### Doctorate

**Functional Analysis **(4 Credit Units, 60 Credit Hours)

Syllabus: Regulated vector spaces. Banach spaces. Space quotient. Linear operators and their adjuncts. Hahn-Banach theorem. Uniform limitation theorem. Closed-graph theorem. Open application theorem. Weak topology. Banach-Alaoglu's Theorem. Reflective spaces. Hilbert Spaces. Orthonormal Sets. Riesz's representation theorem. Compact operators. Spectral theory of Self-attached Compact Operators.

**EDP I Recent Aspects **(4 Credit Units, 60 Credit Hours)

Syllabus: Theory of the Degree in Finite Dimension: Definition of the Topological Degree for functions of class C ^ 2; Extension of degree to functions of continuous; Basic Properties of the Degree Theory, Applications - Brower Fixed Point Theorem; Theorem of Invariance of Brower Domains; Jordan Curve Theorem. Theory of the Degree in Infinite Dimension: degree of Laray-Schauder, definition and properties, applications - Dirichlet problem; Second order quasilinear problems; Global Results on Problems of Nonlinear Eigenvalues, Bifurcations. Galerkin method: applications to EDP Ellipticals.

**EDP II Recent Aspects** (4 Credit Units, 240 Credit Hours)

**EDP III Recent Aspects **(4 Credit Units, 240 Credit Hours)

**Distributions and Partial Differential Equations** (4 Credit Units, 60 Credit Hours)

Syllabus: Functions Tests. Distributions. Fourier transform. Sobolev spaces. Immersions. Trace Theorem. Non-Homogeneous Elliptic Problems.

**Nonlinear Partial Differential Equations **(4 Credit Units, 60 Credit Hours)

Syllabus: Compassion Method-Aubin-Lions Theorem. Nonlinear Wave Equations. Potential well. Navier-Stokes systems. Schroedinger Nonlinear Equations. Method of Monotony. Pseudo Laplacian. Monotonous Operators. Monotone Parabolic Equations. Viscosity Hyperbolic Equations.

**Partial Differential Equations**** **(4 Credit Units, 60 Credit Hours)

Syllabus: First order nonlinear equations. Cauchy's problem for quasi-linear equations. Burgers equation and shock condition (Rankine-Hugoniot condition). Waves of shock and waves of rarefaction. Buckley-Leverett equations. Second order Hyperbolic equations. Propagation of singularity. The wave equation. Shallow Water Equations. The Cauchy-Kowalevski theorem, Green's identity and Holmgren's uniqueness theorem. Weak solutions; distributions. Elliptic equations. The Laplace equation. The Poison equation for the pressure or current function. Equation of the wave in spatial variables. Method of spherical means, principle of Duhamel in methods and energy. Parabolic equations. Principle of the maximum. Analysis of uniqueness and regularity.

**Elliptical Equations **(4 Credit Units, 60 Credit Hours)

Syllabus: Harmonic functions; Example of Zaremba; Dirichlet problem in Rectangle, Method of separation of variables, Candidate to solution; Solutions Existence Theorem; Solution regularity: other models; Dirichlet Problem in Disk, Solution Candidate; Theorem of Existence of Classical Solution; General comments on other types of solutions.

**Integral Equations** (4 Credit Units, 60 Credit Hours)

Syllabus: Hilbert Spaces. Sesquilinear forms and linear applications. Compact Operators. Limited symmetric operators. Operators of Fredholm. Calculation of the Vectors and Values of the Integral Operator. General nuclei. Methods of successive approximations. Fredholm's alternative and applications to the study of Dirichlet-Neumann problems.

**Teaching Internship in Mathematics** (2 Credit Units, 30 Credit Hours)

Syllabus: Assistance in undergraduate courses offered by the Faculties of Mathematics and Statistics, aid to students with learning difficulties, resolution of exercises and related activities.

