Lie theory seminar

Abstract. Finite dimensional representations of quantum affine algebras have a behavior similar to the Kazhdan-Lusztig conjecture. Nakajima showed that characters of standard modules can be expressed as linear combinations of characters of simple modules of "lower weight". Moreover, the coefficients are non-negative and can be expressed as evaluation of some polynomials.
In our work, we want to obtain the same type of results for the category $\mathscr O$ of representations of a Borel subalgebra (introduced by Hernandez-Leclerc). In this talk, we will present the motivation behind the definition of our asymptotical standard modules, as well as an idea for their construction in the case where $\mathfrak{g}=\hat{\mathfrak{sl}}_2$ (upported by the ERC Grant Agreement no. 647353 "Qaffine").

Abstract. We will report some recent progress on the problem of determining the centers of small quantum groups at a root of unity. This is joint work with A. Lachowska.

Abstract. Many questions about the representation theory of a complex semisimple Lie group can be understood in terms of the category $\mathcal{O}(\mathfrak{g})$ associated to its Lie algebra.
In analogy, Soergel constructed a modular category $\mathcal{O}(G)$ of representations of a reductive algebraic group $G$ over a field in characteristic $p$, which was recently used by Williamson to construct counterexamples to Lusztig's conjecture ("Williamson's Torsion Explosion").

Both categories are intimately related to the mixed geometry of the flag variety.

In characteristic $0$, categories of certain mixed $\ell$-adic sheaves, mixed Hodge modules or stratified mixed Tate motives provide geometric versions of the derived graded category $\mathcal{O}(\mathfrak{g})$ (Beilinson, Ginzburg, Soergel and Wendt).

Using the work of Soergel, we prove analogous statements in characteristic $p$. First, we construct an appropriate formalism of "mixed modular sheaves", using motives in equal characteristic. We then apply this formalism to construct a geometric version of the of the derived graded modular category $\mathcal{O}(G)$. (This is joint work with Shane Kelly).

Abstract. In this talk we will characterize the irreducible quasifinite highest weight modules of some subalgebras of the Lie algebra of $N\times N$ matrix quantum pseudodifferential operators.

In order to do this, we will first give a complete description of the anti-involutions that preserve the principal gradation of the algebra of $N\times N$ matrix quantum pseudodifferential operators and we will describe the Lie subalgebras of its minus fixed points. We will obtain, up to conjugation, two families of anti-involutions that show quite different results when $n = N$ and $n < N$. We will then focus on the study of the "orthogonal" and "symplectic" type subalgebras found for case $n = N$, specifically the classification and realization of the quasifinite highest weight modules.

Abstract. We study the family of prime representations of quantum affine $sl_{n+1}$ introduced in the work of Hernandez and Leclerc which are defined by using an $A_n$ quiver. We show that such representations admit a BGG-type resolution where the role of the Verma module is played by the local Weyl module. This leads to a closed formula (the Weyl character formula) for the character of the irreducible representation as an alternating sum of characters of local Weyl modules.

Abstract. In the representation theory of $\mathrm{GL}_n(\mathbb{C})$, there are correspondences between sets of simple roots, flags in $\mathbb{C}^n$, and parabolic subgroups, that are well-behaved with respect to actions by $\mathrm{GL}_n(\mathbb{C})$. These objects index a simplicial complex $\mathcal{B}(\mathrm{GL}_n(\mathbb{C}))$, called the building of $\mathrm{GL}_n(\mathbb{C})$. There is an affine version of the building, called the Bruhat-Tits building, whose simplices correspond to sets of simple affine roots, lattice chains, and parahoric subgroups. The primary goal of this expository talk is to give lots of low rank examples (i.e., for $\mathrm{SL}_2$, $\mathrm{SL}_3$, and $\mathrm{Sp}_4$) for each of these correspondences. If time permits, I will describe the relation between the Bruhat--Tits building and a well-behaved class of filtrations on the loop algebra called Moy-Prasad filtrations. These filtrations will be used in my next talk to define a geometric analogue of fundamental strata (originally developed by C. Bushnell for $p$-adic representation theory).

