Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, over-determined, under-determined) systems. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, time-delay operators), we first concentrate on the computation of morphisms fr...

Here BΓ is the classifying space of the group Γ, and we denote by K−∞(R) the non-connective algebraic K-theory spectrum of the ring R. The homotopy groups of this spectrum are denoted Kn(R) and coincide with Quillen’s algebraic K-groups of R [Qui73] in positive dimensions and with the negative K-groups of Bass [Bas68] in negative dimensions. The homotopy groups of the spectrum X+∧K(R) are denot...

The aim of this paper is to show one more facet of the role of crossed complexes as generalisations of both groups (or groupoids) and of chain complexes. We do this by defining and establishing the main properties of a classifying space functor B: mia -> SToft from the category of crossed complexes to the category of spaces. The basic example of a crossed complex is the fundamental crossed comp...

1. Let M be a compact, oriented, connected, ^-dimensional differential manifold with dM (boundary M) homeomorphic to the n—1 sphere 5~". Then dM represents an element [dM] of r~ , the group of differential structures (up to equivalence) on 5~. We consider the (much studied) problem of expressing [dM] in terms of "computable" invariants of M. Let 7Tn-i be the n — 1 stem, Jo: 7r»(BSO)—»7rw-i the ...

This is an overview and a preview of the theory of mixed motives of level ≤ 1 explaining some results, projects, ideas and indicating a bunch of problems. Dedicated to Jacob Murre Let k be an algebraically closed field of characteristic zero to start with and let S = Spec(k) denote our base scheme. Recall that Murre [42] associates to a smooth n-dimensional projective variety X over S a Chow co...

The permanent-determinant method and its generalization, the HafnianPfaffian method, are methods to enumerate perfect matchings of plane graphs that were discovered by P. W. Kasteleyn. We present several new techniques and arguments related to the permanent-determinant with consequences in enumerative combinatorics. Here are some of the results that follow from these techniques: 1. If a biparti...

Hitchin proved that if M is a spin manifold with positive scalar curvature, then the A^O-characteristic number a(M) vanishes. Gromov and Lawson conjectured that for a simply connected spin manifold M of dimension > 5, the vanishing of a(M) is sufficient for the existence of a Riemannian metric on M with positive scalar curvature. We prove this conjecture using techniques from stable homotopy th...

For a smooth subvariety X ⊂ P , consider (analogously to projective normality) the vanishing condition H(P , I X (k)) = 0, k ≥ 3. This condition is shown to be satisfied for all sufficiently large embeddings of a given X, and for a Veronese embedding of P. For C ⊂ P, the canonical embedding of a non-hyperelliptic curve, this condition guarantees the vanishing of some obstruction groups to defor...

Lecture Notes taken by Nilufer Koldan In this lecture I will describe one of the most significant achievements of the second half of the XX century – the Atiyah-Singer index theorem. I will also discuss some more recent developments in the area as well as some open problems. 1. The definition of the index 1.1. The finite-dimensional case. Let A : V 1 → V 2 be a linear map between finite dimensi...

We approach the problem of the computation of persistent homology for large datasets by a divide-and-conquer strategy. Dividing the total space into separate but overlapping components, we are able to limit the total memory residency for any part of the computation, while not degrading the overall complexity much. Locally computed persistence information is then merged from the components and t...