whittaker modules because of their connection with the Whittaker equations that arise in the study of the associated Lie group. Since then, a number of others have further developed the idea of Whittaker modules for Lie algebras. Recently, in a joint work with K. Christodoulopoulou and E. Wiesner, we have adapted some of these ideas to the setting of Lie superalgebras.

Abstract. I will give an overview of the 2007 paper by S. Witherspoon, "Twisted Graded Hecke Algebras." Given a finite group $G$ acting on a finite dimensional complex vector space $V$ and a 2-cocycle $\alpha$, we may form the twisted crossed product algebra $TV \#_\alpha G$. The main theorem gives conditions for a PBW basis for certain quotients of this algebra. A twisted graded Hecke algebra is one that satisfies these conditions. One can think of these algebras as certain deformations of $SV \#_\alpha G$. I will describe several examples, including symplectic reflection algebras and the case $G \cong (\mathbb{Z}/m \mathbb{Z})^n$.

Abstract. Given an abelian category, one of the natural questions to be addressed is that of understanding the space of extensions between its simple objects. For the category of finite-dimensional representations of an affine Kac-Moody algebra, this question has been answered in the last few years. The quantum version of this category is far more complicated and the answer to this question remains open. We shall discuss some ideas towards the answer with the help of the concept of $q$-characters and show, via examples, that the quantum answer is different from the classical one in an essential way. An interesting feature of the category of finite-dimensional representations of a quantum affine is that it has simple objects which are not prime, i.e., which are isomorphic to a tensor product of two nontrivial simple objects. It is then natural to try to classify the prime ones. Although this classification is also unknown, the amount of known examples of prime modules has been growing. In the main part of this talk we shall present results from a joint paper with V. Chari and C. Young relating the study of prime representations to that of the space of extensions between simple modules. In particular, we show that, if the underlying simple Lie algebra is $\mathfrak{sl}(2)$, then a simple representation is prime if and only if the space of its self extensions is one-dimensional. It is tempting to conjecture that this is true in general and we construct a large class of prime representations satisfying this homological property.

Abstract. Complexity is an established invariant of modules in non-semisimple settings. For a given module $M$ it is defined as the rate of growth of the minimal projective resolution of $M$. Hence, in a sense, it is a measure of how far the module is from being projective. I'll introduce the complexity and its geometric interpretation in the setting of finite groups as an example. I'll also give the results of the recent calculation of the complexity of the simple modules for the complex Lie superalgebra $\mathfrak{gl}(m|n)$ (along with an intriguing geometric interpretation). I'll explain all the necessary background as we go, so all are welcome. Our work is joint with Brian Boe and Dan Nakano.

Abstract. We will discuss a realization of the crystal $B(\infty)$ adapted for the action of an admissible diagram automorphism, with the aim of describing a natural subcrystal for folded Cartan datum.

Abstract. We study finite-dimensional representations of twisted current algebras, especially Demazure and (twisted) Weyl modules. First we identify these Weyl modules with corresponding affine Demazure modules, then we give an explicit construction from untwisted Weyl modules which generalize the fusion

Abstract. I will talk about recent joint work with V. Futorny in which we prove that the quantized enveloping algebra $U_q(\mathfrak{gl}_n)$ has the structure of a Galois order, a certain subring of invariants in a skew group algebra. As an application we describe explicitly the structure of the division ring of fractions of $U_q(\mathfrak{gl}_n)$, in particular obtaining a new proof of the quantum Gelfand-Kirillov conjecture for $\mathfrak{gl}_n$. Secondly we prove that the Gelfand-Tsetlin subalgebra of $U_q(\mathfrak{gl}_n)$ is maximal commutative, and obtain a parametrization of irreducible Gelfand-Tsetlin modules over $U_q(\mathfrak{gl}_n)$.

Abstract. We will define and study deformations along several parameters of the enveloping algebra of a semisimple finite dimensional Lie algebra, called the Generalized Quantum Enveloping (GQE) algebras. We will see how GQE algebras can be used to define Langlands Interpolating Quantum (LIQ) groups and solve conjectures motivated by the geometric Langlands program. These conjectures have been suggested by E. Frenkel and D. Hernandez in an original work on different LIQ groups.

Abstract. The Kostant partition function can be used to determine the weight multiplicities associated to irreducible representations of Kac-Moody algebras. Its $t$-analog was used by Lusztig to define a $t$-analog of weight multiplicity. We study Lusztig's $t$-weight multiplicities associated to the level one representation of twisted affine Kac-Moody algebras. We will derive a closed form expression for the corresponding $t$-string function using constant term identities of Macdonald and Cherednik. We describe how generalized exponents of certain representations of the underlying finite dimensional simple Lie algebra enter the picture.

Abstract. In my talk (based on a joint work with David Kazhdan) I will introduce a class of quantum Hankel algebras which are flat deformations of polynomial algebras and admit a number of automorphisms and same number of derivations. The simplest example is the quadratic algebra $H_1$ generated by $\{X_n\}$, where $n$ runs over integers, with a single relation $X_1X_0=qX_0X_1$, where $q$ is not a root of unity and the remaining relations coming from an automorphism and a derivation of $H_1$ both sending $X_n$ to $X_{n+1}$. Quite surprisingly, $H_1$ is a flat deformation of polynomials in infinitely many variables and:

• admits a canonical basis,
• has a quantum cluster structure,
• contains a $q$-deformation of the so called $Q$-system of type $A$ (the latter one is the set of characters of Kirillov-Reshetikhin modules over affine quantum groups of type $A$),
• each member of this $q$-deformed $Q$-system is a quantum Hankel determinant in $\{X_n\}$,
• each subalgebra of $H_1$ generated by $X_1,\dots,X_n$ is a flat deformation of polynomials in $n$ variables.

I will also define the "$k$-dimensional" quantum Hankel algebra $H_k$ whose generators are labeled by the $k$-dimesnional lattice $\mathbb Z^k$ and whose relations are determined by some basic ones and by $k$ automorphisms and $k$ derivations; and will demonstrate that these algebras share many properties of $H_1$. Ultimately, I will explain that the flatness of $H_k$ and its generalizations follows from the (no less surprising) observation that Hecke algebras "look like" Hopf algebras, which allows to produce many new solutions of the quantum Yang-Baxter equation (QYBE) out of a given initial one.

Abstract. Among finite dimensional modules of a quantum affine algebra, there is a distinguished family called KR modules. It is known that, by taking the restricted classical limit, a KR module becomes isomorphic to a certain Demazure module. In this talk, I will generalize this result to a tensor product of KR modules. In this case its restricted classical limit becomes isomorphic to a certain generalized Demazure module defined using Joseph functor. If time permitted, I will introduce some application of this result.

