1. Contact Information
  2. Prerequisites
  3. Course Description and Credit Hours
  4. Required Texts
  5. Course Objectives
  6. Student Learning Outcomes
  7. Other Course Materials
  8. Outline Of Topics
  9. Exams and Assignments
  10. Grading Policy
  11. Policy on Missed Exams and Coursework
  12. Attendance Policy
  13. Notification of Changes
  14. Custom Sections
  15. Statements on Academic Misconduct
  16. Statement On Disability Accommodations
  17. Severe Weather Protocol
  18. Pregnant Student Accommodations
  19. Religious Observances
  20. UAct Statement

Seminar: Topics in Analysis

MATH 688-001Fall 2018 | 3 Credit Hours

Seminar

Shibin Dai

Contact Information

UA Campus Directory:

Prerequisites

UA Course Catalog Prerequisites:

MATH 681

This course has two prerequisites.

1. Basic knowledge of PDE (Math 541 or 642).

2. Real Analysis (Math 580 and 681).

Course Description

Course Description and Credit Hours

Advanced course in real analysis. Topics may include harmonic analysis (the Fourier transform, Hardy-Littlewood maximal operator, interpolation, singular integral operators, BMO and Hardy spaces, weighted norm inequalities) or analysis and PDEs (Sobolev spaces, weak solutions to PDEs, Lax-Milgram theory, the Fredholm alternative, existence and regularity for elliptic and parabolic equations).

MODERN THEORY OF PARTIAL DIFFERENTIAL EQUATIONS

This course concerns the modern theory of partial differential equations (PDE). The classical approach of PDE is the search for explicit formulas for the solutions. However, many times formulas do not provide as much useful information as we hope. What’s more, most PDEs do not possess a formula for their solutions. Hence it is essential to understand the properties of their solutions without a formula.

In this course, we will abandon the search for explicit solutions. Instead we will concentrate on modern techniques in the theoretical study of linear and nonlinear PDEs. This is a course that provides students the opportunity to learn PDE in topics beyond the classical theory covered in Math 541 and 642. And students will be prepared for research in PDE-related areas. In his now definitive textbook Partial Differential Equations, L. Craig Evans described six principles:

  1. PDE theory is (mostly) not restricted to two independent variables.

  2. Many interesting equations are nonlinear.

  3. Understanding generalized solutions is fundamental.

  4. PDE theory is not a branch of functional analysis.

  5. Notation is a nightmare.

  6. Good theory is (almost) as useful as exact formulas.

These principles are also the guidelines for our course.

Required Texts

Required Texts from UA Supply Store:
  • EVANS / PARTIAL DIFFERENTIAL EQUATIONS (V19) (Required)
  • EVANS (RENTAL) / (RENTAL) PARTIAL DIFFERENTIAL EQUATIONS (V19) (RENTAL)

Student Learning Outcomes

  • Students will analyze the properties of PDEs.

Other Course Materials

Weak Convergence Methods for Nonlinear Partial Differential Equations, (Regional Conference Seriess in Mathematics, No 74) CBMS/74, by Lawrence C. Evans, 1990

Elliptic Partial Differential Equations, Second Edition, by Qing Han and Fanghua Lin

Outline of Topics

Sobolev spaces; weak solutions (existence, uniqueness, and regularity) for linear and nonlinear PDEs; calculus of variations; non variational techniques for nonlinear PDEs.

Exams and Assignments

There will be no written homework or written exam. Rather, each student will give a presentation on some problems chosen from the textbook.

Grading Policy

The grade will be determined by class participation, and the quality of each student's presentation.

Policy on Missed Exams and Coursework

Each student will give at least one presentation on topics from the textbook. If a student has to miss a scheduled presentation due to legitimate reasons or emergencies, he/she should inform the instructor as soon as possible.

Attendance Policy

Attendance is required.

Notification of Changes

The instructor will make every effort to follow the guidelines of this syllabus as listed; however, the instructor reserves the right to amend this document as the need arises. In such instances, the instructor will notify students in class and/or via email and will endeavor to provide reasonable time for students to adjust to any changes.

Statement on Academic Misconduct

Students are expected to be familiar with and adhere to the official Academic Misconduct Policy provided in the Online Catalog.

Statement On Disability Accommodations

Contact the Office of Disability Services (ODS) as detailed in the Online Catalog.

Severe Weather Protocol

Please see the latest Severe Weather Guidelines in the Online Catalog.

Pregnant Student Accommodations

Title IX protects against discrimination related to pregnancy or parental status. If you are pregnant and will need accommodations for this class, please review the University’s FAQs on the UAct website.

Religious Observances

Under the Guidelines for Religious Holiday Observances, students should notify the instructor in writing or via email during the first two weeks of the semester of their intention to be absent from class for religious observance. The instructor will work to provide reasonable opportunity to complete academic responsibilities as long as that does not interfere with the academic integrity of the course. See full guidelines at Religious Holiday Observances Guidelines.

UAct Statement

The University of Alabama is committed to an ethical, inclusive community defined by respect and civility.  The UAct website (www.ua.edu/uact)  provides extensive information on how to report or obtain assistance with a variety of issues, including issues related to dating violence, domestic violence, stalking, sexual assault, sexual violence or other Title IX violations, illegal discrimination, harassment, hate or bias incidents, child abuse or neglect, hazing, threat assessment, retaliation, and ethical violations or fraud.