Mathematics

In this course we will introduce some basic models in life insurance, and the method for calculating net premium and reserve.

The course covers two or three of the following main topics: Dynamical systems, Group Theory, or Complex Analysts. Dynamical systems include ordinary differential equations, phase plane analysis, stability analysis, linearization, limit cycles, Poincaré-Benedixson Theorem. Group theory is a natural setting in which to learn styles of proof-writing and abstract thought characteristic of much of modern mathematics. Complex analysis includes the calculus of complex-valued functions and power series, geometric properties of analytic functions, the Cauchy-Riemann equations, topological properties of integration in the complex plane, Cauchy’s Theorem, Cauchy’s Formula. Which of the above topics are covered may vary from year to year. This course replaces the former Math labs UCSCIMATL5, UCSCIMATL3, and UCSCIMATL6.

This course introduces the students to the basics of quantitative finance and targets all students who have an interest in building a foundation in quantitative finance. Topics include term structure of interest rates, fixed income securities, risk aversion, basic utility theory, single-period portfolio optimization, basic option theory. Mathematical rigor will be emphasized. The course requires students to take prerequisites.

This course examines power series methods (ordinary and regular singular points, Bessel's equation); boundary value problems and separation of variables (Fourier series and other orthogonal series), applications to the vibrating string, heat flow, potentials.

This course provides basic set operations in set theory and examines how to prove propositions, with a focus of setting operations and links to the proofs. Students study systematic operations in set theory and apply such operations to extended mathematical proofs. 

The goal of this course is to familiarize students with the key analytical methods and leading applications in the field of game theory. At the end of the course, students should be able to identify and formalize problems that involve strategic interaction between different economic agents, and to analyze them with game-theoretical thinking.

This course offers a study of stochastic processes. Topics include: discrete time Markov chains; renewal theory and Poisson process; continuous time Markov chains; Brownian motion. Pre-requisite: Probability 

Through software-assisted learning, you will be able to intuitively understand fundamental mathematical concepts such as linear algebra, and delve into advanced topics like the PageRank algorithm, Analytic Hierarchy Process (AHP), function fitting, numerical integration, solution of ordinary differential equations, and machine learning methods.

This course provides in depth knowledge of fundamental results and methods in discrete dynamical systems, knowledge of the concrete dynamical systems presented during the course, and an understanding of the many and diverse appearances and applications of discrete dynamical systems. It develops skills to analyze and argue for results on discrete dynamical systems, produce proofs for theorems, and solve exercises posed during the course.

This course introduces the mathematical theory of probability, starting with the definition of probability spaces to the fundamental limit theorems, namely the law of large numbers and the central limit theorem. The following concepts are covered: measurement theory, probability spaces; conditional probability and independence; random variables, discrete random variables; density random variables; discrete random vectors; density random vectors; notions of convergence for sequences of random variables; limit theorems; Gaussian vectors.