Abstract. The Lie algebra $\mathfrak g[t] = \mathfrak g\otimes \mathbb C[t] $ associated to a simple Lie algebra $\mathfrak g$ is called the current algebra, and has connections to the representation theory of the loop and affine Lie algebras associated to $\mathfrak g$. Of particular interest is the graded representation theory of $\mathfrak g[t]$. In a joint work with V. Chari and N. Manning we proved a BGG-type reciprocity formula for the graded representations of $\mathfrak{sl}_2[t]$ which suggested the presence of a highest weight category theory and of tilting modules. In this talk I will outline a construction of tilting modules for $\mathfrak {sl_2}[t]$ and explain a sufficient condition for the construction to work in general.

Abstract. A Loewy series of a module of finite length is by definition a semisimple filtration whose length is smallest, and it is a fundamental problem to determine the Loewy series of a module. In this talk, we study this problem for Weyl modules of a current algebra for ADE type. In this case, we can show the Loewy series is unique and coincides with the natural filtration given by its grading. As an application of this result, we can show that the Weyl module is isomorphic as graded modules to a standard modules, which are defined as the homology groups of quiver varieties. Hence we can study the Poincare polynomials of quiver varieties using Weyl modules. This talk is based on the joint work with Ryosuke Kodera.

Abstract. For any finite dimensional basic associative algebra, we study the presentation spaces and their relation to the representation spaces. We prove two propositions about a general presentation, one on its subrepresentations and the other on its canonical decomposition. As a special case, we consider rigid presentations. We show how to complete a rigid presentation and study the number of nonisomorphic direct summands and different complements. Based on that, we construct a simplicial complex governing the canonical decompositions of rigid presentations and provide some examples.

Abstract. I will present some systems of short exact sequences in the categories of finite-dimensional representations of quantum affine algebras of types $A$ and $B$. These systems contain the $T$-system of relations among Kirillov-Reshetikhin modules, and extend it to include, for example, all minimal affinizations. I will outline the proofs, which use the theory of $q$-characters, and comment on what can be expected in other types. This is joint work with E. Mukhin.

Abstract. We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux, which are periodic continuations of ordinary skew tableaux, employing a statistical lattice model and non-intersecting paths. We show that the cylindric Macdonald functions appear in the coproduct of a commutative Frobenius algebra, which can be interpreted as a one-parameter deformation of the $\mathfrak{sl}(n)$ Verlinde algebra, i.e. the structure constants of the Frobenius algebra are polynomials in a variable t whose constant terms are the Wess-Zumino-Novikov-Witten fusion coefficients. The latter are known to coincide with dimensions of moduli spaces of generalized theta-functions and multiplicities of tilting modules of quantum groups at roots of unity. Alternatively, the deformed Verlinde algebra can be realised as a commutative subalgebra in the endomorphisms over a Kirillov-Reshetikhin module of the quantum affine $\mathfrak{sl}(n)$ algebra. Acting with special elements of this subalgebra on a highest weight vector, one obtains Lusztig's canonical basis.

Abstract. I will give a brief introduction to (infinitesimal) symplectic reflection algebras. Hochschild cohomology provides information about the deformation theory of an associative algebra. Etingof and Ginzburg have computed the Hochschild cohomology of symplectic reflection algebras. In the hope of obtaining an analogous result for the infinitesimal case, I will make a first step of computing the Hochschild cohomology for the undeformed algebra $SV \rtimes U(\mathfrak{sl}_2)$.

Abstract. We consider a twisted loop superalgebra built from a Lie superalgebra of type $Q$. After presenting some of its properties, we will introduce a quantization of a certain bisuperalgebra structure and we will explain how this new quantized enveloping algebra is related to affine Hecke-Clifford algebras. This is a $q$-version of previous work of M. Nazarov about the Yangian attached to Lie superalgebras of type $Q$.

Abstract. Complex semisimple Lie algebras, as well as a variety of other Lie algebras including the Virasoro algebra, possess a triangular decomposition: $\mathfrak g=\mathfrak n^- \oplus \mathfrak h \oplus \mathfrak n^+$ where $\mathfrak h$ is a Cartan subalgebra and $\mathfrak n^{\pm}$ are maximal nilpotent subalgebras. Whittaker modules are defined in terms of this decomposition and sit naturally inside a larger category of modules that I refer to as a Whittaker category. I'll discuss some of the historical development of these ideas, as well as my own work on the Whittaker category for the Virasoro algebra.

Abstract. Vertex algebroids can be regarded as distant relatives of rings of twisted differential operators (TDO) on smooth varieties. The latter are employed in the classical Beilinson-Bernstein equivalence result that relates $\mathfrak g$-modules to twisted $D$-modules on the flag variety of $\mathfrak g$. I will make a quick introduction to vertex algebroids and show how they can be used to construct a version of Beilinson-Bernstein localization for a class of modules over affine Lie algebras at the critical level.

Abstract. Global Weyl modules, for generalized loop algebras $\mathfrak g\otimes A$, where $\mathfrak g$ is a simple finite dimensional Lie algebra and $A$ is an associative commutative algebra, have been defined and studied for any dominant integral weight $\lambda$. We show that the space of morphisms between global Weyl modules shares some properties with the space of morphisms between Verma modules.

Abstract. Generalized Weyl algebras (GWAs) include well-known examples such as the Weyl algebra and classical and quantum ${\mathfrak{sl}}(2)$. At the same time, they contain "non-Noetherian examples" such as continuous Hecke algebras (defined by Etingof, Gan, and Ginzburg). We study blocks of the BGG category $\mathscr O$ over a GWA, with finitely many simple objects. We compute the Ext-quiver (with relations) of the endomorphism algebra of the projective generator. We also show that this algebra is Koszul and satisfies the Strong Kazhdan-Lusztig condition.

Abstract. Given a finite dimensional simple Lie algebra $\mathfrak g$ over $\mathbb C$ and a commutative associative $\mathbb C$-algebra with unity $A$, we exhibit a $\mathbb Z$-form for the universal enveloping algebra of $\mathfrak g\otimes A$ and an explicit $\mathbb Z$-basis for this $\mathbb Z$-form. We also produce explicit commutation formulas in the universal enveloping algebra of $\mathfrak{sl}_2\otimes A$ that allow us to write certain elements in Poincaré-Birkhoff-Witt order.

Abstract. Examples of almost commutative algebra are abundant in representation theory. In positive characteristic, these algebras tend to be finite over their centers. In this talk I will discuss Kac-Weisfeiler type estimates for dimensions of irreducible modules of an almost commutative algebra in terms of dimensions of symplectic leaves of the corresponding Poisson variety. Applications to symplectic reflection algebras will be discussed.

Abstract. In this talk I will present combinatorial definitions of Kostka numbers and Kostka polynomials, their connection to the algebra of invariant polynomials and some applications to Lie Theory.

