We give a diagrammatic presentation in terms of generators mod relations of the representation category of $U_q(\mathfrak{sl}_n)$. More precisely, we produce all the relations among $\rm{SL}_n$-webs, thus describing the full subcategory tensor-generated by fundamental representations $\bigwedge^k \mathbb{C}^n$ (this subcategory can be idempotent completed to recover the entire representation category). Our result answers a question posed by Kuperberg [arXiv:q-alg/9712003] and affirms conjectures of Kim [arXiv:math.QA/0310143] and Morrison [arXiv:0704.1503]. Our main tool is an application of quantum skew Howe duality.Comment: 32 pages, added a missing relation which had been implicitly used; this version has the same content as the published versio

Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to sl(2) and its standard representation. Our construction is related to that of Seidel-Smith by homological mirror symmetry. We show that the resulting doubly graded knot homology agrees with Khovanov homology.

Abstract. We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians -these are subalgebras of the Yangian we introduce which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type. Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. This conjecture also implies the conjectural quantization of the Zastava spaces for P GL n of Finkelberg-Rybnykov.

Abstract. Following an idea of A. Berenstein, we define a commutor for the category of crystals of a finite dimensional complex reductive Lie algebra. We show that this endows the category of crystals with the structure of a coboundary category. Similar to the role of the braid group in braided categories, a group naturally acts on multiple tensor products in coboundary categories. We call this group the cactus group and identify it as the fundamental group of the moduli space of marked real genus zero stable curves.

We give an explicit description of the Mirković-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We prove that a polytope is an MV polytope if and only if it is a lattice polytope whose defining hyperplanes are parallel to those of the Weyl polytopes and whose 2-faces are rank 2 MV polytopes. As an application, we give a bijection between Lusztig's canonical basis and the set of MV polytopes.

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra g has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given g, there is a limit crystal, usually denoted by B(−∞), which contains all the other crystals. When g is finite dimensional, a convex polytope, called the Mirković-Vilonen polytope, can be associated to each element in B(−∞). This polytope sits in the dual space of a Cartan subalgebra of g, and its edges are parallel to the roots of g. In this paper, we generalize this construction to the case where g is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(−∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as g. The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott's tilting theory for the category Λ-mod. The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.One of these tools is Buan, Iyama, Reiten and Scott's tilting ideals for Λ [14]. Let S i be the simple Λ-module of dimension-vector α i and let I i be its annihilator, a one-codimensional two-sided ideal of Λ. The products of these ideals I i are known to satisfy the braid relations, so to each w in the Weyl group of g, we can attach a two-sided ideal I w of Λ by the rule I w = I i 1 · · · I i ℓ , where s i 1 · · · s i ℓ is any reduced decomposition of w. Given a finite-dimensional Λ-module T , we denote the image of the evaluation map I w ⊗ Λ Hom Λ (I w , T ) → T by T w .Recall that the dominant Weyl chamber C 0 and the Tits cone C T are the convex cones in the dual of RI defined asWe will show the equality T min θ = T max θ = T w for any finite dimensional Λ-module T , any w ∈ W and any linear form θ ∈ wC 0 . This implies that dim T w is a vertex of Pol(T ) and that the normal cone to Pol(T ) at this vertex contains wC 0 . This also implies that Pol(T ) is contained inWhen θ runs over the Tits cone, it generically belongs to a chamber, and we have just seen that in this case, the face P θ is a vertex. When θ lies on a facet, P θ is an edge (possibly degenerate). More precisely, if θ lies on the facet that separates the chambers wC 0 and ws i C 0 , with say ℓ(ws i ) > ℓ(w), then (T min θ , T max θ ) = (T ws i , T w ). Results in [1] and [27] moreover assert that T w /T ws i is the direct sum of a finite number of copies of the Λ-module I w ⊗ Λ S i .There is a similar description when θ is in −C T ; here the submodules T w of T that come into play are the kernels of the coevaluation maps T → ...

In an earlier work, we proved that MV polytopes parameterize both Lusztig's canonical basis and the Mirković-Vilonen cycles on the Affine Grassmannian. Each of these sets has a crystal structure (due to Kashiwara-Lusztig on the canonical basis side and due to Braverman-Finkelberg-Gaitsgory on the MV cycles side). We show that these two crystal structures agree. As an application, we consider a conjecture of Anderson-Mirković which describes the BFG crystal structure on the level of MV polytopes. We prove their conjecture for sln and give a counterexample for sp 6 . Finally we explain how Kashiwara data can be recovered from MV polytopes.

We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of non-alternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. In the non-alternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3-loop Feynman diagrams of hep-th/9803091 and subsequently in hep-ph/9910223, hep-ph/9910224, cond-mat/9911452 and hep-th/0004010.