Algebra Seminar 

Trying to discover a unifying theory connecting the noncommutative geometry of schematic algebras to the classical basis for quantum mechanics and the noncommutative geometry in terms of C*-algebras, we develop noncommutative topology both in the analytic sense or in the sense of Grothendieck topology. The relation with the lattice of linear closed subspaces of a Hilbert space H is indicated. Function theory in a quaternion variable viewed as two noncommuting complex variables is developed and leads to noncommutative Riemann manifolds and other examples of noncommutative manifolds.

There is a general construction of ''blowing-up'' a manifold $W$ along a submanifold $V$. This construction has applications both in algebraic geometry and in symplectic topology. In this talk we show that the rational homotopy type of such a blow-up is completely determined by the rational homotopy class of the embdedding and the Chern classes of its normal bundle, at least when $\dim (W)\ge 2\dim (V)+4$. In fact a computable model of the rational homotopy type of the blow-up $\tilde{W}$ can be explicitely described. This construction gives many examples of non-formal simply-connect symplectic manifolds.

We study the interplay between the integral closedness - or even the normality - of an $m$-primary ideal $I$ and conditions on the Hilbert-Samuel coefficients of $I$. We relate these properties to the Cohen-Macaulayness of the associated graded ring of $I$.

The purpose of this talk is to provide an introduction to the main techniques and results of rational homotopy theory. No original results will be presented.

We say that a finite CW-complex $X$ embeds up to homotopy in a sphere $S^n$ if there exists a subpolyhedron $K\subset S^n$ having the same homotopy type as $X$. In this talk, I will explain a sufficient condition for the existence of such an embedding in a given codimension that we obtained in the particular case where $X$ is a two-cone (that is, $X$ is the homotopy cofibre of a map between two suspensions). As an application of this result, we will see that there is no rational obstruction to embeddings up to homotopy of a two-cone in codimension 3.

The open unit ball in a complex Banach space $A$ is a bounded complex domain, the holomorphic structure of which leads to the existence of a closed subspace $A_s$ of $A$ and a triple product $\{...\}:A\times A_s\times A\to A$ which is symmetric and bilinear in the outer variables, conjugate linear in the middle variable and, for $a$, $b$ and $d$ in $A_s$ and $c$ in $A$, satisfies the Jordan triple identity,

Let $A_k^n$ denote the affine space of dimension $n$ over a field $k$ and $X\subset A_k^n$ be an arrangement of linear subvarieties. Set $R=k[x_1,\dots ,x_n]$ and let $I\subset R$ denote an ideal which defines $X$. If $k$ is a field of characteristic zero, the local cohomology modules modules $H_I^r(R)$ are known to have a module structure over the Weyl algebra $A_n(k)$, and one can therefore consider their characteristic cycles, denoted $\mathrm{CC}(H_I^r(R))$ in this talk. In either the real or the complex case, we shall determine the Betti numbers of the complement of the arrangement $X$ in terms of the multiplicities of the local cohomology modules $H_I^r(R)$.

A central problem in algebraic topology has been to single out the homotopy theoretical properties which characterize compact Lie groups. The right definition of a $p$-local version of a compact Lie group came with Dwyer and Wilkerson who introduced the so called $p$-compact groups and proved that they possess many nice properties. For example a $p$-compact group has a maximal torus, a maximal torus normalizer and a Weyl group.

For odd primes $p$, we give a classification of p-compact groups by proving that they are determined by their maximal torus normalizers. In particular the Weyl group data gives a bijection between connected $p$-compact groups and finite $p$-adic reflection groups.

A number of corollaries follow easily from this classification, for example we give an affirmative answer to the maximal torus conjecture for finite loop spaces up to $\mathbb{Z}[1/2]$-localization.