PPGMM-MG

Syllabus: Basic Theory: Rings and ideals, formal series and p-adic integers; Modules, graded modules, algebras, free modules; Exact sequences. Operations with Modules and Algebras: direct sum, tensor and exterior product, symmetric product. Localization, global-local properties, Nakayama's lemma, minimal bases; Noether normalization; Noetherian and Artinian rings and modules; Associated primes, primary decomposition; Ring completion, Artin-Rees theorem. Dimension theory, Hilbert functions; Additional topics: Planar rings and modules, depth; Cohen-Macaulay rings, normal and regular rings, Serre-Auslander-Buchsbaum theorem, Koszul complex.

Bibliography:

(a) M. Atiyah, I. Macdonald, Introduction to Commutative Algebra. Reading, Mass., Addison-Wesley, 1969.

(b) H. Borges, E. Tegan, ´Algebra Comutativa em Quatro Movimentos. Projeto Euclides, IMPA, 2015.

(c) H. Matsumura, Commutative Algebra. Reading, Mass., Benjamin-Commings, 1980.

(d) O. Zariski, P. Samuel, Commutative Algebra, vols. 1, 2. New York, Van Nostrand, 1960.

Syllabus: Normed vector spaces, Banach spaces, properties and examples of these spaces, linear forms, Hahn-Banach theorem, operators, uniform boundedness theorem, closed graph theorem, open mapping theorem, weak topology, Hilbert spaces and their properties, Riez and Lax-Milgran representation theorems, compact operators and their spectral theory.

Bibliography:

(a) M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. 1: Functional Analysis. Academic Press, New York, 1979.

(b) E. Kreysig, Introductory Functional Analysis with Applications. Wiley, New York, 1978.

(c) H. Brézis, Analyse Fonctionnelle. Masson, Paris, 1983.

(d) M. Schechter, Principles of Functional Analysis. New York, Academic Press, 1971.

(e) G. Botelho, D. Pellegrino, E. Teixeira. Fundamentos de An´alise Funcional, Textos Universit´arios, SBM, 2015.

Syllabus: Solving linear systems: direct and iterative methods. Solving nonlinear equations: fixed-point methods, Newton's method. Polynomial interpolation (Lagrange and Hermite methods), polynomial splines, error estimation. Numerical integration: methods based on polynomials and splines, Gaussian quadrature, methods based on extrapolation (Romberg method). Numerical solution of ordinary differential equations: initial value problems, single-step and multi-step methods. Numerical solution of ordinary differential equations: boundary value problems, finite difference methods, and/or collocation and/or finite element methods.

Bibliography:

(a) J. Stoer, R. Bulirsch, Introduction to Numerical Analysis. Springer, Berlin, 1980.

(b) E. Isaacson, H. B. Keller, Analysis of Numerical Methods. Wiley, 1966.

(c) H.R. Schwarz, Numerical Analysis - a Comprehensive Introduction. John Wiley & Sons, 1989.

Syllabus: Hyperbolic Periodic Points: Hartman's theorem, stable manifold theorem, stability, Kupka-Smale theorem. Classic examples: linear Anosovs, Smale horseshoe. Hyperbolic Sets: definition, continuity of spaces, hyperbolicity is an open property. Hyperbolic Attractors: Plykin attractor, solenoid. Tools: stable manifold theorem, slope lemma, shading lemma, local product structure and homoclinic classes. Axioms A: spectral decomposition, Omega stability, Omega explosions, Markov partitions. Generic dynamics: Franks' lemma, closing and connecting lemma (statement and consequences).

Bibliography:

(a) A. Katok, B. Hasselblatt, Introduction to Modern Dynamical Systems. Cambridge University Press, 1995.

(b) M. Shub, Global Stability of Dynamical Systems. Springer-Verlag, 1987.

(c) J. Palis, W. de Melo, Introdução aos Sistemas Dinâmicos. IMPA, 1978.

Syllabus: First-order nonlinear equations. The Cauchy problem for quasi-linear equations. Burgers' equation and the shock condition (Rankine-Hugoniot condition). Shock waves and rarefaction waves. Second-order hyperbolic equations. Singularity propagation. The wave equation. Shallow water equations. The Cauchy-Kowalevski theorem, Green's identity, and Holmgren's uniqueness theorem. Weak solutions; distributions. Elliptic equations. Poisson's equation for pressure or stream function. Topics in asymptotic analysis of PDEs: asymptotic analysis of solutions, asymptotic analysis of operators. Applications of conservation laws in various models.

Bibliography:

(a) R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. II, Partial Differential Equations. Intersciense Publisher, 1953.

(b) F. John, Partial Differential Equations. Springer-Verlag, 1982.

(c) G.B. Whitham, Linear and Nonlinear Waves. Wiley-Interscience, 1974.

(d) R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena. John Wiley & Sons, 1960.

(e) J. Smoller, Shock Waves and Reaction-Diffusion Equations, vol. 258. Springer Science & Business Media, 2012.

