Mathematics

Set theory is a beautiful and magical subject stemming from transparent and easy observations leading us to a surprising and somewhat unbelievable logical world on which contemporary mathematics is based. Its controversial and contrasting history attracts our attentions as well. In this beginning course, we focus on set operations, orderings, cardinal and ordinal arithmetics which, as primitive notions, are absolutely necessary in learning almost every subject of mathematics. The course also introduces more mysterious and advanced parts of the subject whose full clarifications can be pursued by interested students in their senior or graduate level courses. We often touch on set theory itself, overview the axiom of foundation, the consistency and independence problems, the theory of large cardinals, descriptive set theory, etc. 

This course is an introduction to topological spaces. It deals with constructions like subspaces, product spaces, and quotient spaces, and properties like compactness and connectedness. The course concludes with an introduction to fundamental groups and covering spaces. The course discusses topics including sets and functions, images and preimages, and finite, countable, and uncountable sets; how the topology on a space is determined by the collection of open sets, by the collection of closed sets, or by a basis of neighborhoods at each point, and what it means for a function to be continuous; the definition and basic properties of connected spaces, path connected spaces, compact spaces, and locally compact spaces; what it means for a metric space to be complete, and characterizing compact metric spaces; the Urysohn lemma and the Tietze extension theorem, and characterizing metrizable spaces; and the construction of the fundamental group of a topological space and applications to covering spaces and homotopy theory.

This course offers a study of the basic concepts of discrete mathematics: graphs, vertex and adjacency relations; trees; existence of Euler and Hamiltonian paths; graph coloring; pairings. It explores ways to perform software modeling and resolution of routing optimization, interconnection, and assignment problems.

This course discussed classical solution methods for first order equations. After this, linear equations of higher order with constant coefficients and first order systems are studied. Power series solutions are introduced for linear equations with variable coefficients. The last part of the course focuses on general theorems about existence and uniqueness. These theorems are important since most differential equations lack explicit solutions.

This course introduces students to techniques and tools in modern analysis which have important uses in a variety of areas of analysis, including the study of partial differential equations and Fourier analysis. Students achieve this in the context of linear analysis, introducing normed linear, inner product spaces and their completions, Banach and Hilbert spaces. The structure and geometry of these spaces are studied as well as continuous linear operators acting on them.