Abstract. One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category $\rm{Vect}$. In the past this problem has been resolved by working with a weaker structure called a "twisted" bialgebra. In this talk we will present a different solution by first switching to a new underlying category ${\rm Vect}^K$ of vector spaces graded by a group $K$ called the Grothendieck group. We equip this category with a nontrivial braiding which depends on the $K$-grading. With this braiding, we find that the Hall algebra does satisfy the bialgebra condition exactly for the standard multiplication and comultiplication in this category, and can also be equipped with an antipode, making it a Hopf algebra object in ${\rm Vect}^K$.

Abstract. Given a finite-dimensional, simple Lie algebra $\mathfrak g$ of rank $n$ with a Chevalley basis $$ \{x_\alpha^\pm,\,h_i:\alpha\in R^+,i\in\{1,\dots,n\}\} $$ H. Garland in 1978 found formulas in the universal enveloping algebra of the loop algebra of $\mathfrak g$ for $$ (x_\alpha^+\otimes 1)^{(k-r)}(x_\alpha^-\otimes t)^{(k)} $$ for all $k,r\in\mathbb Z_{\ge 0}$ with $r\le k$. We have generalized this result for $r\in\{0,1\}$ modulo $\mathbf U(\mathfrak g\otimes A)(\mathfrak n^+\otimes A)$ where $A$ is any commutative algebra with unity. I will discuss this generalization and some applications to Weyl modules.

Abstract. A classical result in Lie theory stipulates that a simple finite dimensional Lie algebra of type BCFG can be constructed as the subalgebra of a Lie algebra of type ADE fixed by an admissible diagram automorphism of the latter. This construction is known as folding and extends to Kac-Moody Lie algebras. Although foldings do not admit direct quantum analogues, it can be shown that there exists an embedding of crystals for the corresponding Langlands dual Lie algebras. The aim of this talk is to introduce algebraic analogues and generalizations of foldings in the quantum setting which yield new flat quantum deformations of non-semisimple Lie algebras and of Poisson algebras (joint work with A. Berenstein).

Abstract. We discuss finite generation of the relative cohomology rings for Lie superalgebras. We formulate a definition for detecting subalgebras and also discuss realizability of support varieties. As an application we compute the relative cohomology ring of the Lie superalgebra $\bar S(n)$ relative to the graded zero component $\bar S(n)_0$ and show that this ring is finitely generated. We also compute support varieties of all simple modules in the category of finite dimensional $\bar S(n)$-modules which are completely reducible over $\bar S(n)_0$.

Abstract. We will discuss representation theory of $\mathfrak{sl}_2$, the current algebra, and a conjecture of Chari and Greenstein.

Abstract. This talk will be based on the paper On Blocks and Modules for Whittaker Pairs by P. Batra and V. Mazorchuk, in which they describe a general framework for the study of Whittaker modules. We will give an overview of their main results.

Abstract. This talk is based on a paper in which V. Ginzburg defines certain pairs of commuting elements in a semisimple Lie algebra. I will present some results with special attention to the case of $\mathfrak{sl}_n$.

Abstract. I will speak on Matumoto's theorem that gives a necessary condition for existence of nonzero Whittaker vectors in terms of the associated variety.

Abstract. In this talk I will present a few results regarding the structure (characters) of minimal affinizations of quantum groups. The results are obtained by applying a strategy developed by Chari and the speaker which consists of comparing the classical limit of the minimal affinizations with certain graded modules for the underlying current algebra. I will explain part of this method. The concept of minimal affinizations was introduced by Chari motivated by the impossibility of defining the concept of evaluation modules in the quantum setting (in case the underlying simple Lie algebra is not of type A). An important subclass of minimal affinizations is that of Kirilov-Reshetikhin modules which appears in the mathematical-physics literature. About 4 years ago, Chari and the speaker obtained several character formulas for Kirilov-Reshetikhin modules using this strategy. The present talk will focus mostly on a new paper by the speaker which focuses on extending the method to more general minimal affinizations.

Abstract. I will briefly summarize what is known and what is not known about the cohomology and representation theory of modular Lie (super)algebras.

Abstract. We give generators and relations for the associated graded module of an irreducible $\mathfrak{sl}_n$-module with respect to the PBW filtration. As an application we obtain a new basis and pattern for the $\mathfrak{sl}_n$-module.

Abstract. In this talk I will explain the classification of unitary irreducible representations in the highest weight category of the rational Cherednik algebra of the symmetric group and how unitarity is preserved by the KZ functor, that maps highest weight modules to modules over the corresponding Hecke algebra.

Abstract. The talk will start with an introduction to cohomology, support varieties for finite groups over a field of positive characteristic. After that we will discuss support variety theories for algebraic structures other than finite groups such as restricted Lie algebras, algebraic groups, quantum groups and Lie superalgebras.

Abstract. In this second talk we will discuss cohomology and support varieties for Lie superalgebras over the field of complex numbers. I will focus on examples and present some results joint with Jonathan Kujawa and Daniel Nakano.

Abstract. We study the variety of $n\times n$ matrices with commutator of rank at most one. We describe its irreducible components; two of them correspond to the pairs of commuting matrices, and $n-2$ components of smaller dimension corresponding to the pairs of rank one commutator. In our proof we define a map to the zero fiber of the Hilbert scheme of points and study the image and the fibers.

Abstract. The colored HOMFLY polynomial is a quantum invariant of oriented links in $S^3$ associated with a collection of irreducible representations of each quantum group $U_q(\mathfrak{sl}_N)$ for each component of the link. We will discuss in detail how to construct these polynomials and their general structure. Then we will discuss the new progress, Labastida-Marino-Ooguri-Vafa conjecture. The LMOV conjecture also gives the application of Lichorish-Millet type formula for links. The corresponding theory of colored Kauffman polynomials associated to quantum group $U_q(\mathfrak{so}_{2N+1})$ and orthogonal version of the LMOV conjecture can also be developed in a same fashion by using more complicated algebra structures.

Abstract. This talk is intended as an elementary and informal introduction to the Beilinson-Drinfeld notion of a chiral algebra.

Abstract. The Duflo theorem is a statement in Lie theory which allows us to compute the ring structure of the center of the universal enveloping algebra of a finite-dimensional Lie algebra. A categorical version of it was used by Maxim Kontsevich to give a spectacular proof of the so-called "Theorem on complex manifolds," which computes the multiplicative structure of Hochschild cohomology of a complex manifold in terms of the algebra of polyvector fields. In Lie theory there are also more general Duflo-type statements (mostly conjectural), which study the case of a pair (Lie algebra, Lie subalgebra). I will explain how these translate into conjectures about the multiplicative structure of the Ext-algebra of the structure sheaf of a complex submanifold of a complex manifold, and how from this interaction we can hope to gain new insights into both algebraic geometry and Lie theory. (Based on discussions with Damien Callaque.)

