4000

The objective of this course is for students to develop a study and research plan proposed in advance under the guidance of a professor from the Department.

Rings and Ideals. Modules. Rings and fraction modules. Primary decomposition. Whole dependence and valuations. Chain conditions. Noetherian rings. Artin rings. Discrete valuation rings and Dedeking rings. Completions. Theory of dimension and, if there is time left, other themes that the professor may deem convenient.

Initiate studies of Model Theory of First-Order Logics. Completitude, Compacity, Lowenheim-Skolem Theorems, Categorical K-Theory, Complete Theories, Decidable and Undecidable Theory, Elementary Equivalence and Summersion. Characterization of Universal Theories, Universal-Existential. Existentially Closed Models, Complete Model Theories, Elimination of Quantifiers, Partial Isomorphisms, Feferman-Vaugth Theorems. Interpolation and Definibility Theorems. Automorphisms, Indiscernible, Ehrenfeucht-Mostowski Theorem. Fraissé Generic Models. Boolean Algebra, Filters, Ultra filters. Ultra products, Ultra product Saturation. Types of Elements, Types Realization and Omission, Saturation, Homogeneity, Universality. Atomic and Prime Models, Omega-Categoric Theories. Type Spaces, Stability, Stable Omega Theories. Keisler-Shelah Theorem, Characterization of Elementary Classes. Morley Categoricity Theorem. Baldwin-Lachlan Theorem. After all of the foregoing, the instructor may concentrate further on themes such as: Laws 0-1 in Finite Models. Finite Models Spectrum. Relations with Complexity.

Algebras and sigma-algebras, measures, Lebesgue Measure, Constitution and regularity. integrated functions: Roles measurable properties almost everywhere, the definition of integral, theorems of boundaries, Riemann integral, complex functions and extent of image. Convergence: Modes of convergence, regulated spaces, Lp spaces, dual spaces. Real and complex measures: Absolute continuity, singularity, functions of bounded variation, dual spaces Lp. Product measures: construction, Fubini Theorem, applications.

Banach Spaces: Definitions and examples. Subspaces, linear transformations, quotient spaces. Duality: Hahn-Banach’s theorem. Banach-Steinhaus’ theorems, the Open Mapping and the Closed Graph theorems. Applications: Adjoint operators. Hilbert Spaces: Definitions and examples, orthogonality. Continuous operators: Operator convergence. Hermetian, normal and unitary operations. Orthogonal projections. Compact operators: Introduction to spectral theory.

Introduce the basic concepts of algebraic topology, and the classical algebraic tools used in the calculation of topological invariants of spaces simplicials. Content: Variety topological. Homotopy group fundamental properties. Building simplicials, homology, Ulama-Borsuk theorem, fixed point theorem Lefschetz, Cohomology, Poincare duality. Theory of beams, Prebeams, Resolutions. Fiber vector.

The objective of this course is to introduce the use of modern and classic tools and machinery of a differential nature used in geometry of varieties. The course contents are: Differential varieties. Vector fields on differential varieties. Differential forms and operations with differential forms. Integration of differential forms and Stokes’ theorem. De Rham’s Theorem and Poincare’s lemma. Riemannian structures, Laplacian operator and Hodge’s theorem. Vector fibrations, connections and characteristic classes.

The objective of the course is to place students in contact with a wide range of advanced mathematical topics and teach them to synthesize and orally explain these topics with clarity and accuracy.

The objective of the course is to place students in contact with a wide range of advanced mathematical topics and teach them to synthesize and orally explain these topics with clarity and accuracy. In this seminar, students decide on the topic in the area they would like to do their graduation thesis and prepare a presentation on this topic with the professor who will possibly be the director of the graduation thesis.

The objective of this course is to introduce students to research activities, through the direct study of specialized mathematical literature and train them, not only to solve problems but also to formulate them properly. Students will present their theses to the Graduate and Research Committee of the Department before the last week of withdrawals for the semester. Students are expected to move forward in their research during the following period.

Students will carry out a research project in one of the mathematical areas offered by the Master’s Program. This is to demonstrate that the author has assimilated and systematized or carried out thorough exploration of a particular topic, showing evidence of a certain degree of creativity and great familiarity with recent information on the topic. The Graduation Thesis must be written in Spanish or English and have the formal organization expected of a scientific paper.