Instructor: Nikhil Srivastava, email: firstname at math.obvious.edu
Lectures: TTh 2:10-3:30pm, Hildebrand B56.
Office Hours: T: 3:40-4:40pm, Th 5-6pm (1035 Evans)
Textbooks. Much of the material will be drawn from the following two books. Both are available to Cal students for free on Oskicat or Springerlink.
Robert Richtmyer, Principles of Advanced Mathematical Physics, Volume I
Reed and Simon, Functional Analysis, Vol I
Jonathan Dimock, Quantum mechanics and quantum field theory [electronic resource] : a mathematical primer
Folland, Intro to PDE, 2e
Trefethen, Approximation Theory and Approximation Practice
I will also draw on several other resources and frequently post lecture notes on this webpage.
Announcements
Syllabus The course will survey methods for solving the fundamental problems of mathematical physics. The overall purpose of the course will be to develop a functional analytic framework for understanding and approximating solutions of differential equations, with an emphasis on physical examples. The content can broadly be divided into three parts:
# | Date | Topics | Readings | Notes | Remarks |
1 | Th 8/29 | Lebesgue integral, monotone and dominated convergence, completeness of L1 | RS I.3 | lec1 | |
2 | T 9/3 | L2, Hilbert spaces, separability, orthonormal bases. | RS II.1,II.3 | lec2 | |
3 | Th 9/5 | Weierstrass thm, separability of L2, projections, dual space, Riesz-Frechet thm | RS II.2-II.3 | lec3 | |
4 | T 9/10 | norm, adjoint, positivity, square root | RS VI.1-VI.2, VI.4 | lec4 | |
5 | Th 9/12 | range and kernel, polar decomposition, compact operators | RS VI.4-5 | lec5 | |
6 | T 9/17 | spectral thm for compact operators | RS VI.5 | lec6 | |
7 | Th 9/19 | consequences of spectral thm, trace class operators, Fredholm alternative | RS VI.6 | lec7 | |
8 | T 9/24 | basic ODE theory | lec8 | ||
9 | Th 9/26 | Green's function, completness of eigenfunctions, regular SL theory | lec9 | ||
10 | T 9/31 | group work | lec10 | ||
Th 10/3 | no lecture | ||||
11 | T 10/8 | oscillation theory | lec11 | ||
Th 10/10 | power outage | ||||
12 | T 10/15 | resolvent, spectrum | RS VI.3 | lec12 | |
13 | Th 10/17 | uniform boundedness, spectral radius, multiplication operators | RS I.4, VI.3, VI.1 | lec13 | |
14 | T 10/22 | cts functional calculus, spectral theorem | RS VII.1-2 | lec14 | |
15 | Th 10/24 | group work | lec15 | ||
16 | T 10/29 | Fourier transform | Dimock 1.1.4 | lec16 | |
17 | Th 10/31 | unbounded operators | Dimock 1.2-1.3.3 | lec17 | RS VIII.1-2 for more detail |
18 | T 11/5 | unbdd spectral theorem, physical applications | Dimock 1.3.3, 4.1-4.4 | lec18 | |
19 | Th 11/7 | tempered distributions | RS V.3 | lec19 | |
20 | T 11/12 | Fourier transform of a distribution, wave equation | lec20 | see also Folland Ch 0 | |
21 | Th 11/14 | Malgrange-Ehrenpreis theorem | see Folland PDE Ch 1F | ||
22 | T 11/19 | orthogonal polynomials, 3 term recurrence, Jacobi coeffs | |||
23 | Th 11/21 | Gauss quadrature, separation theorem | |||
24 | T 11/26 | pseudospectral methods | lec24 | guest lecture by Prof. Wilkening | |
Th 11/28 | thanksgiving | ||||
25 | T 12/3 | Chebyshev polyonomials and series, rates of convergence | Trefethen 3,7,8 | lec25 | see Trefethen Ch 7-8. |
- | Th 12/5 | lecture moved to RRR week | |||
26 | T 12/10 | Chebyshev interpolation, Hermite integral formula | Trefethen 4, 11 | makeup lecture for 12/5 | |
27 | Th 12/12 | Potential Theory, Lebesgue Constants | Trefethen 12,13,15 | makeup lecture for power outage |
Homework. Will be due every two weeks, on Thursday in class. HW assignments will be updated (i.e., problems may be added) until upto a week before they are due. Please write clearly or type your solutions using Latex. Collaboration is allowed but you must list your collaborators in your writeup.
Grading. 100% homework. The bottom assignment will be dropped.