Geometry and Topology

Speaker: Robin Koytcheff (Brown)

"A colored operad for infection of links"

Time: 15:30 - 16:30

Room: MC 107

Ryan Budney recently constructed an operad that encodes splicing of knots and extends his little 2-cubes action on the space of (long) knots. He further showed that the space of knots is freely generated over the splicing operad by the subspace of torus and hyperbolic knots. Infection of knots (or links) by string links is a generalization of splicing from knots to links and is useful for studying concordance of knots. In joint work with John Burke, we construct a colored operad that encodes this infection operation.

Analysis Seminar

Speaker: Ekaterina Shemyakova (Western)

"Completeness of Wronskian Formulas for Darboux transformations or order 2"

Time: 14:30 - 15:30

Room: MC 107

I shall report about my 2011 results, which is the resolution of one long standing problem in the theory of Darboux transformations. It is known that many Darboux transformations can be constructed using Darboux Wronskian formulas. The only known exceptions have been two transformations of order one - Laplace transformations, which are often used in applications. I shall show that for order one there is no other exceptions and that for order two Wronskian formulas are complete. History of the question as well as an introduction into the area will be provided.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"Selberg Trace Formula and Heisenberg Group(2)"

Time: 14:30 - 14:30

Room: MC 107

Algebra Seminar

Speaker: Nicole Lemire (Western)

"Stably Cayley groups over arbitrary fields"

Time: 14:40 - 15:40

Room: MC 107

A linear algebraic group is called a Cayley group if it is equivariantly birationally isomorphic to its Lie algebra. It is stably Cayley if the product of the group and some torus is Cayley. Cayley gave the first examples of Cayley groups with his Cayley map back in 1846. In joint work with Blunk, Borovoi, Kunyavskii and Reichstein, we classify the simple stably Cayley groups over an arbitrary field of characteristic $0$.

Geometry and Topology

Speaker: Andrew Salch (Wayne State University)

"Adams spectral sequences, twisted deformation theory, and nonabelian higher-order Hochschild cohomology"

Time: 15:30 - 16:30

Room: MC 107

Given a graded Hopf algebra $A$, one wants to compute the stable representation ring $Stab(A)$. By work of Margolis, computing all possible Adams spectral sequence $E_2$-terms for finite module spectra over certain commutative ring spectra amounts to computing the cohomology of A with coefficients in each generator for Stab(A), when is a subalgebra of the Steenrod algebra. However, actually computing $Stab(A)$ is (in Margolis' words) "a very difficult problem in general."

In this talk we describe this relationship between Stab(A) and Adams spectral sequences, and we describe a new approach to the computation of Stab(A) which uses a twisted version of the deformation theory of modules. While untwisted first-order deformations of an A-module M are classified by the Hochschild cohomology group $HH^1(A, End(M))$, our twisted deformations instead are classified by a nonabelian (that is, with coefficients in a nonsymmetric module) version of the "higher-order Hochschild cohomology" of Pirashvili. We discuss existence and uniqueness results for these nonabelian higher-order Hochschild cohomologies, and the relative difficulty of actually making these computations (in particular, when they do and do not run up against of the unsolvability of the word problem!).

Analysis Seminar

Speaker: Debraj Chakrabarti (Tata Institute, Bangalore)

"The Hartogs Triangle Revisited"

Time: 14:30 - 15:30

Room: MC 107

We will discuss some recent results on the $L^2$-theory of the $\overline{\partial}$-equation on the domain $\{\vert z_1 \vert < \vert z_2\vert <1\}$ in $\mathbb{C}^2$. This is joint work with Mei-Chi Shaw.

Pizza Seminar

Speaker: Zack Wolske (Western)

"What Is The Least Lonely Number?"

Time: 16:30 - 17:30

Room: MC 107

The rational numbers are dense in the reals, so we can always approximate a real as closely as we'd like to by a rational number. But how close can the rational get if we restrict the size of the denominator? If only a few lucky ones can ever get close, the number will be mighty lonely. Many great mathematicians have worked on this problem, and in this talk we'll give a historical survey of their results, along with examples of transcendental and Liouville numbers - the least lonely of them all.

Colloquium

Speaker: Hal Schenck (University of Illinois Champaign Urbana)

"From Approximation Theory to Algebraic Geometry: the Ubiquitous Spline"

Time: 15:30 - 16:30

Room: MC 107

A fundamental problem in mathematics is to approximate a given function on some region R with a nice function, such as a polynomial. In order to get a good approximation, the standard strategy is to subdivide R into smaller regions Δi, approximate f on those regions, and require compatibility conditions on Δi∩Δj. In the most studied case, the Δi are simplices, and the compatibility condition is Cr-smoothness. The set of piecewise polynomial functions of degree at most k and smoothness r on a triangulation Δ is a vector space, and even when Δ ⊆ R2, the dimension of Crk(Δ) is unknown. Work of Alfeld-Schumaker provides an answer if k≥3r+1, and Billera earned the Fulkerson prize for solving a conjecture of Strang for the case r=1 and a generic triangulation Δ. I will discuss recent progress on the dimension question using tools of algebraic geometry, when Δ is a polyhedral complex. I will also touch on a beautiful connection to toric geometry, provided by work of Payne on the equivariant Chow cohomology of toric varieties.

Algebra Seminar

Speaker: Hal Schenck (University of Illinois Champaign Urbana)

"Toric specializations of the Rees algebra of Koszul cycles"

Time: 14:40 - 15:40

Room: MC 107

We study the linear syzygies of a homogeneous ideal $I$ in a polynomial ring $S = k[x_0..x_n]$, focussing on the graded betti numbers \[ b_i = {\textrm{dim}}_k {\textrm{Tor}}_i(S/I, k)_{i+1}. \] For any projective variety $X$ in $P^n$ and divisor $D$, what conditions on $D$ ensure that $b_i$ is nonzero? Eisenbud has shown that a decomposition $D=A+B$ such that $A$ and $B$ have at least two sections give rise to determinantal equations (and corresponding syzygies) in $I_X$ and conjectured that if the quadratic component of $I$ is generated by quadrics of rank at most four, then the last nonvanishing $b_i$ is a consequence of such a decomposition. We describe obstructions to the conjecture and prove a variant. The obstructions arise from toric specializations of the Rees algebra of Koszul cycles, and we give an explicit construction of toric varieties with minimal linear syzygies of arbitrarily high rank. This leads to a number of interesting open questions. (joint work with M. Stillman).

Geometry and Topology

Speaker: Hiro Tanaka (Northwestern)

"Factorization homology and link invariants"

Time: 15:30 - 16:30

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 - 15:30

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 - 17:30

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 - 14:00

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.