Mathematics

This is a first course in real analysis and a concrete introduction to group theory and the mathematics of symmetry. Students study fundamental concepts of Analysis (completeness, epsilon-N, continuity, epsilon-delta) and Group Theory (groups, group actions, symmetries).

Students learn to identify and remove simple trends and seasonalities from time series data; describe the properties of stationary time series and their autocorrelations; define various time series probability models (ARMA, ARIMA, GARCH); construct time series probability models from data and verify model fit; define the spectral density function and understand it as a distribution of energy in the frequency domain; compute the periodogram and smoothed versions; and analyze multivariate time series.

This course is a basic introduction to the dynamics of time-dependent data. The course starts by discussing the type of data to be analyzed. Apart from typical single number time series such as temperatures or stock prices, students also consider the evolution of geospatial variables, 3D, and text data. This is followed by the basic Exploratory Data Analysis in the context of time-dependent data. The course will then provide insights on how time-dependent data can be analyzed based on real world examples and applications. Areas of applications that might be considered are speech, stock market evolution, music, geospatial data such as MRI scans, and medical time series data used in diagnostics.

This course provides a comprehensive introduction to the modern study of computer algorithms. The course uses Python language as a tool to learn various algorithms in depth. Knowledge in mathematics, especially algebra, is expected and having basic knowledge and experience in Python helps to better understand class content.

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.