Book Review: Differential forms in algebraic topology

Course Notes for Math 753: Differential Forms and Algebraic Topology

Algebraic Topology and Modular Forms

The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many mathematical invariants have expressions in terms of homotopy groups, and at different times the subject has found itself located in geometric topology, algebra, al...

BOOK REVIEW: Quantum Logic in Algebraic Approach

Since the invention of quantum logic by Birkhoff & von Neumann there has been a steady interest in the topic. Quantum logic owes much of its current attention to the rise of experimental foundational physics. Many researchers interested in the foundation of physics begin to view this field as a way to understand the quantum better; or at least from a new, hitherto unknown viewpoint. Yet much is...

Algebraic Multigrid for Discrete Differential Forms

Discrete differential forms arise in scientific disciplines ranging from computational electromagnetics to computer graphics. Examples include stable dis-cretizations of the eddy-current problem, topological methods for sensor network coverage, visualization of complex flows, surface parameterization, and the design of vector fields on meshes. In this thesis we describe efficient and scalable n...

Categorically-algebraic topology and its applications

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...