Mathematics

The course introduces rings, subrings, homomorphisms, ideals, quotients, and isomorphism theorems. It includes integral domains, unique factorization domains, principal ideal domains, Euclidean domains, Gauss' lemma and Eisenstein's criterion. Fields, field of quotients, field extensions, the tower law, ruler and compass constructions, construction of finite fields.  Students state the definitions of concepts and prove their main properties, describe fields and rings and perform computations in them. Students discuss the theoretical results covered in the course and outline their proofs. They perform and apply the Euclidean algorithm in a Euclidean domain, giving examples of sets for which some of the defining properties of fields. They focus on proving the tower law, and use it to prove the impossibility of some classical ruler and compass geometric constructions. Students learn to identify concepts as particular cases of fields, rings, and modules (e.g. functions on the real line as a ring, abelian groups, and vector space).

This course covers number theory. Topics include integers on a ring: completely closed rings, quadratic bodies, norm, trace, discriminant in the case of extensions of bodies. Example of cyclotomic bodies of degree p-1; Dedekind rings: Noetherian property; application to integer elements, fractional ideals, fraction rings, localization, group of fractional ideals, norm of an ideal, multiplicativity; decomposition of ideals in an extension: prime ideal, discriminant and ramification, quadratic and cyclotomic bodies of degree p-1, quadratic reciprocity law; class group and unit theorem: networks, canonical folding, statement and proof of the finiteness of the class group, statement of the unit theorem, illustration in the case of quadratic bodies, Fermat cases (or other Diophantine equations); analytical opening (Riemann zeta function, Dirichlet L-functions, Dedekind zeta functions, link to counting prime numbers and ideals).
 

This course introduces students to more advanced topics in Probability Theory and Statistical Inference. The first part is devoted to investigating mathematical aspects of probability, with a special emphasis on multivariate distributions and limiting theorems. In the second part, students are guided through the methodological core of point estimation (both from a frequentist and Bayesian perspective) and hypothesis testing. These theoretical aspects are complemented by an in-depth presentation of elementary simulation and computational techniques that are routinely used within most popular statistical procedures. Prerequisites: Solid knowledge of calculus and of basic programming tools in R. 

The basic purpose of this course is to deeply understand mathematical thought and values behind classical knowledge in mathematics.

This course provides and understanding of group theory and some of its applications. In this course, students work with cyclic groups, permutation groups, dihedral groups, equivalence classes, cosets, Lagrange's theorem, and direct product groups; are introduced to quotient groups, construct the groups of low order, learn about the conjugation map, and construct conjugacy classes; meet the classical matrix groups, which are examples of continuous (or Lie) groups; work with group homomorphisms, isomorphisms, automorphisms, normal subgroups, kernels of homomorphisms, and prove and make extensive use of the group homomorphism theorem (also known as the first isomorphism theorem); learn about the semi-direct product and semi-direct product groups; construct and investigate the Euclidean group; investigate the geometric structure of some of the classical matrix groups, in particular SU(2)and SO(3); work with group actions on sets, stabilisers and orbits; and prove the Sylow theorems.

This course provides an introduction to sets and functions: defining sets, subsets, intersections and unions; injections, surjections, bijections.; compositions and inverses of functions; an introduction to mathematical logic and proof: logical operations, implication, equivalence, quantifiers, converse and contrapositive; proof by induction and contradiction, examples of proofs. These ideas are then applied in the context of the real numbers to make rigorous arguments with sequences and series and develop the notions of convergence and limits. 

This course explores the quantitative instruments needed to pose and analyze economic problems with the aid of a formal model. Topics include: concepts of matrices and algebra of matrices; analysis of dynamic economic models; differential and difference equations and systems; examination of the qualitative behavior of solutions. 

This course explores the concepts of heuristics and optimization as two means of problem-solving and analysis. Topics include: dynamic programming; linear programming; constrained Boolean satisfiability; constraints programming; search. Pre-requisites: Programming; Algorithms and Data Structures; Discrete Mathematics; Artificial Intelligence. 

This course introduces the key principles of mathematical modelling, which then are explored through several real-world examples in disease modelling, environmental planning, and population dynamics. The techniques of calculus are essential, although core concepts such as differential equations are revisited. Students also touch on the mathematical models used in data science, particularly the techniques of principal component analysis and clustering, both essential tools in machine learning models.  

Graph theory provides a basis for computational sciences, financial engineering, chemistry, and epidemiology, among many others. This course covers graph theory including graphs, paths, cycles, trees, connectivity, Eulerian and Hamiltonian graphs, and planar graphs, as well as some important algorithms.