Syllabus: First-order equations, method of characteristics, collisions, weak solutions. Laplace's equation: maximum principle, Dirichlet's problem, fundamental solution, existence and uniqueness of solutions, Green's function, Poisson's equation: weak solution, existence and uniqueness. The heat equation: fundamental solution, mean value formula, maximum principle, uniqueness, regularity and existence of solutions, energy methods. Wave equation, the method of spherical averages, Duhamel's principle, energy methods. Fourier transform in L2, Plancherel's theorem, basic properties and applications.

Bibliography:

(a) R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. I, II. Wiley- Interscience, New York, 1953 e 1962.

(b) E. DiBenedetto, Partial Differential Equations. Birkhauser, Boston, 1995.

(c) L.C. Evans, Partial Differential Equations, second edition. Graduate Studies in Mathematics volume 19, AMS, 2010.

(d) J. John, Partial Differential Equations, 4th edition. New York, Springer, 1982.

Syllabus: Riemannian Metrics. Affine connection, curvature tensor, covariant derivative of tensors. Geodesics, geodesic flow, exponential mapping, normal neighborhoods, normal coordinates, minimizing properties of geodesics. Jacobi fields and conjugate points. The Riemann curvature tensor, curvature tensor symmetries, sectional curvature, Ricci and scalar curvatures. Isometric immersions: Gauss, Ricci and Codazzi equations. Complete manifolds, Hopf-Rinow Theorem, Hadamard Theorem. Spaces of constant curvature. Variations of arc length, Bonnet-Myers Theorem, Synge Theorem, Rauch Comparison Theorem. Morse Index Theorem.

Bibliography:

(a) M. Do Carmo, Geometria Riemanniana. Projeto Euclides, IMPA.

(b) S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry. Universitext, Springer.

(c) W. Klingenberg, Riemannian Geometry. W. de Gruyter, Berlin.

(d) M. Lee, Riemannian Manifolds: an Introduction to Curvature. Graduate Texts in Mathematics, Springer.

(e) P. Petersen, Riemannian Geometry. Graduate Texts in Mathematics, Springer- Verlag.

Syllabus: Introduction: Purposes of statistical inference. Population and sample. Random sample. Parameters and statistics. Sampling distributions. Families of distributions: location-scale family and exponential family. Principle of data reduction: Sufficient statistics, minimal statistics, complete statistics. Ancillarity. Sufficiency and completeness in the exponential family. Basu's theorem. Point Estimation: Concepts of estimator. Estimation methods - method of moments, least squares method and maximum likelihood method. Desirable properties in an estimator - unbiasedness and minimum variance. Uniformly unbiased minimum variance estimators. Confidence Interval Estimation: Introduction, methods for finding intervals, methods for comparing intervals/optimal intervals and asymptotic intervals. Hypothesis testing: The problem of hypothesis testing. Null hypothesis and alternative hypothesis. Type I and Type II errors. Critical region and acceptance region. Significance level and p-value. Power function. Methods for finding tests - likelihood ratio test and generalized likelihood ratio tests. Methods for evaluating tests - most powerful tests and uniformly most powerful tests. Relationship between tests and confidence intervals. Asymptotic tests.

Bibliography:

(a) P.J. Bickel, K.A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, vol. I, second edition. Prentice Hall, 2000.

(b) G. Casella, R.L. Berger, Statistical Inference, second edition. DuxburyPress, 2002.

(c) R. Hogg, A. Craig, Introduction to Mathematical Statistics, 6th edition. Pearson Prentice Hall, 2005.

(d) H. Larson, Introduction to Probability Theory and Statistical Inference. New York, John Willey, 1969.

(e) A.M. Mood, F.A. Graybill, D.C. Boes, Introduction to the Theory of Statistics, third edition. New York, MacGraw-Hill Publishing Company, 1974.

(f) E. Lehmann, G. Casella, Theory of Point Estimation, second edition. Springer, New York, 1998.

(g) E. Lehmann, Testing Statistical Hypotheses, second edition. Springer, New York, 1997.

(h) Y. Pawitan, In All Likelihood: Statistical Modelling and Inference Using Likelihood. Oxford, Clarendon, New York, Oxford University Press, 2001.

Syllabus: Symmetry groups (Galileo and Poincaré), classical mechanics in the formulation with reparametrization invariance. (Non-)dynamic variational problems, Lagrangian and Hamiltonian formalism. Dirac theory for constrained systems. Elements of General Relativity, smooth manifold, Riemann, symplectic and Poisson geometries. Hamiltonian systems on Poisson manifolds and integrals of motion. Dynamics of particles in curved space.

Bibliography:

(a) V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. Springer, New York, NY, 1989.

(b) D.M. Gitman, I.V. Tyutin, Quantization of Fields with Constraints. Springer, Berlin, 1990.

(c) A.A. Deriglazov, ClassicalMechanics: Hamiltonian and Lagrangian Formalism, 2nd ed. Springer, 2017.

