Syllabi

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- MAT1157 - Single Variable Calculus A
**Syllabus:**Numbers, approximations of real numbers with sequences. Solution of equations and inequalities. Coordinate geometry, equations, lines, parabolas, equilateral hyperbola and circles. Functions and graphs. Affine function, quadratic function, polynomial and rational functions, roots. Algebra of functions. Trigonometry. Derivatives. Derivatives of Trigonometric functions. Differentiation rules, including the chain rule. Applications: Derivatives and shape of a graph, optimization problems, related rates, approximation of a function using polynomials function, Newton’s method.**Bibliography:**1) STEWART, J.,

*Cálculo*. Volumes 1, 6ª ed. Editora CENGAGE Learning, 2010.2) EDWARDS, C.H., PENNEY, D. E.,

*Cálculo com Geometria Analítica*. Prentice-Hall do Brasil, 1997.3) MALTA, I., PESCO, S. e LOPES, H.,

*Cálculo a uma Variável*. Volumes I e II. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, 2002.**Prerequisite:**None - MAT1158 - Single Variable Calculus B
**Syllabus:**Exponential and logarithm functions. Derivatives of inverse functions. L'Hôpital's rule. Definite integrals, indefinite integrals. The Fundamental Theorem of Calculus. Applications of integration.**Bibliography:**1) STEWART, J.,

*Cálculo*. Volumes 1, 6ª ed. Editora CENGAGE Learning, 2010.2) EDWARDS, C.H., PENNEY, D. E.,

*Cálculo com Geometria Analítica*. Prentice-Hall do Brasil, 1997.3) MALTA, I., PESCO, S. e LOPES, H.,

*Cálculo a uma Variável*. Volumes I e II. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, 2002.**Prerequisite:**MAT1157 or MAT1003 or MAT1005 - MAT1161 - Single Variable Calculus
**Syllabus**: Real numbers, decimal representation. Sequences. Functions and graphs. Continuity. Limit of a function, limits at infinity and asymptotes. Differentiability. Derivatives of elementary functions and their graphs. Higher-order derivatives. Optimization problems. Definite integral. The fundamental theorem of calculus, antiderivatives. Applications of integration.**Bibliography**:1) MALTA, I., PESCO, S. e LOPES, H., Cálculo a uma Variável. Volumes I e II. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, ano 2002.

2) STEWART, J., Cálculo. Volumes 1, 6ª ed. Editora CENGAGE Learning, 2010.

3) EDWARDS, C.H., PENNEY, D. E., Cálculo com Geometria Analítica. Prentice-Hall do Brasil, 1997.

**Prerequisite**: None - MAT1181 - Single Variable Calculus (Special Class)
**Syllabus**: Real numbers, decimal representation. Sequences. Functions and graphs. Continuity. Limit of a function, limits at infinity and asymptotes . Differentiability. Derivatives of elementary functions and their graphs. Higher-order derivatives. Optimization problems. Definite integral. The fundamental theorem of calculus, antiderivatives. Applications of integration. Additional topics.**Bibliography**:1) MALTA, I., PESCO, S. e LOPES, H., Cálculo a uma Variável. Volumes I e II. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, ano 2002.

2) STEWART, J., Cálculo. Volumes 1, 6ª ed. Editora CENGAGE Learning, 2010.

3) EDWARDS, C.H., PENNEY, D. E., Cálculo com Geometria Analítica. Prentice-Hall do Brasil, 1997.

- MAT1162 - Multivariable Calculus I
**Syllabus**: Continuity and differentiability of functions of 2 and 3 variables: Graph, domain, image. Linear approximation. Critical points. The Weierstrass's theorem. Lagrange multipliers. Double and triple integrals in Cartesian, polar, cylindrical and spherical coordinates.**Bibliography**:1)CRAIZER, M., TAVARES, G., Cálculo Integral a Várias Variáveis. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, 2002.

2)STEWART, J., Cálculo. Volumes 2, 6ª ed. Editora CENGAGE Learning, 2010.

3)BORTOSSI, H. J., Cálculo Diferencial a Várias Variáveis. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, 2002.

