Fwd: COURSE ANNOUNCEMENT: Mathematics 5102

[Heinz, Uli]

FYI, please.

Ulrich

Begin forwarded message:

From: "Gerlach, Ulrich" <gerlach at math.ohio-state.edu<mailto:gerlach at math.ohio-state.edu>>
Subject: COURSE ANNOUNCEMENT: Mathematics 5102
Date: December 1, 2017 at 6:55:00 PM EST
To: "Heinz, Uli" <heinz.9 at osu.edu<mailto:heinz.9 at osu.edu>>

Dear Ulrich,

I would be grateful if you would forward the course announcement below to the graduate and advanced undergraduate students in the Physics Department.
(If a person wants to take the course s/he needs to sign up by Wednesday, Dec. 6, a fact which is included in the course announcement below)

Sincerely,

Ulrich
P.S. Could you send a copy also to me? Thanks.

Ulrich H. Gerlach
Department of Mathematics
231 West 18th Ave
Columbus, OH 43210
614 292 7235 (office MW 506)
614 292 5101 (Department of Mathematics)
__________________________________________________________________________________________________________
________________________________
From: Gerlach, Ulrich

Subject: COURSE ANNOUNCEMENT: Mathematics 5102


    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: 15437 or 31926 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 dimensions for mathematics 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/

<https://people.math.osu.edu/gerlach.1/math/>
If you want to take the course you need to sign up before Wednesday, Dec. 6.

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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