Abstract. Symplectic singularities were introduced by Beauville in 2000. These are especially nice singular Poisson algebraic varieties that include symplectic quotient singularities and the normalizations of orbit closures in semisimple Lie algebras. Poisson deformations of conical symplectic singularities were studied by Namikawa who proved that they are classified by the points of a vector space. Recently I have proved that quantizations of a conical symplectic singularity are still classified by the points of the same vector spaces. I will explain these results and then apply them to establish a version of Kirillov's orbit method for semisimple Lie algebras.

Abstract. The notion of discriminant is an important tool in number theory, algebraic geometry and noncommutative algebra. However, in concrete situations, it is difficult to compute and this has been done for few noncommutative algebras by direct methods. In this talk, we will describe a general method for computing noncommutative discriminants which relates them to representation theory and Poisson geometry. As an application we will provide explicit formulas for the discriminants of the quantum Schubert cell algebras at roots of unity. If time permits, we will also discuss this for the case of quantized coordinate rings of simple algebraic groups and quantized universal enveloping algebras of simple Lie algebras. This is joint work with Kurt Trampel and Milen Yakimov.

Abstract. Over the summer, I learned about Soergel bimodules at the MSRI from Ben Elias and Geordie Williamson, who were able to use these bimodules to solve a conjecture about Kazhdan-Lusztig (KL) polynomials for an arbitrary Coxeter system $(W,S)$. The aim of these talks will be two-fold. First, I want to define the KL polynomials. This will involve looking in the Hecke algebra of a Coxeter system, and showing how the KL polynomials describe one basis of the Hecke algebra in terms of another. The second goal will then be to describe the category of Soergel bimodules, and to show how they can be used to prove that the coefficients of the KL polynomials are non-negative.

Abstract. The Springer resolution is a resolution of singularities of the variety of nilpotent elements in a semisimple Lie algebra. It is an important geometric construction in representation theory, but some of its features are not as nice if we are working in Type $C$ (Symplectic group). To make the symplectic case look more like the Type $A$ case, Kato introduced the exotic nilpotent cone and its resolution, whose fibers are called the exotic Springer fibers. We give a combinatorial description of the irreducible components of these fibers in terms of standard Young bitableaux. This is joint work with Vinoth Nandakumar and Neil Saunders.

Abstract. Starting from a graded Frobenius superalgebra $B$, we consider a graphical calculus of $B$-decorated string diagrams. From this calculus we produce algebras consisting of closed planar diagrams and of closed annular diagrams. The action of annular diagrams on planar diagrams can be used to make clockwise (or counterclockwise) annular diagrams into an inner product space. I will explain how this gives a graphical realization of the space of symmetric functions equipped with the Jack inner product. This is joint work with Tony Licata and Alistair Savage.

Abstract. In this talk, we first introduce the definition of Yangians for the reductive complex Lie algebra $\mathfrak{gl}(N)$, describe their finite-dimensional representations and give necessary and sufficient conditions for the irreducibility of the Verma modules. Next, the definition of twisted Yangians is introduced. In the end, we give necessary and sufficient conditions for the irreducibility of the Verma modules for twisted Yangian $Y^{tw}(\mathfrak{sp}_2)$.

Abstract. The Springer correspondence relates nilpotent orbits in the Lie group of a reductive algebraic group to irreducible representations of the Weyl group. We develop a Springer theory in the case of symmetric spaces using Fourier transform, which relates nilpotent orbits in this setting to irreducible representations of Hecke al gebras at $q=-1$. We discuss applications in computing cohomology of Hessenberg varieties. Examples of such varieties include classical objects in algebraic geometry: Jacobians, Fano varieties of $k$-planes in the intersection of two quadrics, etc. This is based on joint work with Tsao-hsien Chen and Kari Vilonen.

Abstract. Given a quiver $Q$ with/without potential, one can construct an algebra structure on the cohomology of the moduli stacks of representations of $Q$. The algebra is called Cohomological Hall algebra (COHA for short). One can also add a framed structure to quiver $Q$, and discuss the moduli space of the stable framed representations of $Q$. Through these geometric constructions, one can construct two representations of Cohomological Hall algebra of Q over the cohomology of moduli spaces of stable framed representations. One would get the double of the representations of Cohomological Hall algebras by putting these two representations together. This double construction implies that there are some relations between Cohomological Hall algebras and some other algebras. In the talk we will focus on two specific examples: $A_1$ quiver and the Jordan quiver.

