Fwd: COURSE ANNOUNCEMENT: Mathematics 5102

[Heinz, Uli]

Dear all:

Please see attached announcement from Ulrich Gerlach. Graduate students with an interest or need in deepening their math methods skills for physics might give this course offering serious consideration.

Best regards,
Ulrich

Begin forwarded message:

From: "Gerlach, Ulrich" <gerlach at math.ohio-state.edu<mailto:gerlach at math.ohio-state.edu>>
Date: November 16, 2015 at 12:08:07 AM EST
To: "Heinz, Uli" <heinz.9 at osu.edu<mailto:heinz.9 at osu.edu>>
Subject: COURSE ANNOUNCEMENT: Mathematics 5102

Dear Uli,

I would be grateful if you would forward the e-mail below and the attached PDF to the graduate students and to your colleagues in the Physics Department.

BTW, the principles and methods that are developed in Math 5102 are not chosen at random. Instead, as you can guess, they all are driven by physics. Indeed, their primary purpose is to make the E. & M out of Jackson as well as the mathematization of Q.M.  readily digestible.

Sincerely,

Ulrich

Ulrich H. Gerlach
Department of Mathematics
231 West 18th Ave
Columbus, OH 43210
614 292 7235 (office)
614 292 5101 (Department of Mathematics)
===================================================================


    Note: You are getting this e-mail because the information in it
             is not generally, or only in part, accessible to those of interest.
Subject: Informational summary of Math 5102
   From: Ulrich Gerlach



                                COURSE ANNOUNCEMENT
Mathematics 5102 ("Linear Mathematics in Infinite Dimensions")



                Course Name: Mathematical Principles in Science II:
                                      Linear Mathematics in Infinite Dimensions
 Class Number and Time: 19808 or 20056 MWF at 9:10am



              19808 or 20056 MWF at 9:10pm
                          Credits: 3
                       Web Site: https://people.math.osu.edu/gerlach.1/math/
                 Prerequisites: "Mathematical maturity", e.g. Math 5101 or several
                                       quarters of mathematics at the 21xx-4xxx  level.
                      Audience: See https://people.math.osu.edu/gerlach.1/math/
                        Purpose: Develop the principles and methods of linear
                                       mathematics in infinite dimensionsmathematics and science.
           Follow-up Course: Mathematics 5756-5757 ("Modern Mathematical Methods in
                                      Relativity Theory")



  LIST OF MATH 5102 TOPICS





MATH 5102 :
Linear Mathematics in Infinite Dimensions

("Mathematical principles, methods, and concepts of vibrations, oscillations, waves, and signals")
(SPRING 2016, MWF 9:10am, Instructor:  U. Gerlach)
KEY COURSE TOPICS
         0. OVERVIEW
         I. STURM-LIOUVILLE THEORY
         Sturm-Liouville systems: regular, periodic, and singular
         Eigenvalues and eigenfunctions via phase analysis


        II.  INFINITE DIMENSIONAL VECTOR SPACES
         Hilbert spaces: square summable series and square integrable functions
         Unitary transformation between Hilbert spaces

III. FOURIER THEORY
Fourier series
Dirichelet kernel
Fourier's theorem on a finite domain
Sequences leading to the Dirac delta function
Fourier transform representation
Change of basis in Hilbert space:
Orthonormal wavelet and wavepacket representations

IV. GREEN'S FUNCTION THEORY: INHOMOGENEOUS DIFFERENTIAL EQUATIONS
Homogeneous sytems
Adjoint systems
Inhomogeneous systems
The concept of a Green's function
Solution via Green's function
Integral equation of a linear system via its Green's function
Classification of integral equations
The Fredholm alternative
Green's function and the resolvent of the operator of a system
Eigenfunctions and eigenvalues via residue calculus
Branches, branch cuts, and Riemann sheets
Singularity structure of the resolvent of a system:
Poles and branch cuts
Effect of boundary conditions and domain size

V. THEORY OF SOLUTIONS TO PARTIAL DIFFERENTIAL EQUATIONS
IN TWO AND THREE DIMENSIONS
Partial differential equations: hyperbolic, parabolic, and elliptic
The Helmholtz equation and its solutions in the Euclidean plane.
Geometry of the space of solutions
Plane waves vs cylinder waves:
Why, and when to use them
Sommerfeld's integral representation
Hankel, Bessel, and Neumann waves
Change of basis in the space of solutions: partial waves
Displaced cylinder waves
The cylindrical addition theorem
Method of steepest descent and stationary phase
Analytic behaviour of cylinder waves
Interior (cavity) and exterior (scattering) boundary value problems
Spherical waves: symmetric and nonsymmetric
Cauchy problem and characteristics (time permitting)
Texts: (1) U.H. Gerlach: Linear Mathematics in Infinite Dimensions<https://people.math.osu.edu/gerlach.1/math5102/BVtypset> (Chapter 1,2,3,4,5)
           (2) F.W. Byron  and R.W. Fuller: Mathematics of Classical and Quantum Physics



Questions? Visit --> https://people.math.osu.edu/gerlach.1/math/



Additional questions are welcomed.



Ulrich Gerlach



Telephone: 292-7235
   FAX      : 292-1479
   e-mail   : gerlach at math.ohio-state.edu<mailto:gerlach at math.ohio-state.edu>





       Ulrich H. Gerlach         |       E-mail: gerlach.1 at osu.edu<mailto:gerlach at math.ohio-state.edu>
Department of Mathematics | telephone: 1-614-292-7235 (Office)
   The Ohio State University |                 1-614-292-5101 (Math. Dept.)
    231 West 18th Ave.        |                 1-614-292-1479 (Dept. Fax)
Columbus, OH 43210-1174  |  web page: https://people.math.osu.edu/gerlach.1/math/


















































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