Home page for AMCS 609: Analytic Techniques for AMCS II
Spring 2011
Instructor: Charles L. Epstein
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The Course
The main goal of analysis is the solution of equations. We began by
reviewing the basic results from linear and non-linear finite
dimensional analysis and the solution of "algebraic" equations.
In the first semester we considered the case of 1-variable complex
analysis, for most of this semester we turn to the infinite dimensional
context. Amongst other things, we will develop the framework
needed to find solutions
of ordinary and partial differential equations.
We begin this semester by introducing some of the basic tools of
asymptotic analysis: Laplace's method, steepest descent and stationary
phase. We then turn to analysis in infinite dimensional linear spaces.
We will use Peter D. Lax, Functional Analysis as our
principal text book. Good additional references for functional analysis
are: Walter Rudin: Real and Complex
Analysis and Functional
Analysis; Ward Cheney, Analysis
for Applied Mathematics;
Michael Reed and Barry Simon, Methods
of Modern Mathematical Physics, Vol. 1; Tosio Kato, Perturbation Theory of Linear
Operators. Good references for finite dimensional analysis are: Peter Lax, Linear Algebra;
W. Rudin, Principles of
Mathematical Analysis; Jerrold Marsden, Elementary Classical Analysis;
Michael Spivak, Calculus on Manifolds.
Good references for complex analysis are E. Stein and R.
Shakarchi, Complex Analysis;
Lars Ahlfors, Complex Analysis;
Zeev Nehari, Conformal Mapping.
A problem set
will be
assigned every week on Tuesday, due the following week on Tuesday. I
very much prefer that students do the
problem sets
alone. We may have a
take-home midterm and final exams.
- The class meets on TTh from 10:30-12:00 in room 4C8 of the David
Rittenhouse Labs.
- My office hour is Mondays 3:30-5PM. Contact me by e-mail for an
appointment if you can not come during this time.
- My office in the Math Department is 4E7 DRL, tel. 8-8476.
- email: cle@math.upenn.edu.
Send e-mail if you have a question or need to contact me.
Syllabus
The numbers in parentheses are chapters numbers in Lax, Functional Analysis
- Methods of aymptotic analysis
- Normed linear spaces, definitions and examples (1, 2, 5)
- Convexity, the Hahn-Banach Theorem (3, 4, 13.1)
- Hilbert space and the Riesz Representation Theorem (6, 7)
- Duality (8, 9)
- Weak convergence (10, 11)
- Bounded linear maps (15, 16, 20)
- Compact operators (21, 22)
- Fredholm theory (21)
- Solving elliptic equations using boundary layer theory
- Spectra of compact operators in a Hilbert space (28, 29)
Announcement
- Problem set 11 is due on Thursday April 28.
- I will hold extra office hours on Wednesday, April 27 from 4:00-5:00.
Problem sets
- Problem set 1. Due January 27, 2011.
- Problem set 2. Due February 8, 2011.
- Problem set 3. Due February 15, 2011.
- Problem set 4. Due February 22, 2011.
- Problem set 5. Due March 1, 2011.
- Problem set 6. Due March 15, 2011.
- Problem set 7. Due March 22, 2011.
- Problem set 8. Due April 5, 2011.
- Problem set 9. Due April 12, 2011.
- Problem set 10. Due April 19, 2011.
- Problem set 11. Due April 28, 2011.
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