# Course Review: MATH 418

#### Probability

“I heard you had a lot of difficulty with the last homework. So you will be relieved to note that next homework you will have another opportunity to practice similar problems.”

Text:  A First Look at Rigorous Probability Theory, 2nd ed by J.S. Rosenthal

Prof: Dr Gordon Slade

Prof Slade is very clear and keeps things simple. He also has no problem slowing down to answer questions from students if they are not following along. He has a dry sense of humor, that keeps the class interesting, and he can even by funny when he is talking in earnest.

Difficulty

After taking measure theory, several sections of the course can feel like review. This was a good thing for me, as I found MATH 420 a tad fast. There are a handful of new techniques that you learn in the homework and class, but there are not that many new concepts if you have some background in measure theory and probability. MATH 421 material also comes up in terms of weak convergence.

Key Concepts

Probability Triples

Random Variables

Distributions

Expectation

Borel-Cantelli

Modes of Convergence

Law of Large Numbers

Central Limit Theorem

Characteristic Functions

Hard Concepts

Tail Events: Kind of funny to think about. Also, include definitions of limit supremum and limit infimum for sequences of event which can be difficult to convert to statements about limits of random variables.

Weak Convergence: There are a lot of equivalent statements, and if you pick the wrong one it can be a mission to prove that convergence occurs.

Conclusion

Good review of measure theory, and gives you a mathematical foundation to elementary probability.

# Course Review: MATH 421

Real Analysis II

“If everyone is getting closer together, and you are going to Cleveland, then everyone’s going to Cleveland!”

Text: Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland

Prof: Dr Brian Marcus

Prof Marcus is one of the kindest professors I know. He also explains each concept from multiple perspectives and breaks the concepts down to simple intuition. His notes are super useful and he has a great sense of humor.

Difficulty

The classes moves somewhat slower then 420, so we went a lot more deeper into the details of certain proofs and where they come from. This meant the classes were easier to follow (and in my view more enjoyable!). Further the homework was largely straightforward. The one challenge in this course was the material towards the latter half of the term was slightly more complex and were not assessed on homework. We also may not have covered the material generally covered in a similar course.

Key Concepts

Banach spaces

Hilbert spaces

Linear functionals

Hahn Banach Theorem

Open mapping theorem

Uniform boundedness

Riesz representation theorems

Hard Concepts

Weak Convergence: Can be hard to think about neighborhoods in this topology.

Hilbert Spaces: There are a lot of nifty tricks in Hilbert spaces that are not immediately obvious, and do not work in other spaces.

Conclusion

Really fun course and learnt a lot. Nice conclusion to my studies of analysis in some sense.

# Course Review: MATH 322

#### Introduction to Group Theory

“We say groups have a rich structure because it keeps us mathematicians in business”

Text: Basic Algebra I: Second Edition

Prof:  Dr Vinayak Vatsal

Nike is a humorous professor who cares about his students. The homework he set was also useful in getting a mastery of the material in the course. Unfortunately, the textbook was made up of large, listless walls of text that prove half a dozen theses arbitrarily before deciding to declare that a theorem worthy of a name has been proven. Thus it cannot be readily used as a reference book and is difficult to read. The classes were significantly better, but from time to time also suffered from a lack of direction and interest. Group theory, on the other hand, is a fascinating field and the proofs often use a lot of creativity.

Difficulty

I found the material in this course difficult, but since it was an introductory class, I felt I had enough time to master the relevant concepts. Unfortunately, the relatively few iClickers were weighted 10% of the grade and the exams were not difficult but punished small mistakes a lot. Also, the final exam used a peculiar notation and seemed almost entirely about symmetric groups and had no questions about abstract group theory. As a result, the average was quite low in this course.

Key Concepts

Groups

Subgroups

Action on groups

Orbit-Stabilizer

Class equation

Finite Abelian groups

Isomorphism Theorems

Homomorphism

Free group and generators

Hard Concepts

Action on groups: Useful to get some practice with a couple of these actions before one can start seeing the action that occurs in problems

Proving isomorphisms/finding a bijection: Often one can use the obvious or ‘canonical’ map, other times you need to do some clever

Counting elements of a certain order in abelian groups: Can make careless mistakes, double check your answer!

Conclusion

An interesting topic, the homework problems were fun, wished the textbook and exams were structured differently.

