Functional Analysis and the Related Quantum Theory.
This is an introductory course to modern functional analysis and its use in the study of PDE of Mathematical-Physics. Beginning with a review of Hilbert spaces and linear operators, we then proceed to the basics of spectral theory of self adjoint operators in Hilbert space, and applications to Fundamental mathematical aspects of Quantum mechanics, and more general Dispersive wave equations.
The necessary Analysis will be developed in class.
This course is also available by arrangement.
Functional Analysis, by M. Reed and B. Simon
Recommended Text: Functional Analysis, by Yoshida, Kosaku
Operator Theory, part 4, by Barry Simon
Real Analysis and Linear Algebra are prerequisites.
Sobolev maps with values into the circle
Sobolev functions with values into the real line are very well understood and play an immense role in many branches of Mathematics. By contrast, Sobolev maps with values into the unit circle have been investigated only in recent years.
Such maps occur e.g. in Physics. The Sobolev framework allows maps with singularities such as x/|x| in 2D or line singularities in 3D which appear in physical problems. It turns out that these classes of maps have an amazingly rich structure from the point of view of Analysis, Geometry and Topology.The course will be based on Lecture Notes which are not yet complete but parts will be available to students.
A decent knowledge of standard Sobolev spaces is required; it can be found in my book “Functional Analysis, Sobolev spaces and PDEs” (Springer)
Haim Brezis, Functional Analysis,and PDEs (Springer)
Functional Analysis, Sobolev spaces