Abstract. Symmetry is a concept that most people grasp intuitively, and it has important applications in several branches of mathematics, as well as other sciences. We will focus on a surprising connection between the geometry of subspaces of a vector space (like lines and planes) and an algorithm that was originally defined for purposes of combinatorics (arranging and counting things in various ways).

Abstract. In this talk I will discuss the construction of graded tensor products for the current algebra associated to a Lie (super)algebra. For the ortho--symplectic Lie superalgebra we will show that these representations can be filtered by the corresponding graded tensor products for the underlying reductive Lie algebra. In the second part of my talk, I will discuss the appearence of graded tensor products in PBW theory and categorification. One of the future goals is to understand which 2-representation of the categorified quantum group corresponds to graded tensor products.

Abstract. Molien's 1897 formula for the Poincaré series of the polynomial invariants of a finite group has given rise to many results in combinatorics, coding theory, mathematical physics, algebraic geometry, and representation theory. This talk will focus on analogues of Molien's formula for tensor invariants and will discuss various connections with representation theory, and the McKay Correspondence. The approach is via walking on graphs.

Abstract. Affine highest weight category, introduced by Kleshchev, is a generalization of the notion of highest weight category. For example, some module categories over central completions of (degenerate) affine Hecke algebras (more generally KLR algebras of finite type) and polynomial current Lie algebras are known to be affine highest weight. In this talk, we consider tilting modules in affine highest weight categories and explain that a complete collection of indecomposable tilting modules exists if our category has a large center. As an application, we gives a simple criterion for an exact functor between two affine highest weight categories to give an equivalence. We can apply this criterion to the Arakawa-Suzuki functor on the deformed category $\mathcal{O}$ for $\mathfrak{gl}_{m}$.

Abstract. The representation theory of real reductive Lie groups can be studied by complex algebraic and geometric methods using infinitesimal approximations called Harish-Chandra modules. Harish-Chandra modules are representations of the corresponding complex reductive Lie algebra which are locally finite with respect to the complexification of the maximal compact subgroup of the original real Lie group. Generalized Harish-Chandra modules are a generalization in which the maximal compact group is replaced by an arbitrary reductive subalgebra of the complex Lie algebra. In this talk we will review the geometric classification of simple Harish-Chandra modules, and compare this case to the setting of generalized Harish-Chandra modules. This work is part of a program initiated by Ivan Penkov and Gregg Zuckerman.

Abstract. It is well known that the braid group acts on a quantised enveloping algebra by algebra automorphisms. We discuss the categorification of this braid group action and some of its consequences. Applications include constructing reflection functors for KLR algebras, and a theory of restricting a categorical representation along a face of a Weyl polytope.

Abstract. Dr. Chari and Dr. Pressley introduced the local Weyl$ better understanding of the category of finite-dimensional highest weight representations of quantum affine algebras in 2001. Then this notion of a local Weyl module has been extended to the finite-dimensional representations of current algebras, twisted loop algebras and current Lie algebras on anffine varieties. Dr. Nicolas and the author extended the notion to the finite-dimensional representations of Yangians.
In this talk, we will review the definition of Yangians and their finite-dimensional highest weight representations first. Then we introduce the notion of local weyl module and show that the local Weyl module is finite-dimensional and is isomorphic to an ordered tensor product of fundamental representations of Yangians. In what follows, we can provide a cyclicity condition for a tensor product of fundamental representations of Yangians.

Abstract. We introduce a braid group action for the finite-dimensional representations of Yangians $Y(\mathfrak{g})$, where $\mathfrak{g}$ is a complex simple Lie algebra. It provides an efficient way to compute certain polynomials which allows us to provide a finite set of numbers at which the tensor product of Kirillov-Reshetikhin modules of Yangians may fail to be cyclic.

