In this talk I will describe joint work with Fred Cohen on the geometry of the space of ordered commuting and non-commuting n-tuples in a Lie group G.
In his 1932 book Théorie des Opérations Linéaires,
S. Banach introduced the space of isometry classes [X], of
n-dimensional Banach spaces equipped with the famous Banach-Mazur
metric:
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In this talk we shall present some resent results and open problems related to these interesting objects.
Let x be a smooth vector bundle over a differentiable manifold M. Let h : en-i+1 ® x be a generic bundle morphism from the trivial bundle of rank n-i+1 to x. We give a geometric construction of the Stiefel-Whitney classes when x is a real vector bundle, and of the Chern classes when x is a complex vector bundle. Using h we define a differentiable closed manifold [(Z)\tilde](h) and a map f: [(Z)\tilde](h) ® M whose image is the singular set of h. The i-th characteristic class of x is the Poincaré dual of the image, under the homomorphism induced in homology by f, of the fundamental class of the manifold [(Z)\tilde](h). We extend this definition for vector bundles over a paracompact space, using that the universal bundle is filtered by smooth vector bundles.
Moment angle complexes are universal for toric varieties. There is a free action of a torus so that the quotient by this action gives you a toric manifold. We determine the stable structure of the moment angle complexes and their generalizations.
In their study of differential periodic transformations, Conner and Floyd realized the importance of understanding the global structure of manifolds admitting a group action without stationary points. They succeeded in giving a homotopy characterization, but could only determine the corresponding bordism ideal for the case of a finite elementary abelian p-group of rank two (soon after the general rank case was completed by Floyd). In this talk I address the (2-primary) "non-elementary" situation. Much of the information is derived through a sharp description of the (simultaneous) 2- and v1-divisibility properties of 2-typical formal groups. Although the geometric motivation is no longer valid, I discuss some applications to the motion planning and immersion problems for 2-torsion lens spaces.
This is joint work with Leticia Zárate.
Which finite groups can act freely and smoothly on a product Sn×Sn of two spheres? This talk will describe an approach to solving this problem (joint work with Özgün Ünlü). Important test cases are the non-abelian p-groups of order p3 and exponent p, for p an odd prime.
The rational cohomology of a compact connected Lie group G can be expressed and explicitly determined using invariant theory. One need only determine the rational cohomology of BG, the classifying space of G, and that can be expressed as a ring of invariants determined by the action of the Weyl group W (associated to G) on the rational cohomology of BT, the classifying space of any maximal torus T < G.
When one moves to mod p cohomology the same pattern holds unless the group G has p torsion in its integral cohomology. In the torsion case one needs new tools and a new pattern. We will explore the use of the generalized invariants, as defined by Kac and Peterson, to address this question for both algebra and coalgebra structures.
An intercalate matrix M of type (r,s,n) is an r by s matrix each entry of which is colored by one of n given colors such that
In their seminal paper Chas and Sullivan introduced a new structure in the homology (and equivariant homology) of the free loop space of a smooth manifold. This structure behaves in many ways as a quantum field theory. For an algebraic topologist this has an expression as certain algebraic structures (BV-algebras, Lie algebras, operad actions). In this talk we will introduce the basic ideas in this field, and then explain our generalization (jointly with B. Uribe and M. Xicotencatl) to the case in which the manifold is replaced by an orbifold. An orbifold has an atlas which locally looks like an open set of euclidean space with the action of a finite group. Our generalization could be interpreted as an equivariant version of the theory.
Tropical varieties are finite-dimensional polyhedral complexes which are in general not topological manifolds. However, it is possible to define cycles and the cycle intersections there. We plan to discuss these definitions as well as the tropical counterparts of hp,q.
I will discuss various definitions of the moduli spaces of real algebraic curves.
We prove that every p-compact group is the p-completion of a smooth closed parallellisable manifold.
Joint work with Tilman Bauer.
We will describe the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones and show that the braid groups of the 2-dimensional sphere and the 2-dimensional real projective space in any number of strings satisfy FIC. We will present consequences of this fact in the computation of Whitehead group of the above groups.
Milnor's fibration theorem for complex singularities says that every holomorphic map Cn ® C determines canonically an open book decomposition on the 2n-1 sphere. This result has given rise to a vast literature, both in singularity theory and in knot theory.
In this talk I will present a recent work with Anne Pichon, where we extend Milnor's theorem to meromorphic germs and to certain real analytic mappings.
We will discuss basic geometric and homotopical properties of the space of homomorphisms Hom(G,G) when G is a suitable discrete group and G is a Lie group.
By using the loop orbifold of the symmetric product, we give a formula for the Poincaré polynomial of the free loop space of the Borel construction of the symmetric product. We also show that the Chas-Sullivan product in the homology of the free loop space of the Borel construction induces a ring structure in the homology of the inertia orbifold of the symmetric product. This ring structure is compared to the one in cohomology through Poincaré duality.