# Course Review: MATH 227

“Consider an infinitesimal paddle wheel…”

Text: (none)

Prof: Dr. Joel Feldman

Dr. Joel Feldman strikes a great balance between being really organized, while still pretty relaxed. He is very helpful, in that he is always available between classes for questions. He is also really knowledgeable about the field (as he is a mathematical physicist) which is great, though he did give us a particularly tragic expression when we said we didn’t know what curl was, halfway through the course and he had to explain it.

Difficulty

The weekly assignments are generally all straightforward. The questions vary from computational to small proofs. I missed the first midterm, but a lot of people got wrecked, and it looked tough so watch out. Each question in the second midterm was manageable, though it required thinking. The challenge is that there are only four questions, so the cost of getting one question completely wrong is quite high. The final was similar except it had more questions. The challenge then was that often it was not clear which of the various techniques we had learned in the course was the correct approach to a question.

Key Concepts

Analysing/ parameterizing curves in 2,3-space

Analysing/ parameterizing surfaces in 3-space

Analysing/ parameterizing vector fields in 3-space

Integrating over curves, vector fields, and surfaces

Integral Theorems

Hard Concepts

Applications of integral theorems: Hard to pick the right strategy that is going to work.

Biot-Savart Law: Really abstract, not sure if I understood it.

Conclusion

Feels like a fun physics course. Wish we had more time to discuss differential forms, but other than that pretty interesting class.

# Course Review: MATH 226

“The point of calculus is to understand how and when we can approximate non-linear transformations with linear ones.”

Text: Calculus: Several Variables by Robert Adams

Prof: Dr Julia Gordon

Dr Gordon is a really kind and gentle prof. Her classes focused on explaining the intuitive motivations for various theorems or algorithms. The classes did seem to get bogged down with administrative concerns. Since this was the first time she taught the course, she made a lot small errors, but not more than your average prof. However, she did get rather thrown-off by her errors and this, along with administrative concerns, probably slowed the class to the extent that we had to rush a bit towards the end of term. While her midterms were similar in difficulty to practice midterms, her homework was a great deal easier then what we have come to expect in honours. She generally explained the intuition behind theorems, but omitted proofs because she felt that the definitions we were using were not efficient/well-suited for proving those theorems. She is very helpful during office hours, but I don’t think she approves of too much hand-holding in an honours class, even though she will help you a lot if you ask for it. I also get the sense that her life is very busy!

Some quips:

“Please don’t cheat. In Russia, they never took it very seriously, but for some reason they take it very seriously here.”

“I sense some unrest in the class…”

“I haven’t taught you improper integrals, but you can extrapolate from you understanding of proper integrals.”

“Does everyone believe that this is an honest approximation to the function? Good”

“WAIT!! WAIT!! Before you pack up, let me deliver the punchline!”

“At some point in your mathematical career, you will have to solve a 100 difficult integrals. Here they are.”

“The Fundamental Theorem of Calculus is : ‘Never differentiate in public’. I am now going to ignore that rule.”

Difficulty

I have found this course the easiest honours Math course I have taken at UBC. However, people who are new to honours Math were slightly overwhelmed by a combination of n-dimensional space, epsilon-delta and the scarcity of example-based learning, all of which were prevalent in first year honours calculus. The workload was very manageable, with Webwork and a short written assignment due in two weeks. Scaling was generous, because people did poorly on the midterms. This was partly because of the people that were new to honours Math but also because the homework did not challenge us to demonstrate deep understanding of all the necessary concepts, which comes in handy on honours Math midterms, where conceptual understanding often trumps computation.

Key Concepts

N-dimensional intervals/balls (topology), limits (epsilon-delta), differentiability and continuity

Partial derivative of N-dimensional curve

N-dimensional derivative (Gradient and Jacobian Matrix)

Integration over N-dimensional space

N-dimensional optimization (Lagrange)

Hard Concepts

Implicit Function Theorem: Found explanation in book confusing. Found it easier to think in terms of Jacobian Matrix.

Lagrange: Computationally difficult, have to solve non-linear system of equations.

Topology: Found it difficult to do proofs under time pressure.

Change of variables using Jacobian: Make sure not to get confused between image and source.

Geometry: Have fun visualizing N-dimensional intersections of various curves 🙂

Conclusion

Good class. It was nice to understand calculus from a linear transformation perspective. Felt that my understanding of key theorems of the class is still sketchy though, and that the end of the class was rushed.