(d) S. Weinberg, The Quantum Theory of Fields, vol. 1. Cambridge University Press, Cambridge, 1995.

(e) L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields. Pergamon Press, Oxford, 1980.

(f) P.J. Olver, Applications of Lie Groups to Differential Equations. Springer, New York, NY, 1986.

(g) A.T. Fomenko, V.V. Trofimov, Integrable Systems on Lie Algebras and Symmetric Spaces, Series: Advanced Studies in Contemporary Mathematics. Gordon and Breach Science Publishers, 1988.

(h) A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, vol. 1. Birkhauser Verlag, 1990.

Syllabus: Families of sets and the Monotone Class Theorem. Measurable functions and probability spaces. Probability measures, random variables and their distribution functions. Carathéodory's extension theorem, integration and properties of the integral. Mathematical expectation. Convergence theorems. Independence. Kolmogorov's extension theorem. Conditional expectation. Radon-Nikodym theorem. Convergence in probability and almost certain convergence. Weak Law of Large Numbers, Borel-Cantelli lemmas and Strong Law of Large Numbers. Convergence in distribution, characteristic functions and their properties, uniqueness and inversion. Lyapunov's theorem. Lindeberg-Feller theorem.

Bibliography:

(a) P. Billingsley, Probability and Measure. Wiley, New York, 1995.

(b) K.L. Chung, A Course in Probability Theory, second edition. Academic Press, 1974.

(c) R. Durrett, Probability: Theory and Examples. Duxbury Press, 1988.

(d) W. Feller, An Introduction to Probability Theory and its Applications, vol. I, II, second edition. Wiley, 1971.

(e) B. James, Probabilidade: um Curso em Nível Intermediário. Projeto Euclides, 2004.

Syllabus: Rings of algebraic integers, discriminants, integral bases, quadratic and cyclotomic fields, Dedekind domains, ideal decomposition, Galois extensions of number fields, geometric methods, Dirichlet's unit theorem, finiteness of the class group.

Bibliography:

(a) D. Marcus, Number Fields. Springer-Verlag, 1977.

(b) O. Endler, Teoria de Números Algébricos. Projeto Euclides, 2014.

(c) A. Fr¨olich, M.J. Taylor, Algebraic Number Theory. Cambridge, 1991.

Syllabus: Basic concepts of dynamical systems: orbits, limit sets, non-wandering sets, transitivity. Examples. Invariant measures. Weak topology in probability space. Krillov-Bogolyubov theorem. Poincaré theorem. Uniquely ergodic systems. Equidistribution. Examples: irrational rotation, skew products. Ergodicity. Definition and equivalent characterizations. Examples: Torus automorphisms. Bernoulli shifts. Ergodic decomposition. Koopman operator. Von Neumann ergodic theorem. Birkhoff ergodic theorem. Applications: strong law of large numbers. Mixing systems. Spectral equivalence of dynamical systems. Spectral theorem for unitary operators. Systems with absolutely continuous spectrum. Stochastic processes and representation of dynamical systems. Ergodic classification and hierarchy. Kolmogorov automorphisms. Entropy. Definition and basic properties. Sinai-Kolmogorov theorem. Entropy calculation. Shannon-McMillan-Breiman theorem.

Bibliography:

(a) C. Gutierrez, Introducion a la Teoria Ergodica. Sociedad Matematica de Peru, 2009.

(b) R. Mané, Ergodic Theory and Differentiable Dynamics. Springer-Verlag, 1987.

(c) K. Petersen, Ergodic Theory. Cambridge University Press, 1983.

(d) P. Walters, An Introduction to Ergodic Theory. Spinger-Verlag, 1982.

Syllabus: Differentiable manifolds, submanifolds, Whitney Embedding Theorem, differential forms, integration of forms, Stokes' Theorem, de Rham cohomology, Brower degree, homotopy invariance, homotopic classification of maps, number of intersections, singularity indices of vector fields, Poincaré-Hopf Theorem, Lefschetz Theorem, bundles and vector bundles, operations with vector bundles, curvatures and connections.

Bibliography:

(a) M. Hirsch, Differential Topology. Graduate Texts in Mathematics, 33. Springer- Verlag, New York, 1994.

(b) E.L. Lima, Introdução à Topologia Diferencial. Rio de Janeiro, IMPA, 2005.

(c) I.B. Madsen, J. Tornehave, From Calculus to Cohomology. Cambridge University Press, 1997.

(d) J. Milnor, Topology from the Differentiable Viewpoint. Charlottesville, Princeton Univ. Press, 1969.

Syllabus: Differentiable manifolds, manifolds with boundary, orientable manifolds, immersions and submanifolds, transversality, partitions of unity and applications such as the existence of Riemannian metrics, examples of vector bundles, tangent bundles and cotangent bundles, Lie groups, quotient manifolds, homogeneous manifolds. Vector fields and Frobenius' Theorem. Additional topics: Whitney topology, Sard's Theorem and Whitney's Embedding Theorem.

