Differential forms in algebraic topology. Quantum physics

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...

Quantum Methods in Algebraic Topology

In this paper, we present a new version of cochains in Algebraic Topology, starting with “quantum differential forms”. This version provides many examples of modules over the braid group, together with a control of the non commutativity of cup-products on the cochain level. If the quantum parameter q is equal to 1, we essentially recover the commutative differential graded algebra of de Rham-Su...

Topology Change and Quantum Physics

The role of topology in elementary quantum physics is discussed in detail. It is argued that attributes of classical spatial topology emerge from properties of state vectors with suitably smooth time evolution. Equivalently, they emerge from considerations on the domain of the quantum Hamiltonian, this domain being often specified by boundary conditions in elementary quantum physics. Several ex...

Quantum Differential Forms

§