Abstract. The aim of this talk is to describe the analytic nature of the quantum groups. More specifically, I will show how the quantum groups can be interpreted as a natural receptacles for the monodromy representations of certain flat connections arising in the representation theory of Kac-Moody algebras.
This principle first appeared in the famous Kohno-Drinfeld theorem. It states that the universal R-matrix coming from the quantum group of a semi-simple Lie algebra describes the monodromy of the Knizhnik-Zamolodchikov connection (a flat connection over the configuration space of n points in the complex plane with logarithmic singularities on the coordinate hyperplanes).
In this talk, I will explain how this result extends to an interpretation of the quantum Weyl group operators in terms of the monodromy of the Casimir connection (a flat connection over the Cartan subalgebra with with logarithmic singularities on the roots hyperplanes) for Kac-Moody algebras of finite and affine type (joint work with V. Toledano Laredo).

Abstract. The Bernstein-Sato polynomial, or b-function, is an important invariant in singularity theory, which is difficult to compute in general. We describe a few different results towards computing the b-function of the Vandermonde determinant $\xi$. In 1989, Eric Opdam computed the b-function of a related polynomial, and we use his result to produce a lower bound for the b-function of $\xi$. We use this lower bound to prove a conjecture of Budur, Mustata, and Teitler for the case of finite Coxeter hyperplane arrangements, proving the Strong Monodromy Conjecture in this case. In our second result, we use duality of some D-modules to show that the roots of this b-function of $\xi$ are symmetric about -1. Finally, we use results about jumping coefficients together with Kashiwara's proof that the roots of a b-function are rational in order to prove an upper bound for the b-function of $\xi$ and give a conjectured formula. This is a joint work with Asilata Bapat.

Abstract. I will describe the P=W conjecture of de Cataldo-Hausel-Migliorini. It concerns the moduli of local systems and higgs bundles on a smooth curve, and the relation between their cohomology groups. I will then discuss work in progress with Zsuzsanna Dancso and Vivek Shende to formulate and prove an analogous conjecture for nodal curves.

Abstract. We will focus on two kinds of spherical varieties, the L-monoid and the affine closure of G/U for a connected reductive group G and U its unipotent radical. We will first explain the joint work of the speaker with B.C. Ngo and Y. Sakellaridis which gives a way to construct geometrically unramified local L-factors. Nevertheless, the geometric situation is nicely defined only globally as it is also the case for the affine closure of G/U. Locally, we need to consider arc spaces of these spherical varieties which are infinite dimensional and for which there was no theory of perverse sheaves on it. We will then explain the recent work of the speaker with D. Kazhdan which enables to construct such objects and compare it with the global ones.

Abstract. I will begin by recalling the definition and some basic properties of Schur algebras, which play an important role in the representation theory of general linear groups and symmetric groups and are particularly interesting in positive characteristic. I will then describe a geometric interpretation of the category of their representations. Using this description as motivation, I will discuss joint work with Tom Braden, in which we define two new classes of algebras. One is associated to geometric spaces called hypertoric varieties and the other to combinatorial objects called matroids.

Abstract. Nakajima quiver varieties are moduli spaces of certain representations of quivers. They play an important role in Algebraic Geometry, Mathematical Physics and Geometric Representation theory. Their quantizations are noncommutative associative algebras with interesting and rich representation theory conjecturally related to deep geometric properties of the underlying varieties. I will explain some reasons to be interested in that representation theory and also some results in the important special case of quantized Gieseker moduli spaces based on preprint arXiv:1405.4998. All necessary information about quiver varieties and their quantizations will be introduced during the talk.

Abstract. I will briefly motivate and give a gentle introduction to the constructible derived category of sheaves on a (complex) algebraic variety. Next Tuesday, in part 2 of the talk, we will discuss some connections to the representation theory of reductive algebraic groups.

Abstract. We study a subset of a parabolic quotient in a simply-laced Weyl group $W$-stable under an automorphism $\sigma$, which we call the balanced parabolic quotient. This subset describes the interaction between the branching rule for a Levi subalgebra, Demazure modules, and $\sigma$-invariant weight spaces in $\sigma$-stable simple modules for the corresponding Lie algebra. The Hasse diagram of the balanced parabolic quotient under the Bruhat order is a forest with a remarkable self-similarity property. We characterize an element of a balanced quotient on the level of the root system of $W$, and find that the subalgebras of the Borel associated with these elements decompose into the direct sum of two subalgebras: one contained in the Borel for a Levi subalgebra, and another consisting of $sigma$-invariants.

