The problem of solving real polynomial equations long fascinated people. Starting with a finite number of points on the real line one can get many complicated spaces as zero sets of polynomials. For example every closed P.L. manifold occurs as an algebraic set [3]. The main goal of understanding the topology of algebraic sets is to topologically characterize the image of the forgetful functor {...

The development of algebraic topology in the twentieth century is one of the most important advances of its time. However, how did this development get under way? What forces worked to advance the subject, and perhaps more interestingly, to retard the development in its early days? In this talk I will present some historical approaches to these questions and the answers they produce. Toutes les...

We study the Madsen-Tillmann spectrum CP∞ −1 as a quotient of the Mahowald pro-object CP∞ −∞ , which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of moduli space constructed by Kontsevich and Witten.

1 Introduction. Homotopy theory is a subdomain of topology where, instead of considering the category of topological spaces and continuous maps, you prefer to consider as morphisms only the continuous maps up to homotopy, a notion precisely defined in these notes in Section 4. Roughly speaking, you decide not to distinguish two maps which can be continuously deformed into each other; such a wea...

Elliptic curves enter algebraic topology through “Elliptic cohomology”–really a family of cohomology theories–and their associated “elliptic genera”. • Arithmetic aspect: Modularity of elliptic genera, The spectrum TMF of “topological modular forms” and the calculation of π∗TMF →MF (Z), Hopkins’s proof of Borcherds’ congruences. • Physical aspect: Witten’s approach to elliptic genera via string...

This paper is a survey of the new notions and results scattered in [13,11,12]. However the speculations of Section 5 and Section 6 are new. Starting from a formalization of higher dimensional automata (HDA) by strict globular ω-categories, the construction of a diagram of simplicial sets over the three-object small category − ← gl → + is exposed. Some of the properties discovered so far on the ...

The mechanisms by which these open sets may be assigned, is arbitrary as long they satisfy the above properties. In practice, the most common way to define open sets is by way of metrics. However, it is important to note that even in this case, the definitions of continuous maps can be much simplified using open sets especially for large dimensions. For example, the classical definition for con...

I develop algebraic-topological theories, algorithms and software for the analysis of nonlinear data and complex systems arising in various scientific contexts. In particular, I employ discrete Morse-theoretic techniques to substantially compress cell complexes built around the input data without modifying their core topological properties. Recently, I have generalized discrete Morse theory its...