625.601.8VL - Real Analysis

Course Structure

Each module's material will be presented in a virtual-live lecture/discussion on Zoom. Recordings of the lectures will be posted in the corresponding module's folder. Homework assignments will also be posted there, along with course discussion boards for the relevant material and any questions students may have. Each module lasts a week.

Course Topics

• Properties of the Real Numbers
• Metric Space Topology
• Sequences and Series
• Limits and Continuity
• Differentiation
• Riemann-Stieltjes Integral
• Lebesgue Integral

Course Goals

To understand the fundamental principles of the field of real analysis and to use the subject area to practice and refine your mathematical and logical thinking.

Course Learning Outcomes (CLOs)

• Construct proofs of simple theorems
• Apply sophisticated theorems to solve problems in real analysis
• Rigorously explain the main concepts of real analysis, including completeness, limits, continuity, integrability and differentiability

Textbooks

Analysis I and Analysis II, by Terence Tao (Fourth Edition), required but available for free through the JHU Library system (see "eReserves" link on the Canvas site).

Student Coursework Requirements

Homework (30%)The homework for this course is one of the main venues for you to improve your proof-writing and reasoning skills. I will assign several problems a week from the textbook or elsewhere, and they are due the following week. You are encouraged to work together on the homework, but the written work you submit must be in your own words.
Reading Summaries (15%)A reading summary is a brief (approximately 5-7 minutes total) oral recapitulation of the main ideas/definitions/theorems of the assigned reading for the week. Each student will be assigned two reading summaries throughout the course, and these will be prepared with fellow students assigned to the same reading. Your group will share your reading summary with everyone in the course at the beginning of a module's lecture. This is a low-pressure way to encourage you do to the assigned reading, engage with other students, and help each other learn. It also starts the class session off with a summary of what we're going to focus on for the lecture.

Mid-Term Exam (25%)Final Exam (30%)The two exams in the course are week-long, take-home exams with several novel problems. You are not allowed to work with other students on the exams; all of the work you submit must be your own, original work.

Grading Policy

HomeworkEach week, I will select a subset of the homework problems to grade for accuracy; however, all problems will be reviewed for completeness. Half of your homework grade each week is effort-based (put simply, did you genuinely try each problem?), and half is based on the correctness of the answers to the problems I select. I will also drop your lowest homework assignment grade at the end of the semester.

In addition, if you do poorly on a homework question, you will have a one-time opportunity to re-write the proof or answer for up to full credit on (at most) one problem per week. These proof corrections are due a week after the assignment is returned to you, and no late proof corrections are accepted.
Reading SummariesThe reading summaries will be graded on participation and completeness.ExamsUnlike the homework, the exams will be graded completely on the soundness and correctness of your proofs. For the mid-term exam, you will be able to submit a correction for up to full credit for (at most) one exam question response. The correction is due one week after you receive your mid-term exam back. No late mid-term corrections will be accepted.

Course Policies

Late Homework

All homework is due as assigned. A late assignment provided no more than one week late will be given half the credit it otherwise would have received. No credit will be given after one week. Exceptions to the late policy will be made on a case-by-case basis. The student must contact the instructor in advance of the original due date in such cases.

The Use of Large Language Models (e.g., "Artificial Intelligence" models like Chat-GPT, Copilot, Gemini, Claude)

This course's main objective is to improve your mathematical thinking, and the more thinking you do on your own, the more you will get out of the course. That being said, I permit you to use large language models (LLMs) as guides/tutors on homework and for general learning (asking questions like "please explain the concept of ..."). Used in this way, I see LLMs as a powerful companion in helping you refine your thinking. However, as stated above, the written work you submit on homework must be in your own words. You are also, as stated above, permitted and encouraged to work with the other humans in the course on your homework. Any LLM use or collaboration with other humans must be noted on your submission, as a matter of transparency.

By contrast, the use of LLMs on exam questions (on the mid-term and the final) is forbidden; any evidence you have used them in producing your answers on exams will be treated as a violation of the academic misconduct policies linked below.

Academic Policies

Deadlines for Adding, Dropping and Withdrawing from Courses

Students may add a course up to one week after the start of the term for that particular course. Students may drop courses according to the drop deadlines outlined in the EP academic calendar (https://ep.jhu.edu/student-services/academic-calendar/). Between the 6th week of the class and prior to the final withdrawal deadline, a student may withdraw from a course with a W on their academic record. A record of the course will remain on the academic record with a W appearing in the grade column to indicate that the student registered and withdrew from the course.

Academic Misconduct Policy

All students are required to read, know, and comply with the Johns Hopkins University Krieger School of Arts and Sciences (KSAS) / Whiting School of Engineering (WSE) Procedures for Handling Allegations of Misconduct by Full-Time and Part-Time Graduate Students. This policy prohibits academic misconduct, including but not limited to the following: cheating or facilitating cheating; plagiarism; reuse of assignments; unauthorized collaboration; alteration of graded assignments; and unfair competition. Course materials (old assignments, texts, or examinations, etc.) should not be shared unless authorized by the course instructor. Any questions related to this policy should be directed to EP’s academic integrity officer at ep-academic-integrity@jhu.edu.

Students with Disabilities - Accommodations and Accessibility

Johns Hopkins University is committed to providing welcoming, equitable, and accessible educational experiences for all students. If disability accommodations are needed for this course, students should request accommodations through Student Disability Services (SDS) as early as possible to provide time for effective communication and arrangements.  For further information about this process, please refer to the SDS Website.

Student Conduct Code

The fundamental purpose of the JHU regulation of student conduct is to promote and to protect the health, safety, welfare, property, and rights of all members of the University community as well as to promote the orderly operation of the University and to safeguard its property and facilities. As members of the University community, students accept certain responsibilities which support the educational mission and create an environment in which all students are afforded the same opportunity to succeed academically.  For a full description of the code please visit the following website: https://studentaffairs.jhu.edu/policies-guidelines/student-code/

Classroom Climate

JHU is committed to creating a classroom environment that values the diversity of experiences and perspectives that all students bring. Everyone has the right to be treated with dignity and respect. Fostering an inclusive climate is important. Research and experience show that students who interact with peers who are different from themselves learn new things and experience tangible educational outcomes. At no time in this learning process should someone be singled out or treated unequally on the basis of any seen or unseen part of their identity.    If you have concerns in this course about harassment, discrimination, or any unequal treatment, or if you seek accommodations or resources, please reach out to the course instructor directly. Reporting will never impact your course grade. You may also share concerns with your program chair, the Assistant Dean for Diversity and Inclusion, or the Office of Institutional Equity. In handling reports, people will protect your privacy as much as possible, but faculty and staff are required to officially report information for some cases (e.g. sexual harassment).

Course Auditing

When a student enrolls in an EP course with “audit” status, the student must reach an understanding with the instructor as to what is required to earn the “audit.” If the student does not meet those expectations, the instructor must notify the EP Registration Team [EP-Registration@exchange.johnshopkins.edu] in order for the student to be retroactively dropped or withdrawn from the course (depending on when the "audit" was requested and in accordance with EP registration deadlines). All lecture content will remain accessible to auditing students, but access to all other course material is left to the discretion of the instructor.