Mathematics

This linear algebra course offers a study of the following topics: vector space, linear independence, algebraic basis, dimension, linear mapping, kernel and image, bilinear form, dot product, diagonalization, and Jordan canonical form.

The course gives an introduction to probability theory in a measure-theoretic setting. Among the topics discussed are: Probability measures, σ-algebras, conditional expectations, convergence of random variables, the law of large numbers, characteristic functions, the central limit theorem, filtrations, and martingales in discrete time. Recommended prerequisites include calculus, linear algebra, and probability and statistical modeling. 

This course focuses on the main concepts and tools of Bayesian inference. It explores computational algorithms for the Bayesian analysis of some specific models, such as linear regression and simple hierarchical models.

As many natural and social phenomena are described by functions, fundamental laws are formulated through differential and integral calculus. This course covers the basics of differential and integral calculus for single-variable functions. It is recommended to take this "Introduction to Mathematics" course first, followed by the study of more general multivariable calculus in the subsequent “Calculus 1, 2". No prerequisite knowledge of Mathematics III in high school, IB Higher Level, or AP Calculus is required.  

This course studies topological spaces and continuous maps. Main topics include: topological spaces; subspace, order, product, metric and quotient topologies; continuous functions; connectedness and compactness; countability and separation axioms. Secondary topics include: retractions and fixed points; Tychonoff Theorem; compactifications; and vistas of algebraic topology. 

The course introduces the notions of Fourier series and Fourier transform and to study their basic properties. The course is devoted to the one dimensional case in order to simplify the definitions and proofs. Many multidimensional results are obtained in the same manner, and those results may also be stated. The Fourier technique is important in various fields, in particular, in the theory of (partial) differential equations. It is explained how one can solve some integral and differential equations and study the properties of their solutions using this technique.

This course provides a fundamental overview of mathematical finance. It begins with an overview of financial contracts, interest rates, and the value of money. Specifically, it discusses what constitutes a fair price for a contract and explains why fair prices are rarely used in everyday transactions. After that, students investigate financial markets in a discrete-time setting, with the help of some revision on basic probability theory. The concept of risk-neutral asset pricing is discussed with reference to pricing stocks and options in the exchange. The last part of the course introduces the fundamental concepts of stochastic calculus and concentrates on continuous time finance with the widely used Black-Scholes model. The goal of this course is to provide students with a broad understanding of the application to finance theory, while setting a solid theoretical foundation to the field. 

This course offers a comprehensive exploration of scientific computing, covering essential topics crucial for solving mathematical problems encountered in scientific and engineering fields. Beginning with an introduction to the fundamentals of numerical methods, including error analysis and computational complexity, students explore solving systems of linear equations using various techniques such as direct and iterative methods. The course further aims to eigenvalue computation methods and approaches for solving nonlinear equations. Interpolation techniques for approximating functions from discrete data points are also covered in detail. Through hands-on exercises and computational assignments, students develop practical skills in numerical analysis, enabling them to tackle diverse mathematical challenges in scientific computing effectively.

The course reviews number theory including the fundamental theorem of arithmetic, modular, and arithmetic; groups including definition, basic examples of groups, subgroups, normal subgroups, factor groups, isomorphisms, and homomorphisms, Lagrange's theorem, permutation groups, symmetric and alternating groups, finitely generated Abelian groups; rings including definition, basic examples of rings, isomorphisms and homomorphisms, ideals, factor rings, polynomial rings, factorization of polynomials as products of irreducible polynomials; and fields including characteristic, simple field extensions, finite fields.

The main theme of the course is the interplay between Number Theory and rings. Students need to be familiar with the basics of prime numbers, unique factorization of integers and modular arithmetic. This is an advanced course with Fundamentals of Pure Mathematics as a prerequisite.