Bibliography:

(a) J. Lee, Introduction to Smooth Manifolds. Springer-Verlag, 2002.

(b) E.L. Lima, Variedades Diferenci´aveis, MON-15, IMPA,1973.

(c) M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 2. Publish or Perish, 1975.

(d) F. Warner, Foundations of Differentiable Manifolds and Lie Groups. Springer- Verlag, 1983.

Syllabus: Artinian and Noetherian rings; Simple and semisimple rings and modules. Structure of semisimple rings: the Wedderburn-Artin Theorem; Structure of simple Artinian rings; The Jacobson radical: its various characterizations and J-semisimplicity; Simple central algebras: scalar extensions and semisimplicity; the Skolem-Noether Theorem; the Frobenius Theorem. Primitive rings and Jacobson's density theorem. Introduction to PI-theory: algebras with polynomial identities, multilinear polynomials and T-ideals; identities and central polynomials in matrix algebras; Kaplansky's Theorem. Finite-dimensional associative algebras and their representations: path algebras; Morita Theory. Separable algebras and the Wedderburn-Malcev theorem.

Bibliography:

(a) B. Farb, R.K. Dennis, Noncommutative Algebra. Springer-Verlag, 1993.

(b) T.Y. Lam, A First Course in Noncommutative Rings. GTM 31, Springer- Verlag, 2001.

(c) S. Lang, Algebra. GTM 211, Springer-Verlag, 2003.

(d) C.P. Milies, S.K. Sehgal, An Introduction to Group Rings. Kluwer Academic Publishers, 2002.

(e) F. Lorenz, Algebra, vol. 2. Springer-Verlag, 2006.

(f) A. Giambruno, M. Zaicev, Polynomial Identities and Asymptotic Methods. AMS, 2005.

Syllabus: Definitions and initial examples; Ideals and homomorphisms; Solvable and nilpotent Lie algebras; Engel's and Lie's theorems; Semisimple Lie algebras; The Killing form; Cartan's criteria; Root spaces and Dynkin diagrams; Weyl groups; Enveloping algebras; PBW's theorem; Representations of semisimple Lie algebras. Connections with Lie groups. Modular Lie algebras and restricted algebras.

Bibliography:

(a) J.E. Humphreys, Introduction to Lie Algebras and Representation Theory. Springer, 1994.

(b) L.A.B. San-Martin, Álgebras de Lie. Unicamp, 2010.

(c) R. Carter, Lie Algebras of Finite and Affine Type. Cambridge University Press, 2005.

(d) J. Dixmier, Enveloping Algebras. AMS, GSM 11.

(e) H. Strade, R. Farnsteiner, Modular Lie Algebras and Their Representations. Marcel Dekker, 1988.

Syllabus: Tensor Calculus. Laplace-Beltrami Operator. Bochner-Weitzenbock Formula. Laplacian Comparison. Bishop-Gromov Volume Comparison. Cheeger-Gromoll Decomposition Theorem. Gradient Estimation of Harmonic Functions. Yau-Liouville Type Theorem for Manifolds with Non-Negative Ricci Curvature. Laplacian Spectrum. Eigenvalue Comparison Theorems. Cheng Stiffness Theorem. Stability of Minimal Hypersurfaces. Nonlinear Methods: Variational, Sub- and Supersolutions, Continuity Method, Heat Equation, Leray-Schauder Degree, Perturbation Techniques. Applications: Hodge Theory, Prescribed Gaussian Curvature, Dirichlet Problem for Minimal Surfaces, Ricci Flow on Surfaces, Harmonic Maps. Other Topics.

Bibliography:

(a) T. Aubin, Some Nonlinear Problems in Riemannian Geometry. Springer, 1998.

(b) D. Gilbarg, N.L. Trudinger, Elliptic Partial Differential Equations of Second Order, 3rd edition. Springer, 1998.

(c) J. Jost, Riemannian Geometry and Geometric Analysis. Berlin Heidelberg, New York, Springer-Verlag, 1995.

(d) P. Li, Lecture Notes on Geometric Analysis. University of California, 1992.

(e) R. Schoen, S.T. Yau, Lectures on Differential Geometry. International Press, 1994.

Syllabus: Quasi-coherent and coherent bundles; Fiber bundles; Cech and Grothendieck cohomologies, Grothendieck annulment theorem; Cohomology of projective schemes, Serre's theorem, Fiber cohomology. Differential bundle, smoothness criterion, regular immersions and local complete intersection. Canonical bundle, Extgroups and Bundles, Grothendieck duality. Weil and Cartier divisors, Linear systems and morphisms. Blowing-up. Riemann-Roch theorem for curves, Grothendieck-Riemann-Roch theorem. Additional topics.

Bibliography:

(a) R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, Springer Verlag, 1997.

(b) Q. Liu, R. Erne, Algebraic Geometry and Arithmetic Curves. Oxford Graduate Texts in Mathematics, 2006.