**Geometry of Sub-varieties** (4 Credit Units, 60 Credit Hours)

Syllabus: The fundamental equations and the fundamental theorem of isometric immersions. Umbilical and minimal immersions. Convex hypersurfaces. Sub-varieties with non-positive curvature. Reduction of codimension. Isometric immersions between spaces of constant sectional curvature. Local isometric rigidity. Global isometric rigidity. Isometric immersion composition. Subhumanities accordingly Euclidean. Conforming immersions. Other Topics.

**Riemannian Geometry** (4 Credit Units, 60 Credit Hours)

Syllabus: Riemannian metrics. Connection of Levi-Civitta. Geodesics. Normal and totally normal neighborhoods. Torsion of curvature. Covariante derivative of tensors. Jacobi fields and conjugate points. Isometric immersions; Equations of Gauss, Ricci and Codazzi. Complete Riemannian varieties; Theorem of Hopf-Rinow, Hadamard's Theorem. Constant curvature spaces. Variations of arc length; applications. Rauch's comparison theorem; Bonnet-Myers Theorem, Synge's Theorem, and other applications. The Morse Index Theorem. The place of the minimum points. Other topics.

**Riemannian Geometry of Homogeneous Spaces** (4 Credit Units, 60 Credit Hours)

Syllabus: Lie groups and algebras; bi-invariant metrics; enclosed representation; bilinear form of Killing. Homogeneous spaces; metric invariants to the left and bi-invariant. Symmetrical spaces; examples. Geometry of the Laplacian. Other topics.

**Groups and Lie Algebra** (4 Credit Units, 60 Credit Hours)

Syllabus: Lie groups, Lie algebras, exponential application, Lie subgroups and subalgebras, Cartan theorem of the closed subgroup, locally and globally isomorphic groups, simply connected groups, Campbell-Hausdorff formula, and the exponential differential. Homogeneous spaces. Structure: nilpotent and soluble groups, compact groups and introduction to semi-simple groups. Lie and ideal sub-algebras, attached representation, automorphisms and derivations, representations, nilpotent, soluble and semi-simple algebras.

**Riemannian immersions**** **(4 Credit Units, 60 Credit Hours)

Syllabus: Isometric immersions in Riemannian varieties. Theorems of Hartmann-Nireberg and Chern-Kuiper. Reduction of codimension. The fundamental equations and the fundamental theorem of isometric immersions. Totally geodetic, umbilical and minimal immersions. The axiom of r-planes and r-spheres. Convex hypersurfaces. Einstein hypersurfaces. Sub-varieties with non-positive curvature. Reduction of codimension. Isometric immersions between spaces of constant sectional curvature. Flat bilinear forms. Local and global isometric rigidity. Subhumanities accordingly Euclidean. Compliant immersions.

**Introduction to Dynamic Systems **(4 Credit Units, 60 Credit Hours)

Syllabus: Linear fields: hyperbolicity, genericity and stability. Density of stable fields. Local stability. Grobman-Hartman's theorem. Stable variety theorem, Poincaré transformation, transversality. The Kupka-Smale Theorem. Stability and density of the Morse-Smale fields. Peixoto's theorem. Morse functions, Morse-Smale gradient fields.

**Variational Methods**** **(4 Credit Units, 60 Credit Hours)

Syllabus: Functional differentiable in the sense of Frechet and Gateaux. Variation of the gradient of a Functional. Euler equations. Sufficient conditions of extremes. Functional studies of the classical calculation of variations. Minimization of functional values. Initiation of variational inequalities. Lions-Stampacchia Theorem.

**Semigroups and Partial Differential Equations**** **(4 Credit Units, 60 Credit Hours)

Syllabus: The exponential function. Continuous semigroups. Theorem of Hille Yosida. Exponential Formulas. Dissipative operators. Lummer-Phillips theorem. Compact and Holomorphic Semigroups. Theory of Pertubation. Abstract Cauchy Problem. Applications to Partial Differential Equations.

