Homology is easy to compute, thanks to excision, but it isn't very sensitive. It only detects homotopy types. In this talk I'd like to give one answer to the question: Is there a notion of homology theory for manifolds that's sensitive to more? I will present the definition of factorization homology, which Lurie has also called topological chiral homology. Factorization homology generalizes usual Eilenberg-Steenrod homology, and is and invariant of manifolds and stratifications on them. The main result will be a classification of all homology theories, namely by giving an equivalence between the category of homology theories and the category of certain kinds of algebras. I will explain how the theorem in turn gives candidates for new sources of invariants of embedding spaces (and in particular, link invariants). If time allows, I can discuss connections to topological field theories and to Koszul duality. This is joint work with David Ayala and John Francis.

The classical result of H.Poincare states that a local biholomorphic mapping of an open piece of the 3-sphere in $\mathbb{C}^2$ onto another open piece extends analytically to a global holomorphic automorphism of the sphere. This theorem was generalized by H.Alexander to the case of a sphere in an arbitrary $\mathbb{C}^n,\,n\geq 2$, then later by S.Pinchuk for the case of strictly pseudoconvex hypersurface in the preimage and a sphere in the image, and finally by R.Shafikov and D.Hill for the case of an essentially finite hypersurface in the preimage and a quadric in the image. In this joint work with R.Shafikov we consider the - essentially new - case when a hypersurface $M$ in the preimage contains a complex hypersurface. We demonstrate that the above extension results fail in this case, and prove the following analytic continuation phenomenon: a local biholomorphic mapping of $M$ onto a non-degenerate hyperquadric in $\mathbb{CP}^n$ extends to a punctured neighborhood of the complex hypersurface, lying in $M$, as a multiple-valued locally biholomorphic mapping.

In this elementary talk I will discuss the definition and basic properties of continued fractions, a simple and in many respects a convenient way to represent real numbers. I will also give some applications.

In polynomial identity theory, when an associative algebra A has the additional structure of an (associative) group-grading or a G-action, one can often relate the identities of A to the more general graded-identities and G-identities. This technique has proved a powerful method, for example, in discovering a bounded version of Amitsur's celebrated theorem regarding algebras with involution. In this talk we describe several alternate ways to endow a grading on A, namely by considering the induced Lie and Jordan algebras. Moreover, one of these new gradings is used to extend the well known duality between the associative-G-gradings and the G-actions (by automorphisms) of A to include actions by anti-autopmorphisms. We call this new graded structure a Lie-Jordan-G-graded algebra, and mention some of the applications it has to Shirshov bases, polynomial identities, and other topics.