Syllabus: Enumerative Combinatorics and Graph Theory: Paths, cycles, and trees. Matchings. Colorings and chromatic number. Inclusion-exclusion and lattices. Pólya counting. Polynomial invariants of graphs. Extremal Combinatorics and Ramsey Theory: Ramsey's theorem. Mantel and Turán's theorems. Erdős-Stone theorem. Erdős-Simonovits stability. Probabilistic Method: Introduction to discrete probability. The first moment method and modifications. The second moment method. Random graphs. Classical conjectures of modern combinatorics.

Bibliography:

(a) M. Bóna, A Walk Through Combinatorics: an Introduction to Enumeration and Graph Theory. WPSC, 2011.

(b) B. Bollobás, Modern Graph Theory. Springer, 1998.

(c) J.A. Bondy, U.S.R. Murty, Graph Theory. Springer, 2008.

(d) R. Morris, R.I. Oliveira, Extremal and Probabilistic Combinatorics. IMPA, 2013.

(e) N. Alon, J. Spencer, The Probabilistic Method. Wiley, 2004.

(f) R. Stanley, Enumerative Combinatorics, vol. I. Cambridge, 2011.

Syllabus: Sobole spaces, approximations by smooth functions, extension and trace theorems. Gagliardo-Niremberg-Sobolev and Morrey inequalities. Rellich-Kondrachov compactness theorem. Poincaré inequality. H−1 space and spaces involving time. Second-order linear elliptic equations: weak solutions, Lax-Milgram theorem, energy estimates, Fredholm alternative, regularity, maximum principle, Harnack inequality and eigenvalues. Second-order linear parabolic equations: weak solutions, regularity and maximum principle. Second-order linear hyperbolic equations: weak solutions, regularity and propagation of singularities. Semigroups: definition and elementary properties, Hille-Yosida theorem and applications.

Bibliography:

(a) H. Brézis, Analyse Fonctionnelle. Masson, Paris, 1983.

(b) L.C. Evans, Partial Differential Equations. AMS, Graduate Studies in Mathematics, 2010.

(c) D. Gilbarg, N.L. Trudinger, Elliptic Partial Differential Equations of Second Order, 3rd edition. Springer, 1998.

(d) J. Jost, Partial Differential Equations. Springer, Graduate Texts in Mathematics, 2013.

(e) G. Leoni, A First Course in Sobolev Space. AMS, Graduate Studies in Mathematics, 2009.

Syllabus: Holomorphic Foliations: Regular foliations, Pfaff systems and singular foliations, Baum-Kupka theorem, Foliations in projective spaces; Integrability of Pfaff systems: Jouanolou-Ghys integrability theorem, Non-existence theorem of invariant algebraic varieties; Characteristic classes of holomorphic foliations: Baum-Bott theorem, Bott annihilation theorem, The principle of localization of characteristic classes, Generalized Camacho-Sad index theorem; Rudiments of foliated Mori theory: Riccati turbulent foliations and fibrations, Nef model of a foliation, Miyaoka rationality theorem, Rationally connected foliations; Frobenius-Demailly theorem.

Bibliography:

(a) M. Brunella, Birational Geometry of Foliations. Publicações Matemáticas do IMPA, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 2004.

(b) C. Camacho, A. Lins Neto, Teoria Geom´etrica das Folhea¸c˜oes, Projeto Euclides, 1979.

(c) J.-P. Demailly, On the Frobenius Integrability of Certain Holomorphic p-Forms. Complex geometry (Gottingen, 2000), Springer, Berlin, 2002.

(d) J. Jouanolou, Équations de Pfaff Alg´ebriques. Lecture Notes in Mathematics 708, 1979.

Syllabus: Classical Part: Affine and projective manifolds, Noether normalization, Cohen-Seidenberg going-up, Hilbert's Nullstellensatz, morphisms, rational and birational maps, smooth manifolds and tangent space, Jacobian criterion, blowing-up along ideals, Grassmannians, Hilbert polynomials, Bézout's theorem. Projective embeddings. Introduction to schemes: Basic sheaf theory, ring spectra, ringed spaces, schemes, scheme morphisms, projective schemes, Noetherians; reduced and integer schemes; scheme dimension, fiber product and change of basis, separate, proper and projective morphisms. Normal and regular schemes, Jacobian criterion, flat morphisms, fiber dimension, ètale and smooth morphisms.

Bibliography:

(a) R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, Springer Verlag, 1997.

(b) Q. Liu, Algebraic Geometry and Arithmetic Curves. Oxford Graduate Texts in Mathematics, 2006.

(c) D. Mumford, The Red Book of Varieties and Schemes. Lecture Notes in Mathematics, 1999.

Syllabus: Differential calculus on complex manifolds: Holomorphic functions on complex manifolds, Vector bundles, Connections and Curvature, Holomorphic bundles; Characteristic classes via Chern-Wei; Kahler manifolds: Kahler identities, Hodge theory on Kahler manifolds, Lefschetz hard theorem, Kodaira embedding theorem.