**Prerequisite**: MAT1181 or MAT1161 or MAT1158 or MAT1129 or MAT1004 or MAT1151 or MAT1171 - MAT1182 - Multivariable Calculus I (Special Class)
**Syllabus**: Continuity and differentiability of functions of 2 and 3 variables: Graph, domain, image. Linear approximation. Critical points. The Weierstrass's theorem. Lagrange multipliers. Double and triple integrals in Cartesian, polar, cylindrical and spherical coordinates. Additional topics.**Bibliography**:1) CRAIZER, M., TAVARES, G., Cálculo Integral a Várias Variáveis. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, 2002.

2) STEWART, J., Cálculo. Volumes 2, 6ª ed. Editora CENGAGE Learning, 2010.

3) BORTOSSI, H. J., Cálculo Diferencial a Várias Variáveis. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, 2002.

**Prerequisite**: MAT1181 or MAT1161 or MAT1158 - MAT1163 - Multivariable Calculus II
**Syllabus**: Vector functions and their derivatives: the Jacobian matrix. The general chain-rule and the inverse function theorem. Double and triple integrals: the change of variables formula. Parametric curves: velocity and tangent vectors. Path integrals, conservative fields and scalar potentials. Application: work and the kinetic energy theorem. Green´s theorem. Curl and divergence operators; the vector potential. Parametric surfaces: areas, tangent plane and graphs. The implicit function theorem. Surface integrals for scalar fields. Oriented surfaces and surface integrals for vector fields. Stokes and Gauss' theorems.**Bibliography**:1) CRAIZER, M., TAVARES, G., Cálculo Integral a Várias Variáveis. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, 2002.

2) STEWART, J., Cálculo. Volumes 2, 6ª ed. Editora CENGAGE Learning, 2010.

3) BORTOSSI, H. J., Cálculo Diferencial a Várias Variáveis. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, 2002.

**Prerequisite**: (MAT1200 and MAT1182) or (MAT1200 and MAT1162) or MAT1152 or MAT1172 - MAT1183 - Multivariable Calculus II (Special)
**Syllabus**: Vector functions and their derivatives: the Jacobian matrix. The general chain-rule and the inverse function theorem. Double and triple integrals: the change of variables formula. Parametric curves: velocity and tangent vectors. Path integrals, conservative fields and scalar potentials. Application: work and the kinetic energy theorem. Green´s theorem. Curl and divergence operators; the vector potential. Parametric surfaces: areas, tangent plane and graphs. The implicit function theorem. Surface integrals for scalar fields. Oriented surfaces and surface integrals for vector fields. Stokes and Gauss' theorems. Additional topics.**Bibliography**:1) CRAIZER, M., TAVARES, G., Cálculo Integral a Várias Variáveis. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, 2002.

2) STEWART, J., Cálculo. Volumes 2, 6ª ed. Editora CENGAGE Learning, 2010.

3) BORTOSSI, H. J., Cálculo Diferencial a Várias Variáveis. Coleção MatMídia. Edição Loyola, Editora PUC-Rio, 2002.

**Prerequisite**: (MAT1200 and MAT1182) or (MAT1200 and MAT1162) - MAT1154 - Differential and Difference Equations
**Syllabus**: Differential equations of the first order: geometric interpretation in terms of line fields, existence and uniqueness of solutions. Some resolution methods: Separable, exact and linear differential equations of the first order (homogeneous and nonhomogeneous). Linear difference equations of first order with constant coefficients. Linear differential and difference equations of the second order with constant coefficients. Linear systems in the plane. Power series resolution of differential equations.**Bibliography**:1) BOYCE, W. E.; DIPRIMA, R. C., Equações Diferenciais Elementares e Problemas de Valores de Contorno. 9o ed. LTC Editora, 2010.

2) FIGUEIREDO, D. G.; NEVES, A., Equações Diferenciais Aplicadas. IMPA. Colóquio Brasileiro de Matemática, 1979.

3) BRAND, L., Differential and difference equations. New York: J. Wiley, 1966.