**Minimal Sub-varieties** (4 Credit Units, 60 Credit Hours)

Syllabus: Minimum immersions in riemannian varieties. First variation of the volume of a sub-variety. Minimal sub-varieties. Minimal sub-varieties in Euclidean spaces and spheres. Orbits of a group of minimal isometries and sub-variances. Kahlerian geometry and Wirtinger inequality. Second volume variation; the Index Theorem for minimum subvariances; stability. The Plateau Problem and its generalizations. Minimum surfaces. The Chern-Osserman Theorem. The Osserman Theorem on minimum surfaces with finite total curvature. Minimal surfaces layered. Other topics.

**Riemann surface**** **(4 Credit Units, 60 Credit Hours)

Syllabus: Definition of algebraic curves and Riemann surfaces. Meromorphic functions and meromorphic differentials. Singularities of flat algebraic curves, local structure. Theorem of normalization. Dividers, intersection numbers and Bezout's theorem. Hurwitz formula and formula of the flat curves genre. Riemann-Roch's Theorem. Abel-Jacobi's theorem. Applications. Coating spaces and the uniformization theorem. Relation with hyperbolic geometry. Relation between Riemann surfaces and algebraic curves.

**Special Topics of Geometry I-A**** **(4 Credit Units, 240 Credit Hours)

Syllabus: Advanced topics and recent development of Differential Geometry.

**Special Topics of Geometry I-B**** **(2 Credit Units, 120 Credit Hours)

Syllabus: Advanced topics and recent development of Differential Geometry.

**Special Topics of Geometry I-C**

**Special Topics of Geometry II-A**** **(4 Credit Units, 240 Credit Hours)

Syllabus: Advanced topics and recent development of Differential Geometry.

**Special Topics of Geometry II-B** (2 Credit Units, 120 Credit Hours)

Syllabus: Advanced topics and recent development of Differential Geometry.

**Special Topics of Geometry II-C**

**Special Topics of Mathematical Methods I** (4 Credit Units, 240 Credit Hours)

Syllabus: Topics of algebraic topology; Morse theory and applications to elliptical problems; Theory of the Degree of Brouwer and of Leray-Schauder. Applications to Differential Equations; Elliptical problems with lack of compactness. Lebesgue-Sobolev spaces with variable exponents and Applications to elliptic problems involving p (x) -Laplacian and Variable Exponents.

**Special Topics of Mathematical Methods II**** **(4 Credit Units, 240 Credit Hours)

**Special Topics of Mathematical Methods III**

**Special Topics in Applied Mathematics I-A** (4 Credit Units, 240 Credit Hours)

Syllabus: Finite Differences. Explicit Methods. Implicit Methods. Stability. Lax-Milgran equivalence theorem. Semidiscrete methods. Number Power.

**Special Topics in Applied Mathematics II-A** (4 Credit Units, 240 Credit Hours)

Syllabus: Semigrupos de operadores lineares. Geradores infinitesimais. Semigrupo de Contrações. Semigrupo dissipativo. Teorema de Hille-Yosida. Teorema de Lumer-Phillips.

**Special Topics in Applied Mathematics I-B**

**Special Topics in Applied Mathematics II-B**

**Special Topics in Applied Mathematics I-C**

**Special Topics in Applied Mathematics II-C**

**Topology of Varieties **(4 Credit Units, 60 Credit Hours)

Syllabus: Basic group. Coating spaces. CW-complexes. Introduction to Morse Theory: Morse's Lemma; Regular level surfaces; critical level and cell collage. DeRham's Cohomology: Poincaré's Lemma; Sequence of Mayer-Vietoris; applications. Degree of applications. Vector field index. Poincaré-Hopf's theorem. Duality of Poincaré.

**Differential topology**** **(4 Credit Units, 60 Credit Hours)

Syllabus: Differentiable varieties, definitions, examples, varieties with edge, tangent bundle. Differential applications: immersions, submersions, dives. Drive partitions. Whitney Dive Theorem. Function space: topology, approximations. Theorem of Sard. Transversality. Theory of intersection module 2. Oriented intersection theory. Poincaré-Hopf Index. Lefstchetz Fixed Point Theorem.