Computer Science

I enjoyed graduate course work in algorithms and advanced algorithms. The latter I would consider my "keystone". We focused on understanding and rigourously proving bounds, correctness and expected run times (in mostly a probabalistic sense) on industry ubiquities such as hashing, dimensionality reduction, linear programming etc. I was fascinated at the notion of the existence of algorithms that could approximate a solution to a "hard" problem very quickly, whose runtime is parameterized by how accurate that solution needs to be. Moreover sometimes it can be much faster (in expectation) to run an algorithm that only sometimes gets the right answer a bunch of times, than it is to run an algorithm that always gets the answer, only once. The world of algorithms and theory truly becomes infinitely rich and fascinating when we start to consider outputs and runtimes in a continuous, probabalistic way, and relaxing our need for absolute determinism. This class relied on a myriad of maths, including probability, linear algebra, discrete and multivariable calculus. The expected value function and concentration bounds (Chernoff's, Chebyshef's etc) ought to be the most useful tools for algorithm developers.

Other classes

Artificial Intelligence, Object Oriented Design, Database Design, Software Development, Logic and Computation, Theory of Computation.

Maths

pure

In the realm of pure, Group Theory (algebra) and Number Theory (II) have been the most enlightening. Group theory was my first encounter with the language of modern pure maths. This course revolved mostly around classifying the structure and size of instances of the most generalized notion of a mathematical object. The level of generality and power of theorems of Sylow and on Abelian groups was shocking. We lifted back the curtains on some of the mathematical facts we take for granted on a day to day basis. The next pivotal pure math class in many ways was an extension and application of the ideas from Group Theory. At the commencement of Number Theory (II) we worked with some of the quirkier aspects of real numbers, and how our knowledge of integers can help us understand them. One example of this is finding the best rational approximation of irrational numbers through continued fractions. Quickly it became apparant that most of modern number theory required more firepower to work with. We expanded our tools by developing and understanding fields, rings and the structure thereof. It intrigued that after a certain level of complexity, problems in number theory could only be solved with tools from maths that generalized the idea of numbers. This illustrated again the power and implications of results in algebra. We also studied elliptic curves over various domains and their application to cryptography.

applied

The seminal courses in my applied math education are Stochastic Processes, and Financial Derivatives Pricing. These classes built upon the basics of Mutlivariable Calculus and Prob and Stat. Stochastic Processes studied probability as applied to complex systems. One when stuck at a line in a grocery store might wonder how long they expect to stand in line. With two parameters and a simple stochastic process we can rigoursly answer this question. Stochastic Processes culminated in the famous Brownian motion, a model describing how normal random variables act as a function of time. This served as a perfect precursor of course to Financial Derivatives and the Black-Sholes formula. Financial Derivatives was my first oppurtunity to study financial intruments in a deeply mathematical way. It supplied me with a strong intuition of options markets. It was here I learned the mathematical underpinnings and assumptions of how things are priced. Markets in a lot of ways are their own universe. They start with a few assumptions, such as no-arbitrage pricing, independence of returns on an asset in disjoint time intervals, the existence of a risk free rate of return (like Newtons laws of physics, but for markets). Then every we know about how something ought to be priced is merely a mathematical extension of said assumptions (Black Sholes is the perfect example). The most earthshattering result from this class for me was the proof that returns on an asset are lognormally distributed. For the first time I felt like I had gained some ability to quantitatively reason about markets. For any lognormally distributed variable, it happens that the median is less than the mean. This implies that most of the time, a stock earns less than it is expected to. I found this quite comical.

Other classes

Calculus (I - III), Discrete, Linear Algebra/Matrix Theory, Differential Equations.