|30281||MW 11:50-13:05||1101 Hylan Building||Dan-Andrei Geba||MW 13:10-14:10 or by appointment, 806 Hylan Buildingemail@example.com|
Syllabus: Our goal is to cover most of the first seven chapters in the textbook.
Prerequisites: multivariable calculus (MTH 164/174) and, preferably, an introduction to complex analysis (MTH 282).
Textbook: "A guide to distribution theory and Fourier transforms" by Robert S. Strichartz, World Scientific.
Apart from the textbook, the following extra resource will be on reserve at Carlson Library:
Course philosophy: The theory of distributions represents one of the most important discoveries in mathematical analysis of the 20th century. It represents a powerful machinery which can be applied with great success to solving partial differential equations. Even though theory of distributions is usually associated to graduate-level, advanced mathematics, one can use it in a relatively straightforward manner without knowing the formal intricacies of the topic. The aim of this course is to provide an informal introduction to this beautiful subject by explaining carefully its main techniques, the intuition behind them, and the way they are applied.
This course is challenging and requires time commitment. Proficiency will be achieved only by hard work, massive problem solving, and active participation in class discussions. Please take advantage of my office hours.
Quizzes and homework are administered on a weekly basis (with the exception of the 3/5-3/11 week). Typically, quizzes contain 1-2 problems similar to the ones assigned as recommended/homework exercises. There will be twelve quizzes and homework sets, from which the best ten count towards your grade. There will be no make-up quizzes.
The Midterm was an in-class exam on Wednesday, 3/8.
The Final will be on Tuesday, 5/9, 8:30-11:30, in Hylan 1104. It will be mostly on the material covered after the midterm, though there may be some problems on material pertinent to the midterm.
1. The course average is not based on a curve, nor on previously fixed scales. It will reflect how well the class is doing, and it will be high if everyone is working hard on the recommended exercises and is performing well on quizzes and exams.
2. Incomplete "I" grades are almost never given. The only justification is a documented serious medical problem or a genuine personal/family emergency. Falling behind in this course or problems with workload on other courses are not acceptable reasons.
3. If you miss the Midterm with a valid excuse (e.g., illness or emergency), you must notify the instructor and provide supporting documentation verifying your excuse as soon as possible. For a valid excuse with supporting documentation, the Final will count as your make-up test (i.e., the Final will count towards 70% of your grade). If you miss the Final, you are in trouble. No make-up exams will be given for any reason. If you miss an exam without a valid excuse (and supporting documentation), you will receive a score of 0 on that test.
4. You are responsible for knowing and abiding by the University of Rochester's academic integrity code. Any violation of academic integrity will be pursued according to the specified procedures.
|Week of||Topic||Reading assignment||Recommended/homework exercises|
|1/16||Motivation for the study of distributions, notational conventions, test functions (section 1.1 in Strichartz; sections 1.1 and 1.2 in Friedlander)||Sections 1.2 and 1.3 in Strichartz; section 1.3 in Friedlander.|
|1/23||Friedrichs mollifiers, definition of distributions, distributions induced by locally integrable functions, equivalent characterization of distributions, convergence of distributions||Sections 2.2-2.4 in Strichartz; sections 2.1, 2.2, 2.4, and 2.5 in Friedlander.||Problems of section 1.4 in Strichartz|
|1/30||Quiz 1 (2/1), operations with distributions (derivatives for distributions and multiplication of distributions by smooth functions), worked examples of derivatives for certain distributions, weak solutions to the 1+1 dimensional wave equation, and antiderivatives for distributions on the real line||Sections 3.1-3.4 in Strichartz; sections 8.1 and 8.2 in Friedlander.||Problems of section 2.6 in Strichartz|
|2/6||Quiz 2 (2/8), the Fourier transform and its properties, the Schwartz space||Review of problems from sections 1.6 and 2.6 in Strichartz||Problems of section 3.6 in Strichartz|
|2/13||Quiz 3 (2/15), the Fourier transform of a Gaussian, the Fourier inversion formula||Sections 4.1-4.3 in Strichartz; sections 8.3 and 8.4 in Friedlander.||Homework 1 (due 2/24): problems 1-10 from section 3.6 in Strichartz|
|2/20||Plancherel formula, further properties of the Schwartz space, temperate distributions and their relation to classical distributions||Homework 2 (due 3/3): problems 17, 18, and 20 from section 2.6 and problems 13, 14, 16, 20, and 21 from section 3.6 in Strichartz|
|2/27||Quiz 4 (3/1), the Fourier transform for temperate distributions, its properties, and worked examples, convolutions between Schwartz functions and temperate distributions||Sections 5.1 and 5.2 in Strichartz; study for the midterm exam|
|3/6||Properties of the convolutions between Schwartz functions and temperate distributions, Midterm exam (3/8)|
|3/20||Discussion of the midterm exam, further properties of the convolutions between Schwartz functions and temperate distributions, applications of distribution theory to solving partial differential equations||Sections 5.3 and 5.4 in Strichartz||Homework 3 (due 3/30): problems 1-4, 6, 8, 12, and 13 from section 4.4 in Strichartz|
|3/27||Poisson formula and the fundamental solution for the Laplace equation, solution formula and the fundamental solution for the heat equation, solution formula for the wave equation||Homework 4 (due 4/6): problems 1-4, 8, 10, 11, and 18 from section 5.5 in Strichartz|
|4/3||The fundamental solution for the wave equation (explicit computations for one, two, and three spatial dimensions,; general considerations using the Lorentz group and homogeneous distributions), solution formula for the linear inhomogeneous wave equation, the Huygens principle||Sections 6.1 and 6.2 in Strichartz|
|4/10||Domains of determination and the wave equation, localization of distributions, partitions of unity, the notion of support for distributions||Sections 6.3-6.5 in Strichartz||Homework 5 (due 4/20)|
|4/17||Distributions with compact support (properties and equivalent characterization), the space of smooth functions (topology and linear, continuous functionals), distributions with point support||Homework 6 (due 4/27)|