MATH 2315

This is an extremely difficult course. The description on the syllabus that says this class is a guided independent study is accurate. We completed the entire syllabus of 2310, regular calculus 3, in 3 weeks. The assigned reading is mandatory to understand the material as prof Koberda teaches the proofs of calculus 3 in class and less actual calculus 3. Prof Koberda and the small class size make it a great way to learn advanced math with no chance of failing as long as you work hard. However, this class is a huge amount of work, especially if you haven’t taken calculus 3 before. The class starts with set theory and more linear algebra concepts and finishes with the 2310 material. If you can’t attend office hours, I would not recommend taking this class because they really are necessary to understand the material. If you can’t learn by a textbook, I would not recommend taking this class. For me, it wasn’t the right thing, but prof Koberda is an amazing professor and I really enjoyed the opportunity to push myself to the limit. If you love math and can learn well on your own, you will love this class.

I think that the best way to describe this class is ridiculously challenging but 100% worth it. First of all, I read all of the previous Course Forum Reviews multiple times before taking this class, and I found them all to be extremely helpful and accurate, so I would recommend doing that if you haven’t already. I went into this class expecting it to be difficult, but honestly nothing can really prepare you for how fast-paced the class will move. The first semester covers all of Calc 3 and set theory, some Linear Algebra, and a lot of Basic Real Analysis. There are weekly problem sets (I think that we had ten in total) and a take-home midterm and final. I read and worked through How to Prove It by Daniel Velleman the summer before taking this class, and it definitely helped me feel more confident the first two weeks, but as I stated earlier, the class moves so quickly that we covered the entirety of set theory in about two weeks. The first couple weeks of the class can seem really scary (I think we went from about 40 students to 20 in the first week or two) but there are a few things that you need to know going into it. First of all, most everyone in the class will be just as lost as you are. It’s new material that is really hard, but over time you will adjust to the difficulty and start to have a better idea of how to tackle problems on the problem set. Second of all, Professor Koberda can seem really scary but he is honestly the sweetest and nicest person. Go to his office hours and talk to him, he’s always willing to help and talk about anything you want, even if it’s not related to math. And on that note, please do not worry about your grade. Professor Koberda will assure you that if you take care of the learning, he will take care of the grading, and he means it. Lastly, know that this class will be so much fun. I became friends with just about everyone in the class, and it’s really important that you work together. We would have weekly math nights where we met and stayed up ridiculously late working through the problem sets together, and we would often go get dinner together after Monday night discussions. Trust Professor Koberda, go to both his and your TA’s office hours, and become close with your classmates. If you do these things and put in as much effort as you can, you will be completely fine and learn so much math. I don’t think that I’ve ever learned so much in such a short period of time, and as a first-year, I think that taking this class my first semester at UVA was one of the best choices I could have made. Please take it, and please stick with it, and I promise you that it will be worth it.

The first time I looked up a photo of him before joining the course, I thought he was just a geeky, plain math professor, and I could already imagine how little I'd get from an uninteresting lecturer. A few days later, I did see some of his lectures on YouTube for grad students, and I thought he'll just be the decent, bland professor ready to make me sleep in lectures for a whole year. Never have I been so wrong. Never have I met anyone who entirely changed my mind and impression in two weeks without even trying. Never have I met anyone who actually, in his mysterious ways, taught me how to learn, not only in math, but in life. Incredible man.

This class and the follow-up (3315) are by far and away the most challenging classes that I have ever taken. But they are also the most rewarding classes that I have ever taken—I honestly got more out of these two classes than I could have possibly imagined. In essence, the sequence is a intense introduction to higher mathematics. It covers much of the material from Transition to Higher Mathematics (3000) in the first few weeks, and then moves onto a mixture of material from Elementary Linear Algebra (3351), Basic Real Analysis (3310), and Calculus III (2310). By the end of the class, you will have encountered quite a few topics from Elementary Linear Algebra and Basic Real Analysis, and you will have finished Calculus III. In combination with the follow-up to this class, you will finished Elementary Linear Algebra and Ordinary Differential Equations (3255), and believe it or not, most of Advanced Linear Algebra (4651).

This may sound ridiculous, but trust me, it is immensely helpful. The students who came out of this class and entered Basic Real Analysis repeatedly confirmed that the course was significantly easier as a result; indeed, some claim to have already encountered a good portion of the material from the first half of the class. Additionally, this sequence covers an incredible amount of linear algebra—so much so that both of the linear algebra courses in the undergraduate department will be a breeze.

Of course, the class is by no means easy. It requires a ton of dedication, motivation, and focus, and you will absolutely need to work with your classmates and attend office hours (ideally both Koberda's and your TA's). My class would attend every office hours that we could, and then have a weekly "math night" the day before the problem sets were due, often staying up well past midnight. Again, this might sound painful, but most of us found it to be rewarding and enjoyable, and all of us were extremely glad that we pushed through.