Bibliography:

(a) A. Neeman, Algebraic and Analytic Geometry. London Math. Soc. Lec. Note Series 345, 2007.

(b) J.-P. Demailly, Complex Analytic and Algebraic Geometry. Institut Fourier, 2012.

(c) P.A. Grifths, J. Harris, Principles of Algebraic Geometry. Wiley, New York, 1978.

Syllabus: Homotopy, contractible spaces. Paired and relative homotopy. Path homotopy, fundamental group, simply connected spaces, properties of the fundamental group. Fundamental groups of the circle, of real and complex projective spaces. Fundamental group of classical groups. Covering spaces, fundamental lifting theorem, covering automorphisms. Van Kampen's theorem and the existence theorem of covering spaces and applications.

Bibliography:

(a) E.L. Lima, Grupo Fundamental e Espaços de Recobrimento. Projeto Euclides, IMPA, 1998.

(b) W. Massey, A Basic Course in Algebraic Topology. Graduate Text in Mathematics 127, Springer-Verlag, New York, 1991.

(c) E.H. Spanier, Algebraic Topology. Springer, New York, 1966.

Syllabus: Nilpotent groups, characterizations and central series (descending and ascending). p-groups, central series in p-groups. Solvable groups. The Frattini subgroup and its characterizations. Permutation groups, matrix groups, examples of simple groups. Complex representations of finite groups; characters; orthogonality; induced character; Frobenius reciprocity; integrality; semisimplicity and Maschke's Theorem; Clifford's Theorem. Burnside's Theorem on groups of order pa.qb. Modular characters and representations. Sesquilinear forms; classical groups. Group extensions.

Bibliography:

(a) D.J.S. Robinson, A Course in the Theory of Groups, 2nd edition. Springer, 1996.

(b) G. James, M. Liebeck, Representations and Characters of Groups. Cambridge University Press, 2001.

(c) I.M. Isaacs, Character Theory of Finite Groups. Academic Press, 1976.

(d) L.C. Grove, Classical Groups and Geometric Algebra. American Mathematical Society, 2001.

(e) H. Kurzweil, B. Stellmacher, The Theory of Finite Groups (An Introduction). Springer, 2004.

Syllabus: Classical Currie-Weiss theory (mean-field approximation) for ferromagnetic systems, critical exponents. The Ising model: Gibbs states at infinite volume, correlation inequalities, phase diagrams. Equilibrium states and the Dobrushin-Lanford-Ruelle equations: equilibrium states, Dobrushin's uniqueness theorem, exponential decay of truncated correlations, analyticity of free energy, applications to the Ising model at high and low temperatures. Other topics: Series expansion methods, the Lee-Yang theorem, cluster expansion, the Simon-Lieb inequality, models with continuous symmetry.

Bibliography:

(a) S.Friedli, I, Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge University Press, 2017.

(b) B. Simon, The Statistical Mechanics of Lattice Gases, vol. I. Princeton University Press, Princeton, 1993.

(c) H.O. Georgii, Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin, 1988.

(d) R.B. Israel, Convexity in the Theory of Lattice Gases. Princeton University Press, Princeton, 1979.

(e) J. Glimm, A. Jaffe, Quantum Physics: a Functional Integral Point of View, 2nd edition. Springer, New York, 1987.

(f) D. Ruelle, Statistical Mechanics. W. A. Benjamin, London, 1969.

Syllabus: Mathematical foundations of quantum mechanics; The Schrödinger equation; Self-adjoint operators and spectrum; The spectral theorem and applications; Quantum dynamics; Asymptotic behavior of solutions; Schrödinger operators; Many-particle systems; Semiclassical analysis; Quantum statistics; Second quantization; Disordered systems; Spectral analysis; Calculus of variations; Information and computation; Renormalization; Other topics.

Bibliography:

(a) S.J. Gustafson, I.M. Sigal, Mathematical Concepts of Quantum Mechanics, segunda edi¸c˜ao. Universitext, Springer, 2011.

(b) O. Buhler, A Brief Introduction to Classical, Statistical and Quantum Mechanics. Courant Series, Volume 13, AMS, 2006.

(c) C.R. de Oliveira, Intermediate Spectral Theory and Quantum Dynamics. Birkhauser, 2009.

(d) Hislop, Sigal, Introduction to Spectral Theory: With Applications to Schr¨odinger Operators. Springer, 1996.

Syllabus: Variational Methods: Direct Minimization Method, Ekeland's Variational Principle, Deformation Lemma, Mountain Step Theorem, General Minimax Principle, Poincaré and Sobolev Inequalities, Applications: Existence for elliptic PDEs and solution of Yamabe's problem. Topological Methods: Brouwer degree theory: finite dimension, Leray-Schauder degree theory: infinite dimension, Applications: Borsuk-Ulam and Brouwer-Schauder fixed-point theorems, Applications: Existence for elliptic PDEs.

