Geometry and Topology

Speaker: Hiro Tanaka (Northwestern)

"Factorization homology and link invariants"

Time: 15:30

Room: MC 107

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.

Analysis Seminar

Speaker: Ilya Kossovkiy (Western)

"Analytic Continuation of Holomorphic Mappings From Non-Minimal Hypersurfaces"

Time: 14:30

Room: MC 107

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.

Pizza Seminar

Speaker: Rasul Shafikov (Western)

"Introduction to Continued Fractions"

Time: 16:30

Room: MC 107

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.

Ph.D. Presentation

Speaker: Chris Plyley (Western)

"Group-Graded Algebras, Polynomial Identities, and The Duality Theorem"

Time: 13:00

Room: MC 107

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.