MATH 4310

Dr. Ultrasev set an extraordinarily ambitious curriculum for this class. From talking to older students, we went way farther than other Intro Real Analysis classes did in the past. It was super intense. Dr. Ultrasev was an effective teacher, and we learned a lot, but DO NOT enroll in this class unless you have a lot of spare time for the semester. It will take up a lot more of your time than other 4000 level math classes.

This is not a class to take lightly, it will crush you. Gennady lectures well, and he clearly puts tremendous effort into helping you learn this challenging material, but he expects back as much as he gives. You go through allot of material in ALLOT of depth, and the hw workload is likely to be higher than for any other undergrad math class you will take at UVA, it was for me.
You will have a good grasp of undergrad real analysis after completing this class... but at what cost.

I really want to give this professor the benefit of the doubt, especially since it was his first semester teaching at UVA, that too with a fully online course. However, there are several factors that simply discourage me from doing so entirely.

I'll start with the positives. He is pretty lenient with the grading. If you get above an 80% on your final grade you fall somewhere in the A- to A+ range, varying semester to semester. He is active on piazza and is always willing to help you if you have any questions, provided that you have put in some thought into the problem yourself, which is very fair.

However, this class was brutally hard, and the online setting did not help. I don't think it was just the subject matter either. While there are certainly difficult topics to comprehend in Real Analysis, the instruction was incredibly lackluster. Lectures were very disorganized. He would routinely make mistakes in his proofs, second guess himself, and redo parts of his theorems many times over. It was clear he did not come to lectures with a pre-prepared set of notes, and that detracted from our ability to learn as students and follow his work properly. He is very inconsistent in his notation and techniques, and often skips many steps without explaining himself. A lot of his lecture notes were filled with mistakes and algebraic computations that were relatively unnecessary to the subject at large. This lead to a lot of confusing and honestly unengaging lectures.

Onto the homeworks. These were miserable, to put it lightly. They were often riddled with significant typos, and most problems were confusingly notated and very lengthy (for the sake of being lengthy). I felt like half the homework was doing algebraic manipulation instead of actually applying the theorems we covered in lecture. A plurality (if not majority) of the class would routinely spend 15-20 hours on homeworks every week, and still be very unconfident in their answers due to the confusing nature/notation of the problems. There is a certain satisfaction you get when you solve a hw set in a math class, and simply put this class never delivered that feeling. The creme de la creme of this whole experience was the incomprehensible homework answers he posted afterwards. Granted, he did post them, and some were very clever and helpful, but the majority were very disorganized and poorly notated.

We asked him multiple times to be more lenient with the amount/type of homework he assigned, and asked him to be slightly more organized in his assignments and lectures, and he straight up refused for the former. I understand if he felt offended by us criticizing his work, but the students were incredibly confused and discouraged from the homework he assigned. It really feels hard to appreciate the positives of professor Uraltsev when our principal concerns were largely scoffed at.

Again, he is a really cooperative and nice man, and I know from experience that he will help you if you are stuck and reach out. However, this class was pure pain from start to finish. I don't think I learned more in this class than I would have from an easier, yet more straightforward class--the rigor wasn't worth it in the end.

To put my review in context, I actually enjoy analysis a lot, but this class was hellish. I also ended up doing pretty well in this class, so the following review isn't me just being salty from getting a poor grade or something.

First of all, the lectures were poorly organized. It seemed like he was never prepared and would always start proofs with mistakes and then fix them as he was writing the proof. This made for extremely disorganized notes when the structure of the lectures were very disorganized to begin with. He went way too fast and wouldn't write everything down. This meant that I could barely take notes and the notes I did have were missing many important points. He would skip major steps in his algebra and then also copy and paste terms/expressions from step to step, which I obviously couldn't keep up with in my notes. Also the disorganized nature made it impossible to look back upon. I honestly got lost in almost every single lecture for the majority of the lecture. There were maybe 4 lectures where I wasn't completely lost after some point. To be fair, his office hours were actually pretty helpful, but that made it more frustrating when he would just spit out nonsense in lecture because that meant he could be a decent lecturer if he actually put any effort in organization beforehand.

I would've maybe been able to rewatch lectures or go to office hours/recitation more to remedy this if the homework in this class didn't take longer than all my other homework combined. I seriously spent a minimum of 14-15 hours per homework assignment (Basic Real and Survey of Algebra both took me around 5-7 hours on average). We ran numerous polls on how long homework takes, and it seems like around 70-80% of the class takes just as long or longer than I do. We complained numerous times as a class about how unreasonable this workload was, but he ignored all of our complaints. Many students in this class are 4th years and have taken equivalent level math/physics courses or even grad level courses. They've all said that none of those classes have come close to the workload in this class. It is actually ridiculous. There are always a few problems that are too hard to do without going to office hours, which adds another 1-2 hours to the 15 hours already spent. There were also many problems where I actually could figure out the analysis portion in like 10-15 minutes but then had to spend upwards of 2 hours trying to figure out the tedious details/algebra specific to the problem. Maybe the homework wouldn't be so bad if his lectures were organized and I actually understood the material/could reference notes from class. But as mentioned above, this is pretty difficult from his lecture style.

To finish off the semester, he gave us one more 15-hour homework assigned over finals week. We also had an oral exam where for part of it, we were given 15 min to do 2 HW problems we had done before that had solutions posted (around 40-50 long problems), where some of the problems literally took 2-3 pages to write out. He also failed to mention until the day some people already took their exams that he could also ask us optional homework problems and midterm problems.

TLDR: This class was brutal. Never had a class that physically made me curl up in a ball so many times.