# Course Review: MATH 223

Linear Algebra

“This is really useful. You might get a call one day from Downtown, asking you how to take the inverse of a two-by-two matrix.”

Text: Linear Algebra and its Applications by David C. Lay (optional)

Prof: Dr Richard Anstee

Professor Anstee looks kinda like a classic Prof. Fuzzy white hair, check. Kindly yet contemplative expression, check. He is also a natural teacher, having a cavalier attitude yet still able to get the material across. He says the most whimsical things in the middle of lecture, making his classes really entertaining. He spent the first week teaching many of the major concepts intuitively for two-by-two matrices. The following weeks till the second midterm were both applying and more rigorously fleshing out the ideas discussed in the first week, so don’t get psyched by the first week if the material is really new for you. Anstee tends to switch perspectives a lot during lectures, without explicitly informing you. I have a feeling he does this on purpose, to keep us on our toes, because viewing things from multiple perspectives is a core  feature of Linear Algebra. He is also a really kind Prof, holding many-many office hours and always willing to chat to his students. He also brought home-made cookies on the last class :D.

Some quips:

“You never know who you will meet on the bus!”

Difficulty

Since I had learnt this material before, I am not a fair judge. However, for people who have either never taken an honours math class or have no familiarity with matrices this course is probably both challenging and do-able. More then any other course I have taken so far, conceptual understanding is key. In calculus you can memorize a ton of algorithms and you can go pretty far. In linear, the questions are deceptively simple if you can make the connection to theory. On exams, some ten mark questions were literally two or so lines. He also gives you practice exams where the first few questions (these are often the ones that test computation) are exactly the same as the one on the real exam, except with different numbers, so there is really no excuse for losing many marks on the first 40%. Some of the homework questions are significantly more challenging then exam questions, but they are do-able, and he gives you lots of help if you ask him. Conceptually, since linear algebra is about multiple perspectives, to do very well in this course, I would suggest consulting a variety of sources. In terms of marks, after generous scaling, I think that one can get a decent grade in this course, though those who are new to honours math might have to work extra hard.

Key Concepts

Representing linear transformations as matrices

N-dimensional Vector Spaces (generalization of R^n)

Properties of matrices/transformations (characteristic, determinant, trace, rank, eigenvalues)

Types of matrices/transformations (diagonal, diagonalizable, invertible, orthogonal, hermitian)

Hard Concepts

Change of basis: Got really confused by this. What I realized was, depending on the context, one can interpret a matrix as a change-of-basis, or as a linear transformation on a given basis. Another thing to keep in mind is that a change-of-basis matrix has column vectors that form a basis of the domain in the co-ordinates of the image.

Complex conjugate: Didn’t go to deep into complex numbers, but got confused about which theorems apply in the complex/real case respectively.

Entry-wise manipulation of arbitrary matrices: Had difficulty visualizing the a_ij element in a matrix.

Conclusion

Fun, intelligent class. Favourite class this term.

# Course Review: MATH 121

Honours Integral Calculus

“Some of the best ideas in mathematics have been very simple ideas”

Text: Robert A. Adams and Christopher Essex: Calculus: Single Variable, 8th Edition, Pearson, Toronto, 2013

Prof: Dr. Young-Heon Kim

Professor Kim is a boss. He challenges us. He aims to make us understand the mathematical thinking behind a proof. He also promised us that if we worked hard and understood the basics, we should get an A, which was probably realized for most people who worked hard at the end of the term. He answers questions generally patiently and tries to sense if the class is understanding what he is doing, instead of just droning on. Highlights of the class:

On the first class. “I looked at the textbook we are using for this course. It was very disappointing. The questions are too easy.”

“Why are you guys looking so dumb today?”

“Let this by a probability distribution of the midterm. As you can see, it isn’t very high.”

Kim: “Should we do this problem this way or that way? It’s a matter of taste.” Student: “So it doesn’t matter?” Kim: “No. TASTE MATTERS A LOT!”

“Hey! doesn’t this parametric curve look like Picasso?”

Near the end of the course. “I just realized that you guys have been working very hard for this course. Please don’t ignore your other subjects. How come nobody complained? Whenever anyone comes to speak to me, it seems like you are saying “please give me more work”.