Abstract. For the quantum affine superalgebra associated to a general linear Lie superalgebra, we construct inductive systems of Kirillov-Reshetikhin modules. We endow their inductive limits with module structures over the full quantum affine superalgebra in the spirit of Hernandez-Jimbo. Then we indicate some consequences from the asymptotic construction.

Abstract. I will talk about certain graded algebras attached to (multi-)quivers. Examples include the $n$th Weyl algebra, as well as quotients of enveloping algebras of Lie algebras of types $A$ and $C$. These algebras come with a canonical representation by differential operators and provide new solutions to the consistency equations for twisted generalized Weyl algebras. Properties of the quiver (Cartan matrix, equilibrium conditions, acyclicity) are directly related to properties of the corresponding algebra. We also present three new constructions (tensor products, invariant subalgebras, and folding). Finally, we discuss quantum analogs and list some open questions. Part of this talk is based on joint work with Vera Serganova.

Abstract. Beilinson-Lusztig-MacPherson gave a construction of the quantum enveloping algebra $U_q(\mathfrak{sl}_n)$ (and of the $q$-Schur algebras, which are certain finite dimensional quotients) as a convolution algebra on the space of pairs of partial $n$-step flags. I will define a convolution product in the mirabolic setting (triples of two partial flags and a vector) and give some results about the algebra obtained in this way, when $n=2$.

Abstract. In the last few years, a framework was developed (by Church, Ellenberg and Farb) to discover and understand many stabilization phenomenon which occur in sequences of representations for a family of groups. I will give an introduction to this new theory and its applications.

Abstract. If we have a sequence of associative algebras that satisfy certain conditions, we call it a tower of algebras. We will see how, in this setting, induction and restriction functors give us the structure of two dual Hopf algebras on the Grothendieck groups of the two categories respectively of all modules for the algebras and of all projective modules.

Abstract. The first part of the talk will be centered around Nichols algebras in braided tensor categories. They often appear in various algebraic contexts (for example, the upper triangular part of the quantum group, or Lusztig's algebra, is an example of a Nichols algebra). One of their nice properties is that they are generated, as algebras, by their primitive elements which, in principle, allows to find their relations explicitly. Since Hall algebras are known to provide realizations of quantum groups and are algebras and coalgebras in a braided tensor category, it is only natural to ask how far they are from being Nichols algebras. It turns out that Hall algebras of a large class of categories have the Poincaré-Birkhoff-Witt property and are primitively generated. In the second part of the talk we will discuss bosonisation of Nichols algebras and their doubles and a procedure for constructing bases in them, which is in turn motivated by Hall algebras and their properties. (joint work with A. Berenstein)

Abstract. I will talk about the results of a recent joint paper with V. Futorny and E. Wilson where we construct new irreducible weight modules over quantum affine algebras in which all weight spaces are infinite-dimensional. They are obtained by parabolic induction from irreducible modules over the quantum Heisenberg subalgebra.

Abstract. After giving a quick introduction to Lusztig's conjecture on the irreducible characters of reductive algebraic groups we give an overview on recent approaches and results. In particular, we sketch a proof of the conjecture for almost all characteristics. If time permits, we will also talk about the counterexamples to the strong form of Lusztig's conjecture that were found by Williamson in 2012.

Abstract. For a reductive group $G$, the Springer correspondence is an injection from irreducible representations of the Weyl group $W$ to the simple $G$-equivariant perverse sheaves on the nilpotent cone of the Lie algebra (or the unipotent variety of the group). However, in general not all simple perverse sheaves arise in this way. This led Lusztig to define a generalized Springer correspondence, involving the process of inducing cuspidal perverse sheaves from Levi subgroups. The classical correspondence is the part coming from a maximal torus. In the case of the general linear group, though, nothing new arises in this way. In my thesis I studied a modular Springer correspondence, where one takes modular representations of the Weyl group and perverse sheaves with positive characteristic coefficients. In this talk I will explain the modular version of the generalized Springer correspondence, focusing on the case of the general linear group. In the modular case there is something new, namely there is a cuspidal perverse sheaf supported by the regular nilpotent orbit when the rank is a power of the characteristic. I will also mention the most striking phenomena concerning groups of other types. This is joint work with Pramod Achar, Anthony Henderson and Simon Riche.