Abstract. One motivation for studying quantum groups and braided tensor categories is that they provide a method for constructing representations of the braid groups of type $A_n$. It is natural to ask what extra structure on a braided tensor category is required to yield back representations of higher genus braid groups, for example the so-called double affine braid groups, which are $\pi_1$ of the configuration space of points on an elliptic curve. In this talk we explain that in types $A_n$ and $BC_n$, the algebra $\mathscr D$ of quantum differential operators provides this extra structure; more precisely, for any (quantum) $\mathscr D$-module, we construct representations of elliptic braid groups of types $A_n$ and $BC_n$. Connections to classical Lie theory are provided via the theory of double affine Hecke algebras and their degenerations. The $BC_n$ constructions we describe are joint work with Xiaoguang Ma.

Abstract. This will be a series of introductory talks on linear algebraic groups. The goal is to acquire a working knowledge of the subject rather than a systematic development of the theory. Prerequisites will be minimal and I will recall anything from algebraic geometry which we need.

Abstract. We discuss total positivity for matrices and use this as an motivation for the definition of cluster algebras which we will recall. We also introduce some more recent concepts: $g$-vectors and $F$-polynomials.

Abstract. We recall (our version) of Lusztig's construction of the corresponding envelopping algebra $U(\mathfrak n)$ and of Verma and Irreducible modules for the corresponding simple Lie algebra. This is fundamental for the construction of our cluster character.

Abstract. We explain how certain subcategories of the representations of a preprojective algebra categorify the cluster algebra structure for the unipotent cells of the corresponding Lie group - where we come back to total positivity.

Abstract. I will describe some classical results for the boson-fermion correspondence in the context of affine Lie algebras.

Abstract. I will introduce new families of quantum algebras of double affine type which can be seen as Lie algebra analogs of certain algebras of Hecke type which have become of interest in the past ten years, namely Cherednik algebras, symplectic reflection algebras and deformed preprojective algebras. Those quantum algebras are deformations of the enveloping algebra of $\mathfrak{sl}_{n}$ over either $\mathbb C[u,v] \rtimes \Gamma$, where $\Gamma$ is a finite subgroup of $SL_{2}(\mathbb C)$, or over $\Pi(Q)$ (the preprojective algebra of a quiver $Q$). They are related to Yangians.

Abstract. The talk will be based on Kostant's 1978 paper in Inventiones.

Abstract. I will give an interpretation of Littelmann's generalized Gelfand-Tsetlin patterns as saying that computing weight multiplicities for semisimple complex Lie algebras is equivalent to counting points with integral coordinates in certain families of polytopes. The notion of a "chopped and sliced cone" formalizes this kind of families. Using the Blakley-Sturmfels theorem on vector partition functions I obtain properties of functions described by chopped and sliced cones, notably a version of the Duistermaat-Heckman theorem in this context. When applied to semisimple complex Lie algebras this generalizes ideas of Billey-Guillemin-Rassart for $\mathfrak{sl}_{k}(\mathbb C)$. I will present some general results as well as an explicit calculation of the complete character table for $\mathfrak{so}_{5}(\mathbb C)$.

Abstract. I will review some results by Milicic-Soergel on modules induced from Whittaker modules (in the sense of Kostant) in the setting of complex semisimple Lie algebras. Then I will describe some extensions of these results in the context of affine Lie algebras.

Abstract. I will describe the irreducible Whittaker modules for the Lie algebra formed by adjoining a degree derivation to an infinite-dimensional Heisenberg Lie algebra. I will use these modules to construct a new class of modules for non-twisted affine Lie algebras and I will describe an irreducibility criterion for them.

Abstract. In this talk I will introduce co-Poisson module algebras and their quantizations as a framework for studying quantizations of module algebras. Then I will give some examples, ranging from quantizations of semidirect Lie bialgebra structures to quantized symmetric algebras derived from subalgebras of the ad-finite part of quantized enveloping algebra $U_{q}(\mathfrak g)$.

Abstract. We study a family of infinite-dimensional algebras that are similar to semisimple Lie algebras as well as symplectic reflection algebras. Infinitesimal Hecke algebras over $\mathfrak{sl}(2)$ have a triangular decomposition and a nontrivial center, which yields an analogue of Duflo's Theorem (about primitive ideals), as well as a block decomposition of the BGG category $\mathscr O$. These algebras also have a quantized version, with similar representation theory; in particular, category $\mathscr O$ has a block decomposition, even though the center is trivial. Finally, we discuss some questions about the higher rank cases. (Joint with A.Tikaradze, and also with W.L.Gan.)

Abstract. We study a class of structures in groups which provide a powerful tool in classifying the finite simple groups of Lie type. In this talk, we shall discuss the definitions and some basic properties of these systems, provide some examples, and provide further motivation, including a proof that any group with a Tits system admits a Bruhat decomposition.

Abstract. We use some results on Coxeter systems to prove properties of subgroups of $G$ containing $B$ (for the Tits system $(G,B,N,S)$). We define parabolic subgroups of $G$ and show some of their basic properties.

Abstract. Using the results established by the other talks, I will work up to a theorem which establishes a simplicity condition on certain subgroups of a group with a Tits system, and exhibit an example of its use.

Abstract. In this talk, we will discuss some general facts about the group of automorphisms, $\text{Aut}(\mathfrak g)$, of a simple Lie algebra $\mathfrak g$ over $\mathbb C$, including the classification of all finite-order elements in $\text{Aut}(\mathfrak g)$. In particular, we will focus on automorphisms of order 2, also known as involutions, and the associated $\mathbb Z_2$-gradings of $\mathfrak g$ that they induce.

Abstract. In this talk, we begin with the definitions of the degenerated affine Hecke algebra (dAHA) and the degenerated double affine Hecke algebra (dDAHA). We will describe a Lie-theoretic construction of representations of the dAHA of type $A_{n}$ (given by T. Arakawa and T. Suzuki), the dDAHA of type $A_{n}$ (given by D. Calaque, B. Enriquez and P. Etingof) and the dAHA and dDAHA of type $BC_{n}$ (given by P. Etingof, R. Freund and X.Ma). Then we will talk about what kinds of dAHA modules we get from above Lie-theoretic constructions.

Abstract. Let $G$ be an adjoint algebraic group of type $B$, $C$ or $D$ defined over a field $\mathbb k$ of characteristic 2 and $\mathfrak g$ be the Lie algebra of $G$. Let $\mathfrak g^{*}$ be the dual vector space of $\mathfrak g$. We classify the nilpotent orbits in $\mathfrak g$ over a finite field $\mathbb k$ and construct Springer correspondence for the nilpotent variety in $\mathfrak g$. The correspondence would be a bijective map between the set of isomorphism classes of irreducible representations of the Weyl group of $G$ and the set of all pairs $(c,F)$ where $c$ is a nilpotent $G$-orbit in $\mathfrak g$ and $F$ is an irreducible $G$-equivariant local system on $c$ (up to isomorphism). We also classify the nilpotent orbits in $\mathfrak g^{*}$ over an algebraically closed or a finite field $\mathbb k$ and construct Springer correspondence for the nilpotent variety in $\mathfrak g^{*}$.