Difficulty

This course was significantly harder than Math 120. For example. the average on the first homework was 6/15, but it improved dramatically over the course of the term. Midterms were conceptually challenging but only trivial computations. The prof loved to mix multiple concepts into one question. There was generally one proof on every exam. There were often proofs on the weekly homework, and these were significantly more challenging than the ones in the exam. The questions used to nag me for days. The Final exam was 40% of Math 101 stuff and the average was scaled to 75-78. So if you work throughout the course, and make sure you understand the basics well, you should get an A. 16/29 students achieved an A.

Key Concepts

The (Riemann) Integral

Sequences and Series

The Fundamental Theorem of Calculus

Convergence

Hard Concepts

Riemann Integral: Defining integration in terms of epsilons and partitions. Similar to limit definition, but the hard part is the mathematics, not just the logic.

Probability: Some of the harder questions are difficult to wrap one’s head around, especially on the homework, where we had to find the probability of a series converging. One way is to think of probability as mass.

Functions defined by series: Last question on final was a function defined by a series of functions defined by a sequence. Can get pretty meta. One way is to look at how output function is changing with a small change in x, to understand.

Polar Co-ordinates/Parametric: Drawing, finding derivatives etc can get pretty computationally challenging and technical. But there is a step-by-step process one can follow. Also, to find parametric function it is a good idea to divide motion into motion of the centre of mass and motion relative to the centre of mass.

Centre-of-Mass: Some pretty crazy mass distributions. Make sure to define axis etc clearly, and not to change it during the course of the calculation.

Volumes by slicing: Can get pretty hard to imagine. Might require good 3D geometry imagination skills.

Conclusion

Great course.  I wasted a lot of time studying for the first two midterms by practicing too many questions from the book. It was better for me to just try to understand the concepts. The homework problems were challenging but rewarding. The Webwork was a bore, but that is in Math 101 anyway, so you can’t escape it. Some aspects of the course were a bit rushed, but that is to be expected with an honors course. I would recommend it to anyone who wants to really improve their understanding of mathematics.

# Course Review: MATH 120

Honours Differential Calculus

“For any topic in mathematics, you just need to do a few ‘cute’ problems. Otherwise, your mind starts ‘turning'”

Text: Robert A. Adams and Christopher Essex: Calculus: Single Variable, 8th Edition, Pearson, Toronto, 2013.

Prof: Dr. Yue-Xian Li

The thing that I’ll remember the most from Dr. Li’s class were his bizarre one-line quips including:

“You should have asked me that question weeks ago! You are going to fail the midterm tomorrow!”

“How can you confuse a “gamma” with an “r”? The “gamma” looks like the thing you use to hang people!”

“I have some  videos I could show you… people driving on a bridge while it is collapsing… some really great videos”

“There is some theory behind this, but I don’t really understand the theory… let me show you how to get the solution though”

“My third-year students keeping making this mistake, but you guys are honours, so you shouldn’t have a problem”

Dr. Li is actually a mathematical biologist. He was alots of fun and very passionate. He was also really nice during office hours. The best part of his classes were when he deviated from the textbook a bit. That said, he took most of his examples  directly from the assigned readings in the textbook.

Difficulty

I am sure most people who takes this class are a bit apprehensive about the word honours. Is that apprehension justified? I am not so sure. I think that the key to this course is identifying the handful of additional/challenging concepts early on, and being prepared to put some extra effort into them. Both the midterms and the final were really fair, since the prof gave us very similar practice exams. I found the last few questions on the weekly homework pretty hard, though. I often had to think over the questions for a few days to understand them. I also tripped on an optimization question in the final, but I think that was just me…

Key Concepts

Limits

Derivatives

Mean-Value Theorem

Continuity

Hard Concepts

Epsilon-delta Definition of Limit: Arguably  the hardest concept to understand quickly in the course. Make sure you understand the “logic” of it before diving into the math.

Proofs: Proofs often involving mean value theorem on tests, but anything course-related for homework.

Differential  Equations: Lots of Physics often involved, could be hard if that’s not your strongest subject.

Derivatives of inverses: Some of these questions can be computationally difficult and you can get really confused if you don’t notice that a function evaluated at its inverse is x.

Continuous and Differentiable Functions: Often involving piecewise functions with parameters. Its best to go back to definitions when dealing with either or both of these properties.

Chain rule: If you don’t like computation, some chain rule questions could bring you down. Its just a matter of practice and accuracy though. Not really smarts.

Optimization and Related Rates; If geometry’s not your strongest point, these could be a challenge. Once again, practice is key. ( I didn’t practice optimization enough for my final)

Conclusion

Great course, but be prepared to put in extra effort.