Mathematics

This course covers the integral and differential properties of a complex variable. Complex differentiation, Cauchy-Riemann equations, harmonic functions, sequences and series, Taylor's and Laurent's series, and singularities and residues are examined. The class also focuses on complex integration, Cauchy's theorem and consequences, and Cauchy's integral formula and related theorems. Students study residue theorem and the applications used to evaluate integrals and summation of series, along with conformal transformations.

This course offers an introduction to analysis including: sets and relations; demonstrations by induction and contradiction; natural numbers; finite and infinite cardinalities, enumerability; rational and real numbers; limits and convergence of numerical sequences and series; straight topology-- open, closed, compact, connected, dense; Cantor ternary set; continuous functions-- Bolzano-Weierstrass Theorem, Intermediate Value Theorem, uniform continuity.

This course provides an introduction to databases, with a focus on mastering the skills and knowledge required to design and analyze a database and recover and store data in a fast and efficient manner. It reviews key concepts, including the difference between data and information; database design; database implementation (SQL; data manipulation language; normalization; storage and indexation); connectivity; introduction to NoSQL.

This course builds on the fundamentals of calculus, introducing students to hyperbolic functions and their inverses, and the complex exponential functions of two variables. Students are exposed to a wider class of differential equation models, both first and second order, to describe systems such as population models, electrical circuits, and mechanical oscillators. The course also covers convergence and divergence. Calculus topics include the intuitive idea of limits and continuity of functions of one variable, sequences, series, hyperbolic functions and their inverses, level curves, partial derivatives, chain rules for partial derivatives, directional derivative, tangent planes and extrema for functions of several variables. Complex exponential topics include definition, derivative, integral, and applications. Integration topics include techniques of integration and double integrals. Ordinary differential equations topics include first order (separable, linear via integrating factor) and applications, second order constant coefficient (particular solutions, complementary functions) and applications.

This course introduces the theory of functions of a complex variable, starting with an introduction to complex numbers and ending with applications of the Residue Theorem and conformal transformations.

Topics in this course include: vectors, dot products, conics and quadrics, matrices, determinants, and linear equations. Students solve systems of linear equations; prove theorems and identities using induction; discuss terms including determinant, eigenvalue, eigenvector, invertibility, kernel, image, and spectrum; find the eigenvectors and eigenvalues of a square matrix; find the determinant and inverse of a 3 x 3 matrix, and solve associated linear equations; deduce the solvability of a system of linear equations, without finding the solutions, via Gaussian elimination; deduce whether one vector is a linear combination of others, and by the same method deduce the dimension of the kernel and image of any matrix, using Gaussian elimination and examining pivots; and prove the equivalence of the dozen invertibility criteria of Strang's nutshell using the main theorems of linear algebra.

The notion of limit underlies the differential and integral calculus, a central topic in Mathematics. A good understanding of this concept was developed in the early nineteenth century, many years after the calculus was first used, and this is essential for more advanced calculus. The main purpose of this course is to provide a formal introduction to the concept of limit in its simplest setting: sequences and series.

In this course students learn the what the following fundamental concepts involve: numbers, a.o. integers, natural numbers, real numbers and prime numbers and properties that these classes of numbers have; logic, drawing conclusions and how to use logic to prove mathematical statements; sets, properties of sets and concepts related to sets, like intersections, and unions; mathematical relations, a comparing mechanism for elements in a set, several relations and their properties; functions, mapping from one set to another, properties of functions, like invertibility; combinatorics, the science of smart counting, permutations and combinations. Almost every time mathematics is used, it concerns some of the above concepts. Students who are interested in (applied) mathematics, computer science and/or econometrics might find this course particularly useful.

In this course, students study the following topics: rings, integral domains, unique factorization domains, modules, ideals homomorphisms, quotient rings, submodules quotient modules, fields, maximal ideals, prime ideals, principal ideal domains, Euclidean domains, rings of polynomials, Gauss's lemma, Eisenstein's criterion, field extensions, Noetherian rings and Hilbert's basis theorem, dual vector space, tensor algebra and Hom, basics of homological algebra, and complexes and exact sequences.