Abstract. To a group $G$ acting discretely on a simply connected complex manifold $X$, I will attach a Hecke algebra $\mathcal H_{q}(G,X)$, which is a deformation of the group algebra of $G$. We will see that if $H^{2}(X,\mathbb C)=0$ then this deformation is flat. We will also see that this setting unifies many known types of Hecke algebras - usual (finite), affine, double affine (Cherednik), Hecke algebras of complex reflection groups (Broue-Malle-Rouquier), and many others. In particular, there are orbifold Hecke algebras which provide quantization of Del Pezzo surfaces and their Hilbert schemes.

Abstract. Conformal superalgebras describe symmetries of superconformal field theories and come equipped with an infinite family of products. They also arise as singular parts of the vertex operator superalgebras associated with some well-known Lie structures (e.g. affine, Virasoro, Neveu-Schwarz). In joint work with Arturo Pianzola and Victor Kac, we classify forms of conformal superalgebras using a non-abelian Cech-like cohomology set. As the products in scalar extensions are not given by linear extension of the products in the base ring, the usual descent formalism cannot be applied blindly. As a corollary, we obtain a rigourous proof of the pairwise non-isomorphism of an infinite family of N=4 conformal superalgebras appearing in mathematical physics.

Abstract. I will give a quick introduction to the theory of cluster algebras introduced by Fomin and Zelevinsky. I will illustrate it by examples like coordinate rings of unipotent groups and flag varieties.

Abstract. I will introduce the notion of monoidal categorification of a cluster algebra, and will give examples coming from the representation theory of quantum affine algebras.

Abstract. Fomin and Zelevinsky invented cluster algebras in 2000. Soon, it became clear that these new algebras were intimately related to quiver representations. Cluster categories, introduced in 2004, have provided a beautiful framework for making this relation precise. However, cluster categories are only defined for quivers without oriented cycles. Building on Derksen-Weyman-Zelevinsky's fundamental work on quivers with potentials Claire Amiot has recently been able to extend the construction of the cluster category to a large class of quivers admitting oriented cycles and endowed with a potential, namely the so-called Jacobi-finite quivers with potential. I will report on her results and their links to previous work, due notably to Geiss-Leclerc-Schroer and Buan-Iyama-Reiten-Scott.

Abstract. It is a well known result due to D. Peterson that the number of abelian ideals in the positive roots of a simple Lie algebra of rank $n$ is $2^{n}$. In this talk, I will discuss general results for ideals in simple Lie algebras including generalizations to $k$-nilpotent ideals. Furthermore, I will give the details of a simple proof of Peterson's theorem and give a method for explicitly defining all such ideals.

Abstract. Let $U_{q}$ denote a quantum group associated to a finite dimensional semi-simple complex Lie algebra.The Ringel-Donkin theory of tilting modules gives for each dominant weight $\lambda$ a unique indecomposable tilting module $T(\lambda)$ with highest weight $\lambda$. In the generic case these modules are just the finite dimensional irreducible modules but when $q$ is a root of unity we get new interesting modules for $U_{q}$. We shall show that they play a crucial role for instance in the theory of quantum invariants for 3-manifolds, in the theory of (quantum) Schur algebras at roots of unity.

Abstract. For each partition we construct a natural representation of the Lie algebra of matrix-valued polynomials. We discuss universality properties of these repreresntations as well as combinatorics of their characters. We present explicit answers for currents in up to three variables.

Abstract. Let $G$ be a connected unipotent group over an algebraically closed field $k$ of characteristic $p>0$. We define a collection of irreducible conjugation-equivariant perverse sheaves on $G$, which we call character sheaves. The set of all character sheaves naturally decomposes as a disjoint union of finite subsets, called $L$-packets of character sheaves.
We will explain a construction of a large collection of $L$-packets of character sheaves on $G$ (conjecturally, all of them) in terms of very concrete geometric objects related to $G$, namely, pairs consisting of a connected subgroup $H$ of $G$ and a multiplicative (rank 1) local system on $H$, satisfying a suitable analogue of Mackey's irreducibility criterion. This construction can be viewed as a geometric analogue of the classical result that all irreducible representations of a finite nilpotent group are induced from 1-dimensional representations of suitable subgroups.
If time permits, we will discuss the case where $k$ is an algebraic closure of a finite field $F$, and the unipotent group $G$ can be defined over $F$. In this case, there exists a relationship between irreducible characters of the finite group $G(F)$ and character sheaves on $G$ (just as in Lusztig's theory for reductive groups). In particular, the notion of an $L$-packet of irreducible characters of $G(F)$ can also be defined.

Abstract. The Bethe Ansatz is a method to find eigenvectors of a certain family of commutative matrices. This method is often more complicated than the standard methods of linear algebra, moreover, sometimes it fails to produce the complete set of the eigenvectors. However, the attempts to understand it lead to a number of interesting connections with surprisingly many areas of mathematics - and to new results in those areas. In this talk I will try to give an introduction to the Bethe Ansatz method.

Abstract. We will survey some results on tensor density modules of $Vec(\mathbb R)$, $Vec(S^{1})$, and the Virasoro Lie algebra, beginning with the work of Kaplansky-Santharoubane, Chari-Pressley, Martin-Piard, and Mathieu on Kac' conjecture concerning irreducible modules of the Virasoro Lie algebra, and the work of Goncharova and Feigin-Fuchs on cohomology. The focus will be on more recent results of Cohen-Manin-Zagier, Duval, Lecomte, Ovsienko, Roger, and others on modules of differential operators between tensor density modules. We will conclude with a brief look at similar problems over higher dimensional manifolds which are still open.

Abstract. Let $gA$ be the graded multi-loop Lie algebra and $gA(\mu)$ be the graded twisted multi-loop Lie algebra, associated with the simple finite dimensional Lie algebra $g$ over $\mathbb C$. In this talk, we describe the isomorphism classes of irreducible integrable $gA$-modules with finite dimensional weight spaces. We also describe the isomorphism classes of irreducible integrable $gA(\mu)$ -modules which are obtained from the above $gA$-modules by considering the restriction action. The talk is based on a joint work with Punita Batra

Abstract. Inhomogeneous Lie groups and algebras play an important role in physics, and so do some inhomogeneous quantum groups. In this talk I will introduce the notions of a inhomogeneous Lie algebra and Lie bialgebra and show how one can obtain classification results for the inhomogeneous quantum groups by studying inhomogeneous Lie bialgebras. If time permits I shall explain how one obtains quantum symmetric algebras from inhomogeneous quantum groups.

Abstract. Title: The talk will focus on finite-dimensional representations of hyper loop algebras over arbitrary fields. Hyperalgebras are certain Hopf algebras related to algebraic groups. When the field is of characteristic zero, a given hyper loop algebra coincide with the universal enveloping algebra of a certain "classical" loop algebra. The main results we will discuss are: the classification of the irreducible representations, construction of the Weyl modules, a study of base change (forms), and tensor products of irreducible modules. Some of these results are more interesting when the field is not algebraically closed and are beautifully related to the study of irreducible representations of polynomial algebras and field theory.

