By Shotaro Beppu

First and foremost, I thank FUTI inc. to support my short study abroad program without which I could not have done.

I attended one class (4 units worth) during UC Berkeley summer session C (6/21-8/13). MATH 104, Introduction to Analysis, is an upper-division mathematics class where Berkeley students are exposed to a rigorous proof-based class for the first time^{1}. The topics covered correspond to this wiki page (Real Analysis). The class was online due to the pandemic. It was an ideal opportunity for me, a former humanities student majoring in engineering, to obtain rigorous training to prepare for advanced mathematics such as Lebesgue measures and complex analysis. It was also a chance to strengthen my graduate application to the U.S. in related fields such as economics and statistics as taking this class and obtaining an A is a good signal that one is adept at mathematics.

The class was quite intensive. This is partly because the class covers the same material in the course of 1 and a half months that would be done in a usual semester. Thus, the timeline looks like below for each week (for me in Japan). I may have felt the class being more intensive since the class schedule directly intersected with UTokyo’s spring semester, but this is one of the main advantages of online classes and I am thankful for that.

Monday

7-10 hours depending on the difficulty, solving the rest of my homework each week due Tuesday 16.00 in Japan. The homework was quite hard (maybe because I did not form study groups but judging from questions at the office hours people more or less struggled as I have) since one has to complete a proof which I didn’t have experience with and some of the questions were quite conceptual and hard to grasp in a short period (such as compactness for me).

- Tuesday

1 hour of Office Hours (2.00-3.00 in the morning…). This was mainly about the homework.

2-4 hours solving the rest of my homework each week. 1 hour of lecture (Recordings) - Wednesday

1 hour of Office hours (which I did not attend) 1 hour of Discussion (6.00-7.00 in the morning) 1 hour of lecture (Recordings) - Thursday

1 hour of Discussion (6.00-7.00 in the morning) 1 hour of lecture (Recordings) - Friday

1 hour of office hours (which I occasionally attend) 1 hour of Discussion (6.00-7.00 in the morning) 1 hour of lecture (Recordings) - Sat/Sun

Catching up with whatever materials for classes at UTokyo

Preparing for mid-term (at the beginning of week 4) or the final (at the end of week 8)

As you may have already noticed taking this class was different from a typical UTokyo class. One that I was hugely impressed with was the amount of time spent on exercises. The discussions which took place three times a week were where we gathered as a group and solve questions related to the topics covered. At the end of each section, the lecturer (who was by the way fantastic and very helpful to someone like me) and we went through the answers together. For those of you who plan to take the class really should attend this as much as possible (although there were recordings of the materials from last week). I have two reasons for this; one is that it forces you to solve problems which are sometimes hard if you are busy, and you can ask questions directly to the lecturer at that time. The second is that the materials covered are most useful for preparing for exams. I remember that one of the problems worth 20% of the final exam (so 10% in the total grade) could only be solved by what we learnt in the discussion session (if I am not mistaken. At least the main insight was not in the lecture nor class handouts). I would also strongly recommend going to the office hours which you can do nothing even if you are there but ask questions any time. It is also good because you could listen to what others are asking. This was an ideal time for asking questions about homework and other questions I solved. With these two supplementary classes, discussion and office hours, I think one learns a lot and naturally achieve good grades. One thing I was surprised by was that the time for these is not pre-specified and staying in Japan, some of it was impossible for me to attend (office hours at 3 in the morning). So, I suggest you contact the lecturer beforehand to check for the time because I could have easily gotten a horrible time schedule in Japan (I was fortunate).

Another notable point was the amount of feedback you get. For every homework and both mid-term and the final, you get feedback accompanied by grades. One can request a regrade which I benefitted once hugely. The feedback itself is crucial for classes like this because one can improve on constructing a proof the next time. These feedbacks are often hard to get in UTokyo classes because the only assignments are the final exam and those grades come out a month later with no feedback. There isn’t much feedback on weekly assignments either.

The class also gave me a place to experiment with how I should study mathematics in the future. Before, I struggled with these mathematics classes since I did not know how I could utilise proofs presented during lectures to solve problems or come up with proofs on my own. Now, I realise proofs in class are a great way to further your understanding and for solving other questions on your own, provided that you study in a way that fits you. For me, writing down the theorems on paper and drawing the relationships of the theorems (there were a lot I can tell you that) encountered was a good way to grasp the material. For most of the important theorems, I remembered the proof or the outline of proof at least and this was itself sometimes sufficient to solving a homework problem which everyone seemed to not get judging from office hours (there is sampling bias of course). The frequent and hard homework and discussion/office hours certainly helped in forming my study pattern. Advice

from lecturers was also spot-on. Figuring out how to study is probably why I got a much better grade in the final than in the mid-term.

All in all, it was intense, and I really enjoyed it. I noticed that there are not many intense classes such as this in summer classes in other universities and this class is highly recommended for those who want to do rigorous mathematics for their first time in a good environment.

1 You can see the Redditt discussions which gives some idea how people struggle with this class. Just type in MATH 104 berkeley