Bibliography:

(a) K. Deimling, Nonlinear Functional Analysis. Springer, 1985.

(b) L.C. Evans, Partial Differential Equations. AMS, 1998.

(c) I. Fonseca, W. Gangbo, Degree Theory in Analysis and Applications. Clarendon Press, Oxford, 1995.

(d) O. Kavian, Introduction a la Theorie des Points Critiques et Applications aux Problemes Elliptiques. Springer, 1993.

Syllabus: Markov Chains: Transition matrix, Markov property, examples, recurrence, transientness, and classification. Positive recurrent and zero recurrent states. Equilibrium, invariant measures. Convergence to equilibrium, coupling. The Ergodic Theorem for Markov Chains. Martingales: basic properties, the Optional Sampling Theorem, Doob's Inequality, Lp inequality. Convergence theorems. L1-bounded martingales. Closed and uniformly integrable martingales. Lévy's Theorem for reverse martingales. Applications: Strong Law of large numbers. Doob decomposition. Stationary processes: measure-preserving transformations, Recurrence Theorem. Birkhoff's Ergodic Theorem. Ergodic and mixing transformations. Applications. Weak convergence: Motivation: random walk and Brownian motion. Probability measures in metric spaces. Portmanteau's Theorem, Prohorov's Theorem. Weak compactness and rigidity in C[0,1]. Construction of the Wiener measure and Donsker's Invariance Principle.

Bibliography:

(a) P. Billingsley, Convergence of Probability Measures. Wiley, New York, 1968.

(b) P. Billingsley, Probability and Measure. Wiley, New York, 1995.

(c) L. Breiman, Probability. Classics in Applied Mathematics 07, SIAM, 1992.

(d) R. Durrett, Probability: Theory and Examples. Duxbury Press, 1988.

Syllabus: Continuous-time martingales: definitions, examples, convergence theorem, stopping times, optional sampling theorem, fundamental inequalities and applications; Markovian processes: the strong Markov property. Feller processes; Stochastic integration: the construction of the stochastic integral, Ito's formula, Girsanov's theorem; Notions of stochastic differential equations.

Bibliography:

(a) R. Durrett, Stochastic Calculus: a Practical Introduction. Probability and Stochastic Series, CRC Press, 1996.

(b) H.-H. Kuo, Introduction to Stochastic Integration. Universitext, Springer, 2000.

(c) P. M¨orters, Y. Peres, Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge, 2010.

Syllabus: Characteristic exponents and linear cocycles. Lyapunov exponents. Oseledets' theorem. Examples: Schrödinger cocycles and the derivative cocycle. Pesin theory. Invariant manifolds and non-uniformly hyperbolic systems. Examples. Hyperbolic flows. The geodesic flow on surfaces of negative curvature. Ergodicity: Hopf's argument. Hamiltonian and Lagrangian systems. Integrable systems. Twist mappings. Examples and properties. Homoclinic tangencies. The Newhouse phenomenon. Robustly transitive systems. Blenders.

Bibliography:

(a) R. Mané, Ergodic Theory and Differentiable Dynamics. Springer-Verlag, 1987.

(b) M. Viana, Lectures on Lyapunov Exponents. Springer, 2014.

(c) V.I. Arnold, Mathematical Methods of Classical Mechanics. Springer-Verlag, 1989.

(d) C. Ragazzo, M. Carneiro, S. Zanata, Introdução à Dinâmica de Aplicações Tipo Twist. CBM-IMPA, 2005.

(e) B. Hasselblatt, A. Katok, Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, 1995.

Syllabus: Definition of algebraic curves and Riemann surfaces, meromorphic functions and meromorphic differentials, Hurwitz formula, Riemann-Roch theorem, Abel-Jacobi theorem, applications, covering spaces and the uniformization theorem, relation to hyperbolic geometry, relation between Riemann surfaces and algebraic curves.

Bibliography:

(a) H. Farkas, I. Kra, Riemann Surfaces. Berlin, Springer-Verlag, 1980.

(b) P.A. Griffiths, Introduction to Algebraic Curves. Boston, AMS, Trans. Math. Monographs, 76, 1989.

(c) O. Forster, Lectures on Riemann Surfaces. Graduate Texts in Mathematics, 81, Springer-Verlag, New York-Berlin, 1981.

(d) R. Miranda, Algebraic Curves and Riemann Surfaces. Graduate Studies in Mathematics, 5. American Mathematical Society, Providence, RI, 1995.