Abstract. Let $\mathfrak g$ denote a Lie algebra over a field of characteristic zero, and let $P(\mathfrak g)$ denote the tensor product of g with a ring of truncated polynomials. The Lie algebra $P(\mathfrak g)$ is called a polynomial Lie algebra, a truncated current Lie algebra, or a generalized Takiff algebra. In this talk, we develop a highest-weight theory for $P(\mathfrak g)$ when the underlying Lie algebra $\mathfrak g$ possesses a triangular decomposition. We describe a reducibility criterion for the Verma modules of $P(\mathfrak g)$ for a wide class of Lie algebras $\mathfrak g$, including the symmetrizable Kac-Moody Lie algebras, the Heisenberg algebra, and the Virasoro algebra.
If time permits, we may discuss applications of this result to the study of "exponential-polynomial modules".

Abstract. In this talk we discuss two parallel geometric constructions of the basic representation of the affine Lie algebra $\widehat{\mathfrak{gl}(r)}$, one using the Hilbert scheme of points on the ALE space $\widetilde{\mathbb C^2/\mathbb Z_r}$ (the original construction of Nakajima and Grojnowski) and the other using the moduli space of rank $r$ framed torsion-free sheaves on $\mathbb CP^2$. These constructions give a geometric interpretation of level-rank duality in the representation theory of the affine Lie algebra $\widehat{\mathfrak{gl}(r)}$.

Abstract. Minimal affinizations of representations of quantum groups introduced by Chari are relevant modules for quantum integrable systems. We present new results on their structure: we prove that all minimal affinizations in types $A$, $B$, $G$ are "special" in the sense of monomials (an analog property is also proved for a large class in types $C$, $D$, $F)$. As an application, the Frenkel-Mukhin algorithm works for these modules, and then we prove previously predicted explicit $q$-character formulas.

Abstract. Schubert varieties and their singularities are important in the study of representation theory and algebraic groups. In this talk I will describe one aspect of this story which involves singular Chern classes, characteristic cycles, and (small) resolutions of singularities. For concreteness, I'll focus on the case of Schubert varieties in the Grassmannian. In this context there is an open "positivity conjecture" which is interesting from both the geometric and combinatorial points of view.

Abstract. I will give an overview of the various statements which are called the Kirillov-Reshetikhin conjecture. These describe the structure of special modules of the Yangian of a Lie algebra $\mathfrak g$ or the associated quantum affine algebra. I'll explain how to prove that all these conjectures are equivalent (and hence are now proven), and why it implies the Feigin-Loktev conjecture for the fusion product of the corresponding modules defined by Chari for the algebra of polynomials with coefficients in $\mathfrak g$, $\mathfrak g[t]$.

Abstract. We will discuss the classification problem of irreducible Gelfand-Tsetlin modules for Yangians and finite $W$-algebras associated with the Lie algebra $\mathfrak{gl}(n)$.

Abstract. Let $U$ be a quantum group. In this talk, I will discuss a geometric realization of certain simple $U$-modules and their canonical bases, via certain perverse sheaves on open subvarieties of the representation spaces of a quiver.

Abstract. For an associative algebra $A$ negative cyclic chains $CC^{-}(A)$ form a module over the DG Lie algebra $C(A)$ of Hochschild cochains. In recent preprint arXiv:0802.1706 A. Cattaneo and G. Felder consider this DG Lie algebra module for $A$ being the algebra of functions on a smooth real manifold equipped with a volume form. Using an interesting modification of the Poisson sigma model the authors construct a curious L-infinity morphism (not a quasi-isomorphism!) from the DG Lie algebra module $CC^{-}(A)$ to a DG Lie algebra module modeled on polyvector fields using the volume form. The authors also apply this result to a construction of a specific trace on the deformation quantization algebra of a unimodular Poisson manifold. Although this trace can be constructed using the formality quasi- isomorphism for Hochschild chains, the relation of the L-infinity morphism of A. Cattaneo and G. Felder to the formality quasi-isomorphism is a mystery.

Abstract. In this talk, we will explore a generalization of symmetric Frobenius algebras (i.e., where the inner product is symmetric) to the case where the pairing is symmetric after some homological shift. We will explain how this property closely resembles the Calabi-Yau property for infinite-dimensional algebras, and will call such algebras "Calabi-Yau Frobenius algebras". It turns out that the Hochschild (co)homology of such algebras has a very nice structure, and is best described by a $\mathbb Z$-graded version of Hochschild (co)homology, which is a Hochschild analogue of Tate cohomology. The Hochschild cohomology is then a Frobenius algebra. In the case of periodic algebras (algebras which have a periodic bimodule resolution), we obtain a Batalin-Vilkovisky structure on Hochschild cohomology, which is conjecturally selfadjoint with respect to the Frobenius structure.

We will explain these results in detail in the case of preprojective algebras of Dynkin quivers, giving a full computation of their Hochschild (co)homology over the integers.

Abstract. The "chiral de Rham complex" of Malikov-Shechtman-Vaintrob is a sheaf of vertex superalgebras associated to any manifold $X$. We will show how, in the smooth context, extra geometric data on $X$ (e.g. having special holonomy) translates into extra symmetries of the corresponding vertex superalgebras of global sections.

Abstract. I am going to talk about recent preprint arXiv:0708.2725 "Hochschild cohomology and Atiyah classes" by D. Calaque and M. Van den Bergh. In this paper they proved a multiplicative version of Caldararu's conjecture which describes the Hochschild cohomology of a smooth algebraic variety as a graded ring. I will formulate the result of Calaque and Van den Bergh and explain how they proved it using Kontsevich's formality quasi-isomorphism.

Abstract. In my talk (based on the joint paper with Vladimir Retakh) I will introduce a version of Lie algebras and Lie groups over noncommutative rings.

For any Lie algebra $\mathfrak g$ sitting inside an associative algebra $A$ and any associative algebra $F$, I will define a Lie algebra $(\mathfrak g, A)(F)$ functorially in $F$ and $A$. In particular, if $F$ is commutative, the Lie algebra $(\mathfrak g, A)(F)$ is simply the loop Lie algebra of $\mathfrak g$ with coefficients in $F$.

In the case when $\mathfrak g$ is semisimple or Kac-Moody and $F$ is noncommutative, I will explicitly compute $(\mathfrak g, A)(F)$ in terms of commutator ideals of $F$ (surprisingly, these ideals have previously emerged as building blocks in M. Kapranov's approach to noncommutative geometry).

Furthermore, to each Lie algebra $(\mathfrak g, A)(F)$ one associates a "noncommutative algebraic" group which naturally acts on $(\mathfrak g, A)(F)$ by conjugations. I will conclude my talk with examples of such groups and with the description of "noncommutative root systems" of rank 1.