**Prerequisite**: (MAT1200 and MAT1181) or (MAT1200 and MAT1161) or (MAT1200 and MAT1158) or (MAT1151 and MAT1200) or (MAT1151 and MAT1215) or (MAT1158 and MAT1215) or (MAT1161 and mat1215) or (MAT1002 and mat1200) or (MAT1004 and MAT1215) or (MAT1171 and MAT1200) or (MAT1171 and MAT1215) or (MAT1181 and MAT1215) or MAT1129MAT1200 - Linear Algebra I

- MAT4200 - Linear Algebra I
**Syllabus**: Systems of linear equations. Cartesian coordinates in two and three dimensions. Vectors, scalar product, determinants, vector product, triple product. Linear subspaces, basis. Linear maps, matrices. Eigenvalues and eigenvectors.**Bibliography**:1) ANTON, H.; RORRES, C., Álgebra Linear com Aplicações. Bookman, 2004.

2) LIMA, E. L., Coordenadas no plano: Geometria analítica, vetores e transformações geométricas. (Coleção do professor de matemática). Rio de Janeiro : Sociedade Brasileira de Matemática, 1992.

3) LIMA, E. L., Coordenadas no espaço. (Coleção do professor de matemática). Rio de Janeiro : Sociedade Brasileira de Matemática, IMPA, 1993.

**Prerequisite**: None - MAT4202 - Linear Algebra II
**Syllabus**: Invariant subspaces. Kernel, image and the rank–nullity theorem. LU decomposition. Least squares. The Gram–Schmidt process. QR decomposition. Eigenvalues and eigenvectors with numerical methods.**Bibliography**:1) ANTON, H., RORRES, C., Álgebra Linear com Aplicações. Bookman, ano 2004.

2) STRANG, G., Linear Algebra and its Applications. Harcourt Brace Jovanovich, 1988.

3) STEINBRUCH, A., WINTERLE, P., Álgebra Linear. Mc Graw-Hill, 1987.

**Prerequisite**: MAT4200 or MAT1200 or MAT1250 or MAT1260 or MAT1343

Courses with same content in the undergraduate and graduate programs are indicated by "* SD*" in each syllabus.

- MAT1060 - Introduction to the history of mathematics
**Syllabus**: A detailed study of certain key topics, such as Euclid's Elements, the creation of calculus by Newton and Leibniz and the discovery of Non-Euclidean Geometry. Brief presentation of other important moments with the intent of creating a general and broad view of the organic development of Mathematics. Other highlighted topics presented are the origins of concepts and notations, ways of thinking and different mathematicians points of view alongside history.**Bibliography**:1) Katz, “An Introduction to the history of mathematics”

**Prerequisite**: None - MAT1223 - Vector Spaces and Linear Transformations (
*SD*: MAT2215)**Syllabus**: Fields, vector spaces, bases, dimension, matrix algebra, linear operators. n-dimensional real and complex vector spaces as normed spaces. Gaussian elimination, determinants. Invertible matrices. Eigenvalues, eigenvectors, invariant subspaces. Characteristic polynomial. Diagonalization of operators. Real and complex Jordan forms. Inner product. Orthogonal bases. Singular value decomposition. Self-adjoint operators, symmetric matrices. Spectral theorem.**Bibliography**:1) LIMA, E. L,. Álgebra Linear. Rio de Janeiro: Coleção Matemática Universitária (SBM), 2004.

2) HOFFMAN, K., KUNZE, R., Álgebra Linear. Rio de Janeiro: LTC, 1979.

3) LANG, S., GOMIDE, E. F., Álgebra Linear. Brasília: Universidade de Brasilia. São Paulo: E. Blucher, 1971.

**Prerequisite**: MAT4200 or MAT1200 or MAT1250 or MAT1260 or MAT1343 - MAT1224 - Algebraic Structures I (
*SD*: MAT2220)**Syllabus**: Rings, polynomial rings, Ideals. Quotient rings. Homomorphisms. Field of fractions of an integral domain. Euclidian domains. Irreducibility of polynomials. Groups. Permutation groups. Matrix groups. Abelian groups. Homomorphisms and quotient groups. Group actions.**Bibliography**:1) GARCIA, A., LEQUAIN, Y., Elementos de Álgebra. Rio de Janeiro: Projeto Euclides, 2002.

2) ARTIN, M., Algebra. New Jersey: Prentice-Hall, 1991.

3) JACOBSON, N., Basic algebra. San Francisco: W. H. Freeman, 1974-1980.

