Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"On the stability group of a 2-nondegenerate hypersurface in $\mathbb C^3$"

Time: 14:30

Room: MC 107

Real hypersurfaces in a complex space $\mathbb C^N, N \geq 2$, satisfying the Levi non-degeneracy condition, were very well studied in the famous works of Poincare, Cartan, Tanaka, Chern and Moser and in a large number of further papers. The Levi-degenerate case, which is trivial for $N=2$ (all Levi degenerate hypersurfaces in this case are essentially flat), turns out to be absolutely non-trivial for $N=3$. The reason is that a hypersurface in $\mathbb C^3$ can have Levi form of rank $1$ at a generic point, and, in this case, is neither Levi-flat nor Levi non-degenerate. If, in addition, it satisfies some non-degeneracy condition, guaranteeing that it can not be reduced to a product of a hypersurface in $\mathbb C^2$ and a complex line, the hypersurface is called 2-nondegenerate. 2-nondegenerate hypersurfaces in $\mathbb C^3$ were deeply studied in a series of papers by Ebenfelt, Beloshapka, Zaitsev, Merker, Fels and Kaup and many other authors, but a lot of essential questions, concerned with their holomorphic classification and symmetry groups, remained opened. In the present talk we demonstrate a new approach to the study of 2-nondegenerate hypersurfaces, based on the consideration of degenerate quadratic models. This new point of view enables us to give a complete solution for most of the above open questions.

Algebra Seminar

Speaker: Thomas Weigel (Universita' di Milano-Bicocca)

"Profinite groups with cyclotomic $p$-orientations"

Time: 14:30

Room: MC 107

Colloquium

Speaker: Alistair Savage (University of Ottawa)

"A gentle introduction to categorification"

Time: 15:30

Room: MC 107

This will be an expository talk concerning the idea of categorification and its role in representation theory.  We will begin with some very simple yet beautiful observations about how various ideas from basic algebra (monoids, groups, rings, representations etc.) can be reformulated in the language of category theory.  We will then explain how this viewpoint leads to new ideas such as the ``categorification'' of the above-mentioned alegbraic objects.  We will conclude with a brief synopsis of some current active areas of research involving the categorification of quantum groups.  One of the goals of this idea is to produce four-dimensional topological quantum field theories.  Very little background knowledge will be assumed.

Geometry and Topology

Speaker: John Harper (Western)

"Completion with respect to topological Andre-Quillen homology"

Time: 15:30

Room: MC 107

Quillen's derived functor notion of homology provides interesting and useful invariants in a variety of homotopical contexts, and includes as special cases (i) singular homology of spaces, (ii) homology of groups, and (iii) Andre-Quillen homology of commutative rings. Working in the topological context of symmetric spectra, we study topological Quillen homology of commutative ring spectra, E_n ring spectra, and more generally, algebras over any operad O in spectra. Using a QH-completion construction---analogous to the Bousfield-Kan R-completion of spaces---we prove under appropriate conditions (a) strong convergence of the associated homotopy spectral sequence, and (b) that connected O-algebras are QH-complete---thus recovering the O-algebra from its topological Quillen homology plus extra structure. A key problem in usefully describing this extra structure was solved recently using homotopical ideas in joint work with Kathryn Hess that describes a rigidification of the derived comonad that coacts on the object underlying topological Quillen homology, and plays the analogous role (in symmetric spectra) of the Koszul cooperad associated to a Koszul operad in chain complexes. This talk is an introduction to these results with an emphasis on proving (a) and (b) which is joint work with Michael Ching.

Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"On the stability group of a 2-nondegenerate hypersurface in $\mathbb C^3$"

Time: 14:40

Room: MC 107

Real hypersurfaces in a complex space $\mathbb C^N, N \geq 2$, satisfying the Levi non-degeneracy condition, were very well studied in the famous works of Poincare, Cartan, Tanaka, Chern and Moser and in a large number of further papers. The Levi-degenerate case, which is trivial for $N=2$ (all Levi degenerate hypersurfaces in this case are essentially flat), turns out to be absolutely non-trivial for $N=3$. The reason is that a hypersurface in $\mathbb C^3$ can have Levi form of rank $1$ at a generic point, and, in this case, is neither Levi-flat nor Levi non-degenerate. If, in addition, it satisfies some non-degeneracy condition, guaranteeing that it can not be reduced to a product of a hypersurface in $\mathbb C^2$ and a complex line, the hypersurface is called 2-nondegenerate. 2-nondegenerate hypersurfaces in $\mathbb C^3$ were deeply studied in a series of papers by Ebenfelt, Beloshapka, Zaitsev, Merker, Fels and Kaup and many other authors, but a lot of essential questions, concerned with their holomorphic classification and symmetry groups, remained opened. In the present talk we demonstrate a new approach to the study of 2-nondegenerate hypersurfaces, based on the consideration of degenerate quadratic models. This new point of view enables us to give a complete solution for most of the above open questions.