Abstract. I study the Category $\mathscr O$ over the wreath product of $\mathfrak{sl}(2,\mathbb C)$ (15 copies of $U(\mathfrak{sl}(2))$ times $S_{15}$). Complete reducibility and block decomposition hold here, because they hold for $\mathfrak{sl}(2)$. Next, I tensor this algebra 77 times, and ask whether complete reducibility and block decomposition hold in its category $\mathscr O$. Finally, I draw a "commuting cube" involving (sets of) simple modules in various categories $O$, where the "duality functor", "tensor product", and "wreath product induction" form the edges in the $X,Y,Z$-directions.

(The numbers 2,15,77 above can be generalized to any $n,m,k > 0$ - and more.)

Abstract. The formality theorem for Hochschild chains of the algebra of functions on a smooth manifold gives us a version of the trace density map from the zeroth Hochschild homology of a deformation quantization algebra to the zeroth Poisson homology. I will talk about my recent paper with V. Rubtsov in which we propose a version of the algebraic index theorem for a Poisson manifold based on this trace density map.

Abstract. In my talk I will report on commutors for crystals which were introduced and studied by Kamnitzer and Henriques. We will define the commutors associated to tensor products of crystal bases of modules over quantized enveloping algebras. Then, we will lift them to the modules and show how they are related to Drinfeld's unitarized $R$-matrix, as shown recently by Kamnitzer and Tingley.

Abstract. In this talk we discuss the relation between the following objects: rank 1 projective modules (ideals) over the first Weyl algebra $A_{1}(\mathbb C)$, simple modules over deformed preprojective algebras $\Pi_{\lambda}(Q)$, and simple modules over the rational Cherednik algebras $H_{0,c}(S_{n})$ associated to symmetric groups. The isomorphism classes of each type of these objects can be geometrically parametrized by the same space, the Calogero-Moser algebraic varieties. We will give a conceptual explanation of this bijection by constructing a natu- ral functor between the corresponding module categories. This is joint work with Y. Berest and O. Chalykh.

Abstract. We will discuss the "Factorization Phenomenon" which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into smaller representations of the subalgebra. The original Lie algebra may be any symmetrizable Kac-Moody algebra (including finite-dimensional, semi-simple Lie algebras). We will provide an algebraic explanation for such a phenomenon using "Spin construction". We will present a few Factorization results for any embedding of a symmetrizable Kac-Moody algebra into another, using Spin construction and give some combinatorial consequences of it. We will extend the notion of Spin from finite-dimensional to symmetrizable Kac-Moody algebras which requires a very delicate treatment. We will introduce a category of "$d$-finite, Orthogonal Level zero" representations for which, surprisingly, the Spin gives a representation in the Bernstein-Gelfand- Gelfand category $\mathscr O$. We will give the formula for the character of Spin for the above category and refine the factorization results in the case of affine Lie algebras. Finally, we will discuss classification of "Coprimary representations" i.e those representations whose Spin is irreducible.

Abstract. I will talk about recent work by Stroppel (0608234), which relates two algebras. The first is the endomorphism ring of a "minimal" projective-injective progenerator in the principal block of the parabolic BGG Category $\mathscr O$, for $\mathfrak{gl}(2n)$ and the maximal parabolic subalgebra for the partition $(n,n)$. (This will occupy most of the talk.) The second is the Khovanov algebra obtained by considering the $2d$ TQFT associated to the Frobenius algebra of dual numbers. Stroppel establishes an isomorphism of both of these, as graded $\mathbb C$-algebras.

Abstract. Let $\mathfrak g$ be a finite dimensional simple Lie algebra. In this talk we will focus on the category $\mathscr B$ of all bounded weight $\mathfrak g$-modules, i.e. those that are direct sum of their weight spaces and have uniformly bounded weight multiplicities. A result of Fernando implies that bounded weight $\mathfrak g$-modules exist only for $\mathfrak g= \mathfrak{sl}(n)$ and $\mathfrak g= \mathfrak{sp}(2n)$. In the second case we show that the category $\mathscr B$ has enough projectives if and only if $n$>1 and is wild if and only if $n>2$. The case $\mathfrak g= \mathfrak{sl}(n)$ is much more complicated as the description of each block $\mathscr B^\chi$ of $\mathscr B$ depends on the type of the central character $chi$.

Abstract. Let $\mathfrak g_{n}$ be either the $n\times n$ complex general linear Lie algebra $\mathfrak{gl}(n)$ or the $n\times n$ complex orthogonal Lie algebra $\mathfrak{so}(n)$ and let $G_{n}$ be the corresponding adjoint group. Let $P(\mathfrak g_{n})$ denote the algebra of polynomials on $\mathfrak g_{n}$. The associative commutator on the universal enveloping algebra of $\mathfrak g_{n}$ induces a Poisson structure on $P(\mathfrak g_{n})$. Let $J(\mathfrak g_{n})$ be the commutative Poisson subalgebra of $P(\mathfrak g_{n})$ generated by the invariants $P(\mathfrak g_{m})^{G_{m}}$ for $m=1,\dots n$. $J(\mathfrak g_{n})$ gives rise to a commutative Lie algebra of Hamiltonian vector fields on $\mathfrak g_{n}$; $V=\{ \xi_{f}\,:\, f \in J(\mathfrak g_{n})\}$. Choosing an appropriate set of generators for $J(\mathfrak g_{n})$ gives rise to a subalgebra $V'\subset V$. This subalgebra integrates to an action of a commutative, simply connected complex analytic group isomorphic to $\mathbb C^{d/2}$ on $\mathfrak g_{n}$, where $d$ is the dimension of a generic adjoint orbit in $\mathfrak g_{n}$. This fact should then allow one to polarize open submanifolds of generic adjoint orbits. We will discuss the orbit structure of the action of this group on $\mathfrak g_{n}$. In the case of $\mathfrak g_{n}= \mathfrak{gl}(n)$, we will give a description of the work of Kostant-Wallach in the most generic case in such a form that can be used to establish a formalism for dealing with the less generic orbits studied by the speaker. In the case of $\mathfrak g_{n}=\mathfrak{so}(n)$, the speaker will discuss his work on analyzing the orbit structure of the group on certain sets of semi-simple elements.

Abstract. The rational Cherednik algebras are an interesting family of algebras that can be attached to any complex reflection group. In this talk, I will show how to study finite dimensional modules for the rational Cherednik algebras attached to the infinite family of complex reflection groups $G(r,p,n)$ via eigenspace decompositions. Our approach allows us to construct finite dimensional irreducible modules of dimension $m^{n}$ for each integer $m$ coprime to the "Coxeter" number of $G(r,p,n)$, to build "BGG" resolutions of these modules, and to construct a basis of the coinvariant ring for $G(r,p,n)$ generalizing the Garsia-Stanton "descent monomial" basis for the coinvariant ring for the symmetric group $G(1,1,n)$.