Abstract. The category of finite dimensional representations of a quantum loop algebra $U_q(L\mathfrak g)$ is not semi-simple. Moreover, the tensor product of irreducible representations remains irreducible generically. This leads naturally to the definition of prime objects: the factorization of irreducible objects into irreducible primes. We show that there is an interesting connection between the notion of primes and the homological properties of the category, namely, for $\mathfrak g=\mathfrak sl_2$, an irreducible representation $V$ is a tensor product of $r$ prime representations if and only if the dimension of the space of self extensions of $V$ is $r$.

Abstract. We study Demazure modules which occur in a level $\ell$ irreducible integrable representation of an affine Lie algebra. We also assume that they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie algebra. We prove that such a module is isomorphic to the fusion product of «prime» Demazure modules, where the prime factors are indexed by dominant integral weights which are either a multiple of $\ell$ or take value less than $\ell$ on all simple coroots. Our proof depends on a technical result which we prove in all the classical cases and $G_2$. Calculations with mathematica show that this result is correct for small values of the level. Using our result, we show that there exist generalizations of $Q$-systems to pairs of weights where one of the weights is not necessarily rectangular and is of a different level. Our results also allow us to compare the multiplicities of an irreducible representation occurring in the tensor product of certain pairs of irreducible representations, i.e., we establish a version of Schur positivity for such pairs of irreducible modules for a simple Lie algebra.

Abstract. Differential graded categories are widely used in homological algebra, representation theory, and algebraic geometry. The purpose of these talks is to provide an introduction to the theory of DG categories, focusing on definitions, basic properties, main results, and its relationship to derived categories and triangulated categories. To minimize preliminaries, these talks will be self-contained except some elementary category theory and homological algebra such as categories and functors, chain complexes and chain maps.
References:

1. On Differential graded categories by B. Keller (available on arXiv);
2. Lectures on DG-categories by T. Bertrand (available on arXiv);
3. Deriving DG categories by B. Keller (http://www.math.jussieu.fr/~keller/publ/ddc.pdf)

Abstract. We give different constructions of new integrable representations of quantum toroidal algebras, called extremal loop weight representations. Its definition is given by Hernandez in 2009, following the one of extremal weight representations of quantum affine algebras by Kashiwara. The aim, like in the works of Kashiwara, is to construct finite-dimensional representations of the quantum toroidal algebras, but at roots of unity in this case.

Abstract. We give different constructions of new integrable representations of quantum toroidal algebras, called extremal loop weight representations. Its definition is given by Hernandez in 2009, following the one of extremal weight representations of quantum affine algebras by Kashiwara. The aim, like in the works of Kashiwara, is to construct finite-dimensional representations of the quantum toroidal algebras, but at roots of unity in this case.

Abstract. A fundamental module for any finite dimensional, simple Lie algebra $\mathfrak{g}$ is an irreducible module of highest weight $\omega_i$ where $\omega_i$ is a fundamental weight of $\mathfrak{g}$. In the case of $A_n$ it happens that all the fundamental modules are isomorphic to exterior powers of the natural module. In general, such isomorphisms do not always exist. In particular, for $B_n$ and $D_n$, the fundamental modules associated to the short root (in the case of $B_n$) and to the spin nodes (in the case of $D_n$) are called spin modules. This talk will review the construction of Clifford algebras and spin modules. From this construction we will exhibit how the fundamental modules associated to the spin nodes of $D_n$ are spin modules.

Abstract. Categorification is the process of finding higher level structure by replacing sets with categories, functions between sets with functors, and relations between functions with natural transformations of functors. This talk will focus on the categorification of algebras, and representations of these algebras. We will define naive, weak, and strong categorification, and look at some examples.

