Crossings states and sets of states in P\'olya random walks

We consider the P\'olya random walk in $\mathbb{Z}^2$. The paper establishes a number of results for the distributions and expectations of the number of usual (undirected) and specifically defined in the paper up- and down-directed state-crossings and different sets of states crossings. One of the most important results of this paper is that the expected number of undirected state-crossings $\mathbf{n}$ is equal to 1 for any state $\mathbf{n}\in\mathbb{Z}^2\setminus\{\mathbf{0}\}$. As well, the results of the paper are extended to $d$-dimensional random walks, $d\geq2$, in bounded areas.Comment: Dear readers. I made a tremendous work to revise this paper after referee report. There are 30 pages of 11pt format, 4 figures and 1 tabl

Knot invariants and higher representation theory

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is $\mathfrak{sl}_n$, we show that these categories agree with certain subcategories of parabolic category O for gl_k. We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory. The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius. In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.Comment: 99 pages. This is a significantly rewritten version of arXiv:1001.2020 and arXiv:1005.4559; both the exposition and proofs have been significantly improved. These earlier papers have been left up mainly in the interest of preserving references. v3: final version, to appear in Memoirs of the AMS. Proof of nondegeneracy moved to separate erratu

On the cycle class map for zero-cycles over local fields

We study the Chow group of zero-cycles of smooth projective varieties over local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus, over local fields of residue characteristic different from l. The same statement holds for semistable K3 surfaces defined over C((t)), but does not hold in general for surfaces over strictly local fields.Comment: 37 pages (with an appendix by Spencer Bloch); bibliography updated, final versio