Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry and sta...

Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A 'directed space' has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). Applications have been mostly developed in the theory of concurrency. Unexpected links with noncomm...

In this largely expository note we give some homological properties of algebraic maps of complex algebraic varieties which are rather surprising from the topological point of view. These include a generalisation to higher dimension of the invariant cycle theorem for maps to curves. These properties are all corollaries of a recent deep theorem of Deligne, Gabber, Beilinson, and Bernstein which i...

This is a short note intended to explore the applications of duality theory to the study of manifolds. I discuss Alexander duality, Lefschetz duality and Poincare duality, along with applications to the study of compact connected orientable manifolds. 1. Manifolds and points 1.1. The core question. One of the questions we shall be interested in is: Given two manifolds M and N , what are the way...

Perspective (Invariants). The homotopy type of a fixed configuration space is a homeomorphism invariant of the background space, and, in the case of a manifold, it turns out that these invariants remember a rather large amount of information. A simple-minded example is provided by Euclidean spaces of different dimension; indeed, as we will see, there is a homotopy equivalence B2(R) ' B2(R) if a...

Nowadays, Internet appears as a suitable tool for performing scientific computations in a distributed collaborative way. One of the fields where this idea can be applied is that of Symbolic Computation. Computer Algebra packages are being extended to interconnect them. In fact, some Computer Algebra systems are already capable of performing distributed computing. An example is Distributed Maple...

Geometric β ⊆ α T (β) ≤ S(α) V vertices of K dimS = card(α)− 1 dimS = d dimA = maxα∈A(dimα) dimA = maxS∈A dimS Table 1. Analogous Properties of Abstract and Geometric Simplicial Complexes