The authors study Forman's discrete Morse theory from an algebraic viewpoint. Analogous to independent work of Emil Skoldberg, this book shows that this theory can be applied to chain complexes of free modules over a ring and provides four applications of this theory.

A memoir that studies positive definite functions on convex subsets of finite- or infinite-dimensional vector spaces. It studies representations of convex cones by positive operators on Hilbert spaces. It also studies the interplay between positive definite functions and representations of convex cones.

An important theorem by Beilinson describes the bounded derived category of coherent sheaves on $mathbb^n$, yielding in particular a resolution of every coherent sheaf on $mathbb^n$ in terms of the vector bundles $Omega_{mathbb^n}^j(j)$ for $0le j

The second part of a series of papers called "HAG", devoted to developing the foundations of homotopical algebraic geometry, this work defines and studies generalizations of standard notions of linear algebra in an abstract monoidal model category, such as

Develops the limit relations between the errors of polynomial approximation in weighted metrics and apply them to various problems in approximation theory such as asymptotically best constants, convergence of polynomials, approximation of individual functi

It is well known that some compact $3$-manifolds with boundary admit homotopy equivalences that are not homotopic to homeomorphisms. This title investigates a natural question arising in the topological theory of $3$-manifolds, and applies the results to give new information about the deformation theory of hyperbolic $3$-manifolds.

Develops the basic theory of root systems $R$ in a real vector space $X$ which are defined in analogy to the usual finite root systems, except that finiteness is replaced by local finiteness: the intersection of $R$ with every finite-dimensional subspace of $X$ is finite.

Complex symplectic spaces are non-trivial generalizations of the real symplectic spaces of classical analytical dynamics. This title presents a self-contained investigation of general complex symplectic spaces, and their Lagrangian subspaces, regardless of the finite or infinite dimensionality.

Studies the evolution of the large finite spatial systems in size-dependent time scales and compare them with the behavior of the infinite systems, which amounts to establishing the so-called finite system scheme. This title introduces the concept of a continuum limit in the hierarchical mean field limit.

Intends to complete the determination of the maximal subgroups of positive dimension in simple algebraic groups of exceptional type over algebraically closed fields. This title follows work of Dynkin, who solved the problem in characteristic zero, and Seitz who did likewise over fields whose characteristic is not too small.