Abstract. This talk is based on a joint work with D. Jakelic where we study finite-dimensional representations of hyper loop algebras, i.e., the hyperalgebras over a field of positive characteristic associated to non-twisted affine Kac-Moody algebras. In the talk we will go over the classification of the irreducible modules, a version of Steinberg's Tensor Product Theorem, and the construction of positive characteristic analogues of the Weyl modules as defined by Chari and Pressley in the characteristic zero case. We will also discuss reduction modulo $p$ and a Conjecture regarding reduction modulo $p$ of Weyl modules.

Abstract. A. King has proposed a method for organizing the representation theory of wild algebras by using the concept of stability which originally arose in the context of Mumfors's geometric invariant theory. I will talk about a joint work with V. Futorny and M. Jardim where we explore some applications of these ideas to certain categories of modules for Lie algebras.

Abstract. This talk is based on joint works with P. Littelmann. We study finite dimensional modules for current algebras, starting with Demazure modules for (twisted) affine Kac-Moody algebras. A few useful properties of these Demazure modules are found and proved. With these properties one can prove that Weyl modules as well as Kirillov-Reshetikhin modules are (in the simply-laced case) in fact isomorphic to Demazure modules as modules for the current algebra. As some applications of this isomorphism one can prove the conjectured dimension formula for Weyl modules, some conjectures about fusion products, that the conjectural Kirillov-Reshetikhin crystals are unique etc

Abstract. This talk is based on joint works with P. Littelmann. We study finite dimensional modules for current algebras, starting with Demazure modules for (twisted) affine Kac-Moody algebras. A few useful properties of these Demazure modules are found and proved. With these properties one can prove that Weyl modules as well as Kirillov-Reshetikhin modules are (in the simply-laced case) in fact isomorphic to Demazure modules as modules for the current algebra. As some applications of this isomorphism one can prove the conjectured dimension formula for Weyl modules, some conjectures about fusion products, that the conjectural Kirillov-Reshetikhin crystals are unique etc

Abstract. I will give an introduction to symplectic reflection algebras. These algebras are closely related to quotient singularities of the form $V/G$, where $V$ is a symplectic vector space, and $G$ is a finite group of automorphisms of $V$.

Abstract. I will present a theorem by Braverman and Gaitsgory that characterizes what Koszul algebras generated by "quadratic" relations, have a PBW-type theorem. The usual PBW theorem for Lie algebras is an example, as are Weyl and Clifford algebras.

Abstract. Quotient categories were introduced to extend to the categorical setting the notion of a quotient module and to provide a formal way of "getting rid of" subquotients in module categories. In this talk we will discuss an application of quotient categories to structural properties of a special class of highest weight categories.

Abstract. We will give a construction of a knot homology theory using the derived category of coherent sheaves of a certain variety arising in geometric representation theory. We conjecture that our knot homology is related to Khovanov homology (joint work with Sabin Cautis).

Abstract. The BGG (after Bernstein-Gelfand-Gelfand) category $\mathscr O$ is an important category of modules over a complex semisimple Lie algebra, that contains all finite-dimensional and all Verma modules. We show how category $\mathscr O$ decomposes into a direct sum of subcategories, and explore some of their properties. We further use category $\mathscr O$ to obtain certain well-known formulae, such as Weyl character formula and BGG reciprocity formula.

Abstract. We give a fairly simple proof for formulas for multiplicities for restricting representations of $GL(N)$ to $O(N)$ using Brauer algebras and fusion categories. The result is stated in tems of certain reflection groups. This formalism and results about tilting modules of quantum groups also suggest a description of the non-semisimple Brauer algebra in terms of certain parabolic Kazhdan-Lusztig polynomials.

Abstract. The equivariant cohomology ring $H_G(M)$ is an algebraic invariant one can attach to a smooth manifold M equipped with an action of a compact Lie group G. The chiral equivariant cohomology is a "chiralization" of $H_G(M)$, that is, a vertex algebra which contains $H_G(M)$ as the subspace of conformal weight zero. I will give a brief introduction to vertex algebras, and then discuss the construction of the new cohomology and some of the basic results and examples. This a joint work with Bong Lian and Bailin Song

Abstract. A block decomposition of the category $\mathscr C$ of finite-dimensional representations of a twisted affine Lie algebra is examined. The result is an extension of one given by Chari and Moura for the untwisted affine Lie algebras, in which the blocks of $\mathscr C$ are shown to be in bijective correspondence with the spectral characters of the Lie algebra.

Abstract. A block decomposition of the category $\mathscr C$ of finite-dimensional representations of a twisted affine Lie algebra is examined. The result is an extension of one given by Chari and Moura for the untwisted affine Lie algebras, in which the blocks of $\mathscr C$ are shown to be in bijective correspondence with the spectral characters of the Lie algebra.

Abstract. For every two sided cell in a Weyl group Lusztig attached a tensor category (via suitably truncated convolution of perverse sheaves on the corresponding flag variety). Moreover, Lusztig proposed a precise conjecture which describes this category in elementary terms. In this talk we report on recent joint work with R.Bezrukavnikov and M.Finkelberg where this conjecture was proved for almost all two sided cells. On the other hand the conjecture fails in the remaining cases.

Abstract. Dunkl operators are differential-difference operators introduced by Charles Dunkl in 1989. Due to the work of Opdam, Heckman and others, they are now a key tool in the theory of multivariable orthogonal polynomials. A major development was Cherednik's discovery of their intimate connection with degenerate affine Hecke algebras. I will give an introduction to Dunkl operators and some of their applications.

Abstract. M. Kontsevich showed that one can deform any Poisson algebra whose spectrum is a manifold or a variety. His results, however, leave the following natural questions:
1) Is it possible to find a presentation for the deformed algebra?
2) Can one deform Poisson algebras with zero-divisors?
In this talk I will discuss these questions in the case of r-matrix brackets on the symmetric algebra of a Lie bialgebra module. I will classify all modules for which these brackets are Poisson and then explicitly describe their deformations as quantum symmetric algebras. I will then explain, why I conjecture that quantum symmetric algebras provide an affirmative answer to questions 1) and 2).

Abstract. I will introduce the quantum flag variety and quantum $D$-modules on it. These are noncommutative algebro-geometric objects. In roots of unity they localize to the Springer resolution and allow for a computation of dimensions of modules over the quantum group.

Abstract. Joint with D. Maulik. Let $X$ be a resolution of the ADE singularity $\mathbb C^2/\Gamma$. We formulate the conjectural description of the structure of the ring of quantum equivariant cohomology of $Hilb_n(X)$. In particular the generators of the ring are given in terms of the loop algebra of the corresponding type. In the case of $A_n$ singularity the conjecture is a theorem. In my talk I will mostly discuss the case of $A_1$ singularity. All necessary geometric definitions unfamiliar to the audience will be reminded.