**Additional Bibliography**:1) LANG, S., Algebra. 3ª ed. Reading, Mass.: Addison-Wesley, 1993.

**Prerequisite**: None - MAT1225 - Algebraic Structures II (
*SD*: MAT2221)**Syllabus**: Fields and Field extensions. Algebraic number fields. Finite fields. Characteristic of a field. Constructions by ruler and compass. Galois Theory. Examples of low degree. Resolution of polynomials equations of degree 3 and 4 in one variable. Solvable groups, resolution by radicals. Examples of equations that cannot be solved by radicals.**Bibliography**:1) Edwards, H. M., Galois Theory. New York: Springer, 1984

2) STEWART, I., Galois Theory. 3rd ed. Boca Raton, Fla.: Chapman & Hall, 2004.

3) JACOBSON, N., Basic algebra. San Francisco: W. H. Freeman, 1974-1980.

**Prerequisite**: MAT1224 - MAT1231 - Computational Linear Algebra (
*SD*: MAT2229)**Syllabus**: Vector and matrix norms, orthogonal projections. Matrix algebra algorithms with rounding error analysis. System of linear equations: LU decomposition, positive definite systems, band symmetric, bloc and sparse matrices. QR and SVD decompositions with applications. Iterative methods, Krylov subspace methods, conjugate gradient and related methods. Algorithms for eigenvalue decomposition.**Bibliography**:1) DEMMEL, J., Applied Numerical Linear Algebra. Philadelphia: SIAM, 1997.

2) GOLUB, G., VAN LOAN, C., Matrix Computations. Baltimore: Johns Hopkins University Press, 1989.

3) PENNY, J. E. T., LINDFIELD, G. R., Numerical Methods using Matlab. New York : E. Horwood, 1995.

**Prerequisite**: (MAT1202 and INF1001) or (MAT1223 and INF1001) or (MAT1202 and INF1005) or (MAT1223 and INF1005) - MAT1303 - Mathematical Elements for Computer Graphics (
*SD*: MAT2461)**Syllabus**: Geometric transformations; 3d interface; arcball and quaternions; curve drawing; sampling; basis of geometric data structure; rendering and shading; graphs of 2d and 3d functions; programming notions in C/C++ or python; openGL basics.**Bibliography**:1) KERNIGHAN, B.W., RITCHIE, D. M., C - A Linguagem de Programação - Padrão ANSI. Ed. Campus, 1990.

2) RUAS, V., Curso de Cálculo Numérico. Rio de Janeiro: Livros Técnicos e Científicos, 1982.

3) CORMEN, T. H., Algoritmos: teoria e prática. Rio de Janeiro: Campus, 2002.

**Prerequisite**: None - MAT1305 - Geometric Modeling (
*SD*: MAT2462)**Syllabus**: Splines; geometric interpolation; Delaunay triangulations; mesh data structure; parametric and implicit surfaces; boolean operations.**Bibliography**:1) FARIN, G., Curves and Surfaces for CAGD. Waltham: Academic Press. 1993.

2) MORTENSON, M.E., Geometric Modeling. Hoboken: Wiley, 1997.

3) BOTSCH, M., PAULY, M., KOBBELT, L., ALLIEZ, P., LEVY, B., BISCHOFF, S., RÖSSL, C., Geometric Modeling Based on Polygonal Meshes. SIGGRAPH Course Notes, 2006.

4) BOISSONNAT, J.-D., YVINEC, M., Algorithmic Geometry. Cambridge University Press. 1998.

5) FIGUEIREDO, L.H., Geometria Computacional. Colóquio Brasileiro de Matemática, IMPA 1991.

6) SORKINE, O., Laplacian Mesh Processing. Eurographics STAR, 2005.

**Prerequisite**: (MAT1163 and MAT1303) or (MAT1183 and MAT1303) - MAT1330 - Mathematics for Machine Learning (
*SD*)**Syllabus**: Zero-order, first-order and second-order optimization techniques: Random search, Gradient Descent and Newton’s method. Supervised Learning: Linear and Nonlinear regression, Two-Class Classification, Logistic Regression, Support Vector Machines. Unsupervised Learning: Principal Component Analysis, K-means Clustering. Neural Networks: Activation functions, Backