Mathematics

This course gives an introduction to the theory of systems of ordinary differential equations. This includes topics such as existence and uniqueness of solutions, linearization and stability theory, phase space techniques, bifurcation theory and Hamiltonian systems. Emphasis is put on using dynamical systems as a modeling tool in different areas of science, technology, medicine, economy. Students perform numerical studies of complex systems.

This is an introductory course on algebraic geometry. We cover affine/projective varieties, Zariski topology, Hilbert's Nullstellensatz, regular morphisms, Zariski tangent spaces, etc. Topics include Polynomial Rings, Varieties and Ideals, Irreducibility of Affine Varieties, Coordinate Rings, Polynomial Maps, Proof of the Nullstellensatz, Dimension of Affine Varieties, Tangent Spaces and Smoothness, Projective Varieties, Maps of Projective Varieties, Quasiprojective Varieties, and Further Quasiprojective Topics.

Prerequisites: Modern algebra (1), (2), Topology (1)

This course covers various topics on supervised learning (regression, classification) on tabular data, including fundamentals of statistical learning; linear models with and without penalization; course of dimensionality in nonparametric models; additive models; tree based methods and neural networks; and post-hoc interpretability.

This course opens the door to higher mathematics, science and technology, as well as economics and social science. This course emphasizes skills, theory, and applications. The course presents the core of the linear algebra as an axiomatic development of the most important elements of finite-dimensional linear algebra and progresses into more abstract areas as we add structure to our knowledge: Fields and Vector spaces, Linear Operators, Determinants and eigenvalues, The Jordan canonical form, Orthogonality and its most important application of best approximation, spectral theory of symmetric matrices and Hermitian matrices, The singular value decomposition, Matrix factorizations and numerical linear algebra, Infinite dimensional vector spaces and Analysis in vector spaces. Linear algebra forms the basis for much of modern mathematics-theoretical, applied, and computational. 

This course gives an introduction to analytical techniques for partial differential equations, in particular to separation of variables. In addition the course treats qualitative properties of solutions, such as maximum principles and energy estimates. The course also gives a basic introduction to difference methods and their stability analysis.

This course examines some basic notions and techniques of number theory. It focuses on such topics as divisibility, prime numbers, the arithmetic of residues rings, additive properties of integers and their powers and Diophantine approximations. Some applications of number theory to cryptography will be discussed as well. Students taking this course will develop an appreciation of the basic problems of number theory and will learn the interplay between number-theoretic problems and other areas of mathematics.

This course covers the following topics: Calculus of variations: functional, variation, extremals, Euler-Lagrange equation, Beltrami identity, brachistochrone, catenary, natural boundary conditions, isoperimetric constraints; Conservation laws: transport equation, Burgers' equation, method of characteristics, weak solutions, Rankine-Hugoniot jump condition, Oleinik entropy condition; and Wave equation: Spherical means, Euler-Poisson-Darboux equation, Kirchhoff's formula in 3 dimensions, Poisson's formula in 2 dimensions, energy method, finite speed of propagation, domain of dependence.

This course introduces approximation techniques in numerical analysis and scientific computing. It examines numerical methods for solving nonlinear equations, systems of linear and nonlinear equations, and related problems. Techniques for interpolation, approximation, and numerical differentiation and integration are studied.

This course begins with an understanding of the concept of distance in mathematics, and introduces the fundamental concepts and various properties of topological structures. It provides a theoretical foundation for topics previously used in calculus, including the real number system, limits, continuous functions, the extreme value theorem, the mean value theorem, the existence of definite integrals, the fundamental theorem of calculus, and the intermediate value theorem. In addition, the course explores the geometric properties of topological concepts. Topics include Set Theory and Logic, Topological Spaces and Continuous Functions, Connectedness and Compactness, Countability and Separation Axioms, The Tychonoff Theorem, Metrization Theorems and Paracompactness, and Complete Metric Spaces and Function Spaces.