# Course Review: MATH 420

Real Analysis I

“F just comes along for the ride…”

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

Prof: Dr Stephen Gustafson

Prof Gustafson is an engaging professor. He is able to give a quick, intuitive overview of quite complex topics in class, leaving us to work out the technical details on our own. He also is very helpful during office hours. He also posted his notes online.

Difficulty

The classes move very fast so one might find it challenging keeping up with the readings. Also, though the classes are easy to follow, the homework is increasingly challenging and time-consuming. The final exam was quite hard, practice and recalling theorems and definitions are all important. This course covers a lot of material, so watch out!

Key Concepts

Sigma-Algebras

Measures

Differentiation of Measures

Lebesgue integration

Convergence of Functions

LP spaces

Hard Concepts

Lebesgue Radon Nikodyn Theorem: Quite powerful theorem, need to notice where to apply it. Also can be hard to find the decomposition at times.

Fatou’s Lemma: Tricky working with the limit infimum, also quite powerful and pops up in unexpected places.

Conclusion

One of the harder courses I have ever taken. Started learning some really powerful theorems about convergence, for example.

# 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: STAT 406

Methods for Statistical Learning

“To fix ideas…”

Text: An Introduction to Statistical Learning by James, G., Witten, D., Hastie, T. and Tibshirani, R.

Prof:  Dr. Matías Salibián-Barrera

Matías is a very earnest professor who often works through helpful examples in class. He also provides supplementary class notes that are useful to review in your own time. His slides can be a bit confusing from time to time but are generally understandable.

Difficulty

This was my first statistics course. I found the material quite challenging at first because I was unfamiliar with a lot of the vocabulary and notation that was used. Also, we used a lot of quite novel results from advanced linear algebra and probability. However, I soon got used to the notation and realized that to succeed in the course one does not need to fully understand what mathematics operating in the background (just need to understand a little). The first midterm was short, free response and had a low average. The subsequent midterms and the final were multiple choice, largely, and were easier, the only difficulty often being the wording of some of the options. A lot of the material was also reviewed from CPSC 340. One of the aspects of the course that was a bit annoying was the webwork. They had a time limit and would often start by accident, or give you an incorrect mark on your first attempt. The labs were pretty reasonable, though you need to be quite quick and compare answers with your friends.

Key Concepts

Model Selection, esp Cross-Validation

Elastic Net Methods

Tree-based methods

Splines

Bagging and Boosting

Clustering Techniques

PCA

Hard Concepts

AIC Derivation: Can be pretty confusing as each formula makes different assumptions

Cross-Validation as Expectation: Make sure one understands the notation for expectation used

Linear Predictor: Some models may be linear predictors though they do not appear all that linear

Conclusion

A handy review of CPSC 340 except in R. Would have liked to have delved more into the statistics of it and how one chooses an appropriate model for a given data set.

# Course Review: CPSC 421

Introduction to Theory of Computing

“Let me start off by telling you a fact without enough background to prove the fact”

Text: Introduction to the Theory of Computation by Michael Sipser

Prof:  Dr. Nick Harvey

Dr Nick Harvey is a very clear professor. He makes effective use of slides and slowly explains complex topics, making his lectures both interesting and easy to follow. For example, his explanation of the diagonalization argument was much clearer than any of the presentations on the same topic I have received in math classes. He also has a peculiar sense of humour and is a big soccer fan, so the class is interspersed with entertaining distractions.

Difficulty

The material starts with fairly simple with a bit of review from CPSC 121. It picks up a bit when you start performing reductions between problems from various complexity classes though there is a bit of recap from CPSC 320. The first midterm was straightforward, however, given its short length it was possible to not to do as well if one tripped on only one or two questions. The final was a bit harder, however, the previous exam was good preparation.  The homework included some questions that could be genuinely hard to understand. However, he gave generous hints as the deadline approached, and they were the type of questions that once you understood what the question was asking, the proof was often one of the first few obvious approaches that you would think of. Given that he dropped the lowest couple of homework, it’s no surprise that the overall average in the course was 80.

Key Concepts

Finite Automata and Regular Languages

Turing Machines

Non-determinism

Time complexity classes

Mapping Reductions

Probabilistic Turing Machines

Communication Complexity and Fooling sets

Hard Concepts

Context Free Languages: Can be hard sometimes to come up with CFG or PDA for a given context-free language

Probabilistic Turing Machines: Give rise to quite convoluted complexity classes.

NP-Hard reductions: Might require a bit of imagination to come up with the reductions, at times.

Fooling Sets and Covers: One can sometimes get fooled by these, make sure one recalls the definition of rectangles and what exactly needs to hold for a fooling set.

Conclusion

Fun course. Dr Nick Harvey can make the course appear deceptively simple at times, but you actually learned a lot of quite complex material.