Fall 2025

Foreword

“If someone calls formula sheets too much, if they call clarity and care “unprofessional,” then what does that say about what they think education is for? Because if education isn’t for our students, then who is it for?”

— Ivan Z. Feng, “Formula Sheets, We Miss You”

Math 540a: Algebraic Topology — Prof. Aravind Asok
Math 555a: Partial Differential Equations — Prof. Mohammed Ziane

Page Overview: I’ve provided all the information you might need about our discussions on this page, aiming to save you the time of asking questions I should have made clear to you. (But that doesn’t mean you can’t ask me things that can be found on that page; you are still very welcome to ask me anytime!)

Discussion Notes:

For Algebraic Topology, there are no homework submissions, and quiz solutions appear in the Gradescope rubrics. In addition, links to past qualifying exams (with many solutions) are provided below.

Ivan’s Class Emails for MATH 540a

Math 540a: Topology Notes Supported on Qual Solutions (Ivan)

Math 540a: Past Qualifying Exams and Exam Syllabus

Math 540a: Solutions to Past Qualifying Exams

I also want to recommend an excellent free YouTube topology lecture series by Dr. Anthony Bosman to complement your understanding (link). It covers at least 2/3 of the course material for Math 540a and the topology qualifying exam syllabus. The lectures are pretty funny and very easy to understand, since they start with concrete examples before building intuition all the way up (an approach that’s essential but often overlooked in college-level math teaching). Of course, there’s no pressure to watch, but I got a high PhD pass on the Topology qual, and I give most of the credit to this lecture series.

For PDE, I will post detailed, explicit solutions to each HW assignment in the folder below after grading. They include notes on common error-prone spots I noticed while grading, as well as some new methods that came up from your work. Although optional, I encourage you to take a look when you have a chance. I usually give a couple of different methods for each problem (collected from your submissions), written in a very easy-to-follow way.

Math 555a: Ivan’s Notes and HW solutions for Math 555a

Math 555a: Past Qualifying Exams and Exam Syllabus

Math 555a: Solutions to Past Qualifying Exams

Email Responding: Whenever you have any math questions or comments, email me anytime at ifeng@usc.edu. For all emails received by 10 pm each day, I’ll be sure to respond within the same day (usually pretty soon, ASAP). For all emails received between 10 pm and 11:59 pm each day, I’ll be sure to respond by the next noon.

Ivan’s Office Hours*:

Monday: 10-11 am
Tuesday: 2-3 pm
Thursday: 11am-12pm

*You are always welcome to attend my office hours. This is my designated time to assist students, so please don’t feel hesitant—it’s what I’m here for. I’ll be waiting for you in the math center. If my hours don’t fit your schedule, feel free to schedule an appointment with me outside of my regular office hours. We can either do it in the Math Center physically or in my personal zoom room online. Send me an email so we can set up a time, and then I’ll send you my zoom link. You can find my weekly schedule here to check my availability.

Office-Hour Locations:

Math Center [KAP 263] and Zoom*

*If you want to join my office hours by Zoom, click here. The Math Center Assistant on duty will assign you to my breakout room.

Note I: As a rule of the Math Center, TAs are there for *all* students (not just their own students) when holding office hours. So you can go there anytime to do homework or get help from any TAs. And I will surely be there at least during my office hours. You can check out here the Math Center Schedules including all TAs’ office hours this semester.

Note II: The Math Center is run on a drop-in basis without appointments. Thus, if many students show up during my office hours, I’ll try to let you ask one question at a time in order of arrival (unless you have to leave soon) and then continue the cycle, to ensure everyone there has a chance to discuss questions with me instead of waiting for nothing. Feel free to give me any feedback!

Homework/Quiz Grading: If you click into your assignment page (on either Brightspace or Gradescope), you’ll find my comments in the feedback window. In most cases, unless a critical math error occurred, I don’t take off any points for minor mistakes or missing steps; I just noted them in the feedback and comments in your uploaded file. So even if you earned a perfect score, I still recommend entering the assignment to check my comments.

Asking Questions: Feel free to stop by my office hours with any questions! A small note: If you’d like to discuss problems outside the homework or text, I’d appreciate it if you could email me the problem in advance. This way I can prepare and give you the best answer during your visit!

As your TA, I’m your assistant and guide in this course, not a judge. I’ll do my best to help, not to evaluate your questions. To me, there are no “silly” questions; every question is a great question. If my explanations aren’t clear during office hours or by email, that’s on me, not you!

Remember, Isaac Newton asked himself a super silly and childish question in the summer of 1666: Why would an apple fall down instead of falling upwards or sideways? But that childish question finally led him to the discovery of universal gravitation. Thus, you are always welcome to ask everything in your mind, no matter whether the answer can be found on the board or in previous lecture notes. And I’d be very happy to answer anything you ask.

Or if you prefer to work independently and rely on yourself, it’s also great. There’s no single way to find answers to your questions. But, I’m asking you to preserve your curiosity well and never let it go.

Recommended Textbooks: 

Math 540a: Algebraic Topology by Allen Hatcher

Math 540a: An Introduction to Algebraic Topology by Joseph J. Rotman

Math 555a: Partial Differential Equations (Second Edition) by Lawrence C. Evans
(This link is pretty slow, but, just like this course, eventually you’ll get there.)