Abstract. We will follow Anton Khoroshkin's paper -- Highest weight categories and Macdonald polynomials. In the first talk, I will introduce the basic setting, including the graded Lie algebras with anti-involution, category of finitely generated graded modules, and graded characters to the Grothendieck rings. If the module category is stratified, the Macdonald-type polynomials can be realized via taking the character of the (proper) standard modules. This construction can be generalized to any stratified highest weight category. However, there are many interesting nonstratified highest weight categories. For example, the bigraded Lie algebra of currents, which are related to the classical Macdonald polynomials. In those cases, we may not expect that Macdonald type polynomials are the characters of specific modules, but it is possible if we consider complexes of modules. This is the content of my second talk. I will give enough detail for those not familiar with derived categories.

Abstract. Geometric representation theory has a revealed a deep connection between geometry and quantum groups suggesting that quantum groups are shadows of richer algebraic structures called categorified quantum groups. Crane and Frenkel conjectured that these structures could be understood combinatorially and applied to low-dimensional topology. In this lecture we will categorify quantum groups using a simple diagrammatic calculus that requires no previous knowledge of quantum groups. We will explain how the new diagrammatic relations not only lift the quantum group relations to explicit isomorphisms, they also give rise to a representation of the corresponding current algebra.

Abstract. Let $\mathfrak g$ be a semisimple finitedimensional Lie algebra and $W$ its Weyl group. I will introduce Arkhipov's twisting functors $T_w$, $w\in W$. The twisting functors are certain functors on the BGG category $\mathscr O$ that can be used, for example, to construct what is called twisted Verma modules. The functor consists of tensoring with a "semiregular bimodule" and twisting the action by an automorphism corresponding to $w$. I will talk about the construction of $T_w$ and some of the properties of the twisting functors.

Abstract. In these talks we study a family of finite-dimensional graded representations of the current algebra of $\mathfrak{sl}_2$ which are indexed by partitions. We show that for $\ell$ sufficiently large, these representations admit a filtration by submodule where the successive quotients are Demazure modules which occur in a level $\ell$ integrable module for $A_1^{(1)}$. We associate to each partition and to each $\ell$ an edge-labeled directed graph which allows us to describe in a combinatorial way the graded multiplicity of a given level $\ell$-Demazure module in the filtration. In the special case of the partition $1^s$ and $\ell=2$, we give a closed formula for the graded multiplicity of level two Demazure modules in a level one Demazure module. As an application, we use our result along with the results of K. Naoi and Lenart et al, to give the character of a $\mathfrak g$-stable level one Demazure module associated to $B_n^{(1)}$ as an explicit combination of suitably specialized Macdonald polynomials. In the case of $\mathfrak{sl}_2$, we also study the filtration of the level two Demazure module by level three Demazure Modules and compute the numerical filtration multiplicities and show that the graded multiplicites are related to (variants) of partial theta series.

Abstract. Skew group algebras naturally generalize ordinary group algebras. Let $A$ be a finite dimensional algebra and let $G$ be a finite subgroup of the automorphism group of $A$. It has been shown that the skew group algebra $AG$ and $A$ share many important properties (such as finite representation type, finite global dimension, etc) if the order of $G$ is invertible. However, when the order of $G$ is not invertible, many results fail. Therefore, it is natural to ask under what conditions $AG$ and $A$ still share these properties for arbitrary $G$. The answer of this question will be described in this talk.

Abstract. For a quiver with potential, we can associate a vanishing cycle to each representation space. If there is a nice torus action on the potential, the vanishing cycles can be expressed in terms of truncated Jacobian algebras. We study how these vanishing cycles change under the mutation of Derksen-Weyman-Zelevinsky. The wall-crossing formula leads to a categorification of quantum cluster algebras under the assumption of existence of certain potential. This is a special case of A. Efimov's result, but our approach is more concrete and down-to-earth. We also obtain a formula relating the representation Grassmannians under sink-source reflections. In this talk, I will start with basic definitions and examples of vanishing cycles, Hall algebras, and quivers with potentials.

Abstract. An infinite dimensional Lie algebra is locally finite if every finitely generated subalgebra is finite dimensional. On one extreme are the simple, locally finite Lie algebras. We provide structure theorems which describe such algebras over fields of positive characteristic. On the other extreme are the maximal, locally solvable Lie algebras, which are Borel subalgebras. We provide a theorem which shows that such Lie algebras are stabilizers of maximal, generalized flags, which is a generalization of Lie's theorem. We will finish by describing some new directions in the study of these Lie algebras.