Math 497, Fall 2018

Course Title: A Mathematical Journey with Fourier Series

Instructor: Nigel Higson

Assistant: Shintaro Nishikawa

Meeting Times: Mondays, Tuesdays, Wednesdays and Fridays, 1:25-2:15, in 113 McAllister.  Generally I will lecture on Mondays, Wednesdays and Fridays, and Shintaro will hold court on Tuesdays, but we may switch days from time to time.

Office Hours:  My office hours are Tuesdays 3:40-4:40 and Wednesdays 2:30-3:30, in 228 McAllister.  Shintaro’s office hours are 2:30-3:30 Mondays and 3:30-4:30 Wednesdays, in 015 McAllister.

Overview: The theory of Fourier series is about writing possibly complicated periodic functions as infinite linear combinations of simple ones.  The simple ones are sin(x) and cos(x), as well as sin(2x) and cos(2x), and sin(3x) and cos(3x), and so on.  What makes these functions simple is that their derivatives are easy to understand.  What makes Fourier theory useful is that by expressing possibly complicated periodic functions as combinations of simple ones, it becomes possible to say something about their derivatives.  As a result, Fourier theory is useful nearly everywhere derivatives are useful, which is to say nearly everywhere.  In this course we shall develop the mathematical theory of Fourier series and study a diverse range of applications in mathematics and beyond.

We shall very roughly follow this book:

Thomas Korner, Fourier Analysis

Cambridge University Press (1988)

This is not a required text, but I highly recommend that you obtain a copy!

Prerequisites:  The most important requirement will be good familiarity with single-variable calculus, some familiarity with basic point-set topology (open and closed sets in a metric space, convergence, continuity, etc) and with elementary analysis (continuity,  differentiability, convergence of sequences and series, and so on).  Some of these latter topics will be reviewed in the beginning part of the class, as needed. There will be an emphasis on understanding and explanation (that is, proofs) rather than computation (although we shall do computations too).  Contact me if you have any questions or concerns.

MASS Program: This course will form part of the 2018 MASS program at Penn State University.  See the MASS web pages for further information about the program.  Students outside of the MASS program who wish to take this course ought to contact me.

Homework:  There will be weekly homework assignments, due on Fridays just before the start of class.

Midterm:  There will be a midterm exam on Wednesday October 3.  The exact time and format of the exam will be announced later.

Final Exams:  A final oral exam will be scheduled for each student on December 7, 10 or 12.  The exam will follow the standard MASS final exam format.  More about that later.

Project:  Each student will be required to complete a project and present it during the final exam (this is part of the standard MASS exam format).  A list of possible project titles will be distributed at about the time of the midterm, along with further details about the project requirement.

Academic Integrity Statement: Academic integrity is the pursuit of scholarly activity in an open, honest and responsible manner. Academic integrity is a basic guiding principle for all academic activity at The Pennsylvania State University, and all members of the University community are expected to act in accordance with this principle. Consistent with this expectation, the University's Code of Conduct states that all students should act with personal integrity, respect other students' dignity, rights and property, and help create and maintain an environment in which all can succeed through the fruits of their efforts.

Academic integrity includes a commitment not to engage in or tolerate acts of falsification, misrepresentation or deception. Such acts of dishonesty violate the fundamental ethical principles of the University community and compromise the worth of work completed by others.

In order to ensure all students have a fair and equal opportunity to succeed in this course, the Math Department is committed to enforcing the University’s academic integrity policy. Below is a description of academic misconduct and the department’s responsibilities when misconduct is suspected.

Academic misconduct includes, but is not limited to:

Copying the work of another student on an exam, quiz, or assignment;

Passing off the work of another individual as your own;

Using non-approved devices or aids on exams, quizzes, or assignments;

Having unauthorized possession of exams or quizzes;

Engaging in deception in order to extend or reschedule an exam, quiz, or assignment;

Facilitating acts of academic misconduct by others.

If a student is suspected of academic misconduct, the instructor’s duties are to:

Confidentially inform the student of the allegation;

Enter the charge and recommended sanctions on an Eberly College of Science Academic Integrity form;

Ask the student to meet in order to review the form and discuss the charges and sanctions. The student can choose to accept or contest the allegation at this point.

Note that a student’s refusal to meet with the instructor or respond to the charges within a reasonable period of time is construed as acceptance of the allegation and proposed sanctions.

Once the Academic Integrity form has been accepted or contested by the student, it is sent to the College’s Academic Integrity Committee for adjudication. A student cannot drop or withdraw from the course during the adjudication process.

If a student accepts an academic misconduct allegation, or if (s)he is found guilty during adjudication, probable sanctions include:

A warning and

Reduction of the assignment grade to zero or

Reduction of the quiz or exam grade to zero.

Reduction in the final course grade;

An F in the course.

In addition, the student will be unable to drop or withdraw from the course.

(For a more compelling account of what honesty and integrity should mean, at least for a scientist, consider these famous words of Richard Feynman.

Disability Statement: Penn State welcomes students with disabilities into the University's educational programs. Every Penn State campus has an office for students with disabilities. The Office for Disability Services (ODS) Web site provides contact information for every Penn State campus: http://equity.psu.edu. For further information, please visit the Office for Disability Services Web site: http://equity.psu.edu/student-disability-resources/

In order to receive consideration for reasonable accommodations, you must contact the appropriate disability services office at the campus where you are officially enrolled, participate in an intake interview, and provide documentation; see the above link. If the documentation supports your request for reasonable accommodations, your campus’s disability services office will provide you with an accommodation letter. Please share this letter with your instructors and discuss the accommodations with them as early in your courses as possible. You must follow this process for every semester that you request accommodations.

Nigel Higson - 2018