UW-Madison Course
There are a lot of fun learning at UW-Madison. I will sort them based on the subjects.
1. Math
One of the surpising thing is that UW-Madison has a very strong math department. I personally like the probability theory and stochastic process the most (Section 1.2 and Section 1.3). They are all taught by Prof. Erik Bates (https://www.ewbates.com/). Unfortunately, he is no longer in UW-Madison, but in NCSU.
The hardest math course I took is the Modern Algebra II (Section 1.6), taught by Prof. Paul Apisa (https://people.math.wisc.edu/~apisa/).
I did take one of the calculus course (Multivariable Calculus, Math 234), but I did not find it very interesting and worth a section. One of my friends took the honors version (Math 375 + 376) which combines calculus and linear algebra and he said it is much more interesting.
1.1. Linear Algebra (341)
This is the first math course that I found very interesting in UW-Madison. It is taught by Prof. Sigurd Angenent (https://people.math.wisc.edu/~angenent/). The textbook we used is Linear Algebra Done Wrong by Sergei Treil. Prof. Angenent is a very profound math professor. He knows many things about the math history and explains the concept in a very clear way (yet a bit slow, but I won’t complain as I suffers so much in Math 542 which is taught really fast). I find the most rewarding part of the course is that it combines the abstract math theory with concrete usage.
The first part that is very interesting is that it views matrix as the representation of linear transformation, which explains why matrix multiplication is defined the way it is.
Another part I like very much is the spectrum theorem and the eigenvalue decomposition, and how it can be used to compute matrix power.
1.2. Probability Theory (431)
This is by far my favorite math course in UW-Madison. It is taught by Prof. Erik Bates (https://www.ewbates.com/). We do have a textbook, Introduction to Probability, which is also written by professors in our school. However, Erik didn’t really follow the textbook and instead drives the course in a different order and have his own lecture notes (which is hand written in a very interesting font).
It is a fortune for me to have a chance to take the course with Erik. I wasn’t plan to take this course at the first place, as I was afriad whether the courses will be too hard for me as I am having quite a few workload during that semester including Philosophy (Section 3.3), which I believes I need to spend a lot of time on. However, near the start of the semester, I could only waitlist in the probability theory in the statistics department (Math 309), which I audited and thought it was a super easy course. Luckily, the section Erik is teaching has one last spot. At the sudden I saw that he has a 4.9/5 score on RateMyProfessor, I thought it worth the risk to take it. It is one of my best decision in UW-Madison.
Erik is a very nice and interesting guy. He looks very young. At the first time I went to class, I thought I went to the wrong classroom, because the one who stands in front of the blackboard looks like a TA (especially with the red UW-Madison mask). He talks very fast at first, but the lecture is very engaging and I never imagine 50 minutes courses feel so short. He also asks us to write a feedback every two weeks, and read everyone’s feedback (even write comments on it) and literally follow some of the feedback to improve the course. For example, he significantly reduces the speed he talks.
One of the most insightful thing I learnt is what Erik is saying at the first lecture. He throws a die to the ground and told us there is no such things as random. Physics determines what the results of the die will be. If we tracks every motion of the die, then we can know the result of the die. However, why do people still say it is “random”? It is because it is very hard to track every motion of the die, but still want to have a sense about what the die will be. Probability is a theory on top of uncertainty.
One thing Erik is strving us to do is to do sanity check. He wants us to have a “sense” about whether a result is making sense and being intuitive. I think this is something everyone should do when learning math, but nobody has emphasized it before Erik. Another outstanding thing that Erik is doing is that he always motivates us about a theory before introducing complicate concepts. One very interesting example is that he first gives a very complex “proof” of the Central Limit Theorem about Sum of binary random variables, and then introduces the generating function and moment generating function to provide a simpler but more general proof.