Syllabus: Infinite prime numbers, proofs and first bounds for pi(x); Arithmetic, additive and multiplicative functions, functions: Möbius, Euler's Totient, von Mangoldt, divisors and sum of divisors; Dirichlet's product; Asymptotic properties and estimates; Logarithmic integral; Euler's Sum formula, Partial sums, Stirling's formula; Convolution method for approximating arithmetic functions, applications; Hardy-Ramanujan theorem on the normal number of divisors; Dirichlet's hyperbola method, applications; Distribution of primes: Chebyshev's estimate, Mertens' estimate and Prime Number Theorem (statement), consequences; Sieves: of Eratosthenes, Brun's combinatorics; Large sieve; Sieve of Turan, Sieve of Selberg, Applications; Additional topics: proof of the Prime Number Theorem, Methods of Complex Analysis, Riemann Zeta Function; Dirichlet's Theorem on primes in arithmetic progression.

Bibliography:

(a) T.A. Apostol, Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, 2010.

(b) A.C. Cojocaru, M.R. Murty, An Introduction to Sieve Methods and Their Applications. London Mathematical Society Student Texts, 2010.

(c) G.H. Hardy, E.M. Wright, An Introduction on the Theory of Numbers. Oxford University Press, 2008.

Syllabus: Bounded and unbounded linear operators. Integral operators, multiplication operators, and differential operators. The extension theorem for bounded operators. The Fourier transform in L1(Rn), S(Rn), and L2(Rn). L. Schwartz distributions, tempered distributions, and distributions with compact support. The Sobolev spaces Hs(Rn). Applications to linear and nonlinear evolution equations. Closed, closeable, symmetric, and self-adjoint operators. Resolvent and spectrum. Riemann-Stieltjes and Lebesgue-Stieltjes integrals. The spectral theorem for self-adjoint operators in the forms of spectral integrals, multiplication operators, and functional calculus. Stone's theorem.

Bibliography:

(a) M. Reed, B. Simon, Methods of Modern Mathematical Physics, vols. I e II. Academic Press, New York, 1972-1978.

(b) R. Iorio Jr, Fourier Analysis and Partial Differential Equations. Cambridge Studies in Advanced Mathematics, 2001.

(c) C.R. de Oliveira, Intermediate Spectral Theory and Quantum Dynamics. Birkh¨auser, Basel, Switzerland, 2009.

(d) E. Hille, Methods in Classical and Functional Analysis. Reading, Mass., Addison- Wesley Pub. Co., 1972.

(e) W. Rudin, Real and Complex Analysis. New York, McGraw-Hill, 1966.

Syllabus: Variable syllabus, established on a case-by-case basis and approved by the program coordinator.

Bibliography:

Variable bibliography, established on a case-by-case basis and approved by the Program Coordination.

Syllabus: Variable syllabus, established on a case-by-case basis and approved by the program coordinator.

Bibliography:

Variable bibliography, established on a case-by-case basis and approved by the Program Coordination.

Syllabus: Homology and cohomology groups: simplex and complex. Exact sequences. Relative homology. Mayer-Vietoris sequence. Simplicial homology: calculation of simplicial homology groups of the sphere, the projective plane, the Möbius strip, the torus, and the Klein bottle. Singular homology: Homotopic invariance. Excision theorem. Calculation of singular homology groups. Comparison of the Mayer-Vietoris theorem with the Van Kampen theorem. Applications of homology. Simplicial (singular) homology groups with arbitrary coefficients. Cohomology: Simplicial (singular) cohomology groups with arbitrary coefficients. Calculation of simplicial cohomology groups with integer coefficients. Homomorphisms between singular cohomology groups induced by continuous mappings. Mayer-Vietoris sequence for singular cohomology groups. Examples of use of the Mayer-Vietoris sequence. Orientation of topological manifolds. Poincaré duality theorem. Homotopy groups: n-dimensional homotopy groups of topological spaces. Calculation of the 2-dimensional homotopy group of the sphere and compact surfaces. Exact sequences of homotopy groups.

Bibliography:

(a) E.L. Lima, Homologia Básica. Projeto Euclides, 2009.

(b) J. Munkres, Elements of Algebraic Topology. Prentice Hall, 1975.

(c) E.L. Lima, Grupo Fundamental e Espa¸cos de Recobrimento. SBM, 2006.

(d) W. Massey, A Basic Course in Algebraic Topology. Springer-Verlag, 1991.

Syllabus: Elementary properties of holomorphic functions, power series, extensions of holomorphic functions, Hartogs' theorem, the d-operator, cohomology with differential forms, holomorphic domains, holomorphic convexity, plurisubharmonic functions, pseudoconvex domains, Hormander's theorem, pseudoconvex domains with differentiable boundary, Levi form.

Bibliography:

(a) S. Chabat, Introduction a l’Analyse Complexe, vol. 2. Ed. MIR, Moscou, 1990.

(b) R. Gunning, Introduction to Holomorphic Functions of Several Variables, vol. 1. Belmont, Ed. Wadsworth, 1990.

(c) L. Hormander, An Introduction to Complex Analysis in Several Variables. Princeton, Ed. Van Nostrand, 1966.

Syllabus: The syllabus will be determined for each case and must be approved by the doctoral program coordinator.

Bibliography:

The bibliography will be determined for each case and must be approved by the doctoral program coordinator.