Every week, Koberda assigns a new problem set, most of which will go way over your head for the first four days of the assignment, but by halfway through the week, you will miraculously find a way to complete it. This process will, of course, require copious amounts of help from a TA (hopefully you have Alec Traaseth, a gift from God himself), and possibly assistance from random lecture notes, textbooks, and Math Stack Exchange. Every once and a while you will receive a take-home exam, which you will have a week to complete, and the experience should be similar to that of the problem sets (although opening the exam is always an unwanted adventure). There is one thing to keep in mind with all of this: the professor will absolutely understand your struggle, and will reward you for your hard work and dedication, even if you do not do well on the problem sets or exams. That is not to say that you will necessarily get an A in the course, but if you put in all of the effort that this course requires, I expect that you will end with a B at the lowest.

Luckily, this course forces you to make incredible friends, and it teaches you so much about mathematics that you come out with an utterly new outlook on things. The professor may seem scary, but he is actually kind, hard-working, inspiring, and humble, and he has your best interests at heart. Please get to know him. I cannot convince you to take this course, but I can warn you what the experience will be like in the hopes that you enter it prepared. Still, if you are hard-working, ambitious, dedicated, and you really want to learn mathematics, there is no better course. Have at it.

*****SUMMARY*****This course requires immense amounts of effort and quickly makes you feel completely useless, but it is so inspiring, rewarding, and unique that you will 100% be glad that you took it. You cover an insane volume of material from the undergraduate department, and my guess is that you will not encounter another class this difficult for some time (or for all of your time at UVA), but that means it's all uphill from here. If you stick with it, though, the professor will reward you for your efforts and you will pass with a reasonably good grade. You also make a lot of close friends in the process. If this somehow sounds fun to you, then you are perfect for this course; don't think about it anymore, just take it. Even if you have just left BC Calculus from high school, you can make it through the class—plenty of people did. If it sounds fun but you are worried, you can always get these books: "How to Prove It," by Daniel Velleman; and "Linear Algebra Done Right," by Sheldon Axler (actually, pick this up regardless because it is great). Honestly, just take it. I have zero regrets.

This class (and the continuation into Math 3315 the following semester) is easily the best class I have ever taken. It is also the hardest class I have ever taken. It's essentially a year long course, the first semester is 2315 and you get credit for Calc III and the second semester you get credit for ODE and Linear Algebra. But you learn way more than that. 2315 starts out with you learning set theory for the first time, and chances are, you'll have no idea what's going on. That's okay. Then you move into learning some intro to Linear Algebra, and about halfway through you move on to Multivariable Calculus. It's a very proof-heavy course, and most people who take it have no experience with writing proofs. That's okay. Professor Koberda really just wants you to learn math for the sake of learning math. He won't fail you, and as long as you try you will not get below a B- in the course. But don't take that for granted. The problems sets get very difficult, and the lectures become impossible to follow if you let yourself fall behind. Use your TA's office hours as much as you possibly can, they help a lot. Also, do outside research on the topics you learn in class. You'll find yourself going down rabbit holes of Wikipedia articles, textbook pdfs that you downloaded from Springer, and anything else that you can find to learn as much math as possible. Looking back on it, I learned an insane amount of material in a very short amount of time, and things I couldn't even begin to understand a year ago I now know like the back of my hand. If you're willing to put in the work and want to get an intense introduction to higher math with a safety net for your GPA, this is the perfect class for you. I recommend going to as many office hours as you possibly can. Definitely go to Professor Koberda's office hours too. He's honestly the nicest person I've ever met. Also, work in groups with your classmates, you'll become pretty close if you do, we're kind of a cult. If you're smart, you'll work together before the night before an assignment is due, but you probably won't. Between office hours, working with classmates, researching on your own, and reading by yourself, this class is a lot of work, and you get out of it whatever you put into it. It moves at at least 2x the speed of a normal class, so you need to hold yourself accountable and not fall behind. But it's 100% worth it. This class is the reason why I decided to be a math major, and I hope that everybody who takes it enjoys it as much as I did.

This man is incredible. He is literally the reason I want to spend the next 6 years of my life studying nothing but mathematics. Yes this class is a 5 in difficulty. It is also undoubtedly the best class I've ever taken. I wouldn't worry about the grade, just try to learn and understand as much material as possible. Go to office hours, work together on problem sets. I think I learned the entirety of Basic Real Analysis in the first 2 weeks of this course, and I don't feel like I missed a thing. Have I mentioned this professor is a wizard? I genuinely think he may have magical powers. I would take another course with him even if it meant I failed the class. He is just that good.

Don't be scared away by the first few classes or problem sets! Koberda is the GOAT, you will learn a non-negligible amount about almost every field of undergraduate math from this course; it literally turned me into a math major. Work with your classmates and utilize your TA heavily and it'll be the best sequence of courses you'll take (with MATH 3315). Also, don't worry about your grade, just work hard on the problem sets and improve over the course of the semester and you'll be totally fine.