3000

Rings, integral domains and fields. Algebraic and transcendental extensions. Unique factorization domains and Euclidean domains. Field automorphisms, Galois’ theory and solvability of equations.

This course is an introduction with mathematical emphasis to logic. The minimum content of the course includes the study of the calculation of propositions and predicates: symbolization, syntax, semantics, formal deduction, validity and completeness theorems for these calculations. It gives an introduction to calculations: recursive functions, Turing-calculable functions, equivalence between them. We study some links between calculation and formal properties of logical calculations studied.

Introduce students to the basic tools provided by the differential geometry (Riemannian) for the study of differential varieties with metrics. Specially, study in depth fundamental notions as parallel transport, curvature and geodesic and its properties. Study the key results on Riemannian varieties and get to know the classic examples from which we can study the mentioned concepts.

Topological spaces, bases and sub-bases, sub-spaces, continuous functions. Order topology, product topology, quotient topology and metric topology. Connected spaces, arc-connected spaces and local connectedness. Compact and locally compact spaces, compactifications, Tychonoff’s Theorem. Axioms of enumerability. Regular and completely regular spaces. Normal spaces, Urysohn’s Lemma, Tietze’s Theorem of extension. Metrizability.

Introduction. Stochastic processes specifications. Some important classes of processes such as stationary processes. Processes with stationary increments and processes with independent increments. Markov processes. Martingales. MARKOV CHAINS: definitions and examples. Features. Finite Markov chains. Classification of states and chains. Markov chains accounting. Limit theorems. Stationary distribution. POISSON PROCESSES: generalizations of Poisson processes. not homogeneous process. Compound Poisson processes. Conditional processes. Processes of birth and death. MARTINGALA in discrete time: conditional expected value. Definition and examples. Time to Stop. Optional stopping theorem. Inequalities of the Doob Martingale. Convergence Theorem of Martingala. RENEWAL PROCESSES : Renewal Equation. Laws of large numbers. Age and residual life. Applications to the theory of the tail. Brownian motion: Preliminaries. Simple Features of standard Brownian motion. Variations in the Brownian motion. Brownian motion with drift. Kolomogorov equations. Ornstein-Uhlenbeck process.

Estimation methods: time for confidence intervals. Methods of moments, least squares, maximum likeliness. Theory of optimality criteria estimate UMVU, the information. Consistent estimators, asymptotic distribution, efficient estimators, non-biased. Confidence intervals and hypothesis tests. Theme of Neyman-Pearson. Likeliness reason. Tests of fitting, contingency tables. Linear models, Gauss-Markov theorem, Testing in linear models.

Training in teaching methodology. Micro-teaching practice on how to handle questions, the use of the blackboard and diagnostic sessions. Instructions prior to each class, observations on development, creation of exams. Students teach one section of problems relating to a lecture under the guidance of a professor from the Department.

Instructions on how to teach a class, reading and discussion of articles in Mathematical Education, creation of exams, observations. Students teach a course under the guidance of a professor from the Department.

Students write a paper in a specific area of mathematics in which they show their capacity as regards research and presenting a topic with all the requirements of clarity, correctness and proper style.

Students start out in the current events of research in mathematics.

Students start out in research as an independent activity, as well as the proper oral and written communication of mathematics by reading articles and solving problems. This course requires students to choose a topic for their graduation thesis.