Colloquium

Speaker: Thomas Weigel (Universita' di Milano-Bicocca)

"Galois theory: Old stories and modern fashion"

Time: 15:30

Room: MC 108

Algebra Seminar

Speaker: Masoud Khalkhali & Farzad Fathizadeh (Western)

"Curvature in noncommutative geometry"

Time: 15:10

Room: MC 107

Geometry and Topology

Speaker: Parker Lowrey (Western)

"A geometric classifying stack for the bounded derived category"

Time: 15:30

Room: MC 107

We define a classifying stack for the bounded derived category associated to any scheme X. When X is projective, we show that this stack is locally geometric, i.e., we can treat it as a slight abstraction of a scheme. We will also provide some applications of this result.

Analysis Seminar

Speaker: Seyed Mehdi Mousavi (Western)

"An Infinite-Dimensional Maximal Torus and Shur-Horn-Kostant Convexity"

Time: 14:40

Room: MC 107

One of the main notion introduced in the study of finite dimensional compact Lie groups is the so-called maximal torus. In 1997, Bao and Ratiu discovered an infinite dimensional subgroup in the group of the volume-preserving diffeomorphisms of the 2-dimensional annulus that can potentially play the role of a maximal torus. They showed this subgroup is a path-connected submanifold which is flat and totally geodesic with respect to the hydrodynamic metric. Moreover it is a maximal abelian subgroup (with a finite Weyl group). This suggested that part of finite dimensional Lie group theory may be extended to the volume-preserving diffeomorphisms of the annulus. Indeed, in a later work, Bloch, Flaschka and Ratiu showed that after an appropriate completion of the spaces considered, a version of Schur-Horn-Kostant convexity theorem holds. El-Hadrami extended these results to the case of the unit sphere and CP^{2}, found a candidate for the maximal torus in the symplectomorphism group of symplectic toric manifolds, and then conjectured that some results in previous works can be extended to those groups. However, a gap in El-Hadrami’s arguments was later discovered.

In two talks we discuss some possible extensions and corrections to El-Hadrami´s work. We also mention the Schur-Horn-Kostant convexity theorem for the symplectomorphism groups of toric manifolds.

Pizza Seminar

Speaker: Masoud Khalkhali (Western)

"Why 1 + 2 + 3 + 4 + ... = -1/12"

Time: 16:30

Room: MC 107

Colloquium

Speaker: George Pappas (Michigan State University)

"Shimura varieties, their integral models and singularities"

Time: 15:30

Room: MC 107

Shimura varieties are algebraic varieties that play an important role in number theory and the Langlands program. I will discuss constructions of models of Shimura varieties over the integers and recent results about the singularities of their reductions modulo primes that divide the level.

Algebra Seminar

Speaker: Stefan Tohaneanu (Western)

"Spline approximation and homology"

Time: 14:30

Room: MC 107

Let $\Delta$ be a triangulation of a connected region in the real plane. Let $C(r,d,\Delta)$ be the space of piecewise polynomial functions of degree $\leq d$ and smoothness $r$. A major question in Approximation Theory is to find the dimension of this space, which is not known even for the case when $d=3$ and $r=1$. Alfeld and Schumaker give a formula for this dimension, when $d\geq 3r+1$ and any $\Delta$. Using homological algebra, this problem can be translated into finding the Hilbert function of a graded module (the ``homogenization'' of $C(r,d,\Delta)$). I will discuss about this approach and about the Schenck-Stiller conjecture that says that Alfeld-Schumaker formula holds for any $d\geq 2r+1$. I will present a very recent project with Jan Minac where we prove this conjecture for a triangulation that is not trivial, in the sense that the formula does not hold if $d=2r$.