Mathematics

This course is an introduction to computability theory and Gödel's incompleteness theorems. The first half of the course focuses on computability theory, and includes Recursive and primitive recursive functions; Turing machines and computable functions; basic results in computability theory including Kleene's Normal Form Theorem, the s-m-n Theorem, Kleene's Recursion Theorem, Recursively enumerable sets, the halting problem and decision problems in general; as well as hierarchy theory, relative computability, and Turing degrees. The second part of the course focuses on Gödel's first incompleteness theorem, and includes Axiom systems for number theory, representable relations and functions, arithmetization of syntax, the Fixed-Point Lemma, and Gödel's first incompleteness theorem, as well as Gödel's second incompleteness theorem.

Mathematics underpins virtually everything that we take for granted in our daily lives, and it is sometimes referred to as the “Queen of Science” due to its demands of logical rigor and cold calculations. However, despite its intimidating veneer, mathematics is the culmination of millennia of human endeavor. The purpose of this course is to give an accessible overview of some of the key developments in mathematics, covering the period from the time of the ancients, up to the early modern period. The course also provides an opportunity to apply historical mathematical methods to solve problems. While covering the well-known Greek, Chinese, Islamic, and European mathematicians, the course also addresses Japanese mathematics during the Edo period.　
 

While the course includes written assignments, to properly understand and follow the thinking of the mathematicians, the course covers problem solving using historical mathematical methods. While a background in high-school level math is useful, an enthusiasm for critical thinking and problem solving could replace that prerequisite, since the mathematical concepts will be introduced as they were historically considered.

This course delves into matrix algebra, calculus (including differentiation and integration), and introductory optimization techniques, all of which are essential in the social sciences, particularly economics and finance. The primary focus of the course lies in mathematical analysis via programming with Octave. The course employs Octave (the free version of Matlab) to facilitate both analytical calculations and simulations.  

The course does not assume that students have prior proficiency in calculus or programming and will start from the basics, progressing to the theoretical application of calculus, notably optimization. This term, we will emphasize studying static optimization using the Lagrange method. Optimization theory serves as the cornerstone of economics and finance.  

For anyone interested in economics, finance, and programming, this course will be invaluable. 

This course introduces eigenvalues and eigenvectors of matrices, leading to diagonalization of matrices. Furthermore, vector spaces with inner product are treated and applications of linear algebra to various specialized topics are discussed.

Upon completion of the class, students are expected to:

• Compute eigenvalues and eigenvectors of matrices, and diagonalize real symmetric matrices; 
• Understand inner products, orthogonality, and to be able to find orthogonal bases; and,
• Learn applications of linear algebra and perform computations to solve explicit problems.

This course provides an essential toolkit for solving real-world problems that arise in various industries, such as the financial and tech sectors, healthcare, manufacturing, and planning. Through an engaging set of lectures and classes, students develop problem-solving and modelling skills, and learn insights necessary for strategic decision-making.

The course examines the basic paradigms of modern financial investment theory, to provide a foundation for analyzing risks in financial markets and to study the pricing of financial securities. Topics include the pricing of forward and futures contracts, swaps, interest rate and currency derivatives, hedging of risk exposures using these instruments, option trading strategies and value-at-risk computation for core financial instruments. A programming project provides students with hands-on experience with real market instruments and data. This course is for students with an interest in quantitative finance. The course requires students to take prerequisites.

This course introduces the concept of modelling dependence and focuses on discrete-time Markov chains. Topics include discrete-time Markov chains, examples of discrete-time Markov chains, classification of states, irreducibility, periodicity, first passage times, recurrence and transience, convergence theorems and stationary distributions. The course requires students to take prerequisites.

This course offers a study of classical partial differential equations from mathematics and physics. It examines the structures of differential equations and their practical applications. 

This course offers a study of Numerical Methods (NM) to calculate approximate solutions of mathematical models. Topics include: floating point, errors, stability, and algorithms; solution of linear systems of equations; numerical solution of equations and systems of nonlinear equations; interpolation and approximation of functions; least squares problems; numerical optimization; numerical integration; numerical differentiation; fast Fourier transform. 

The course is a rigorous introduction to probability. Students gain a solid grounding on the its foundations, learn how to deal with randomness with the correct mathematical tools and how to solve problems. Course topics include probability; definition and properties; conditional probability and independence; random variables and random vectors; joint and conditional distributions; expectation and moments; integral tranforms; convergence in distribution and the Central Limit Theorum; and modes of convergence and the laws of large numbers. Prerequisites: Set theory, sequences and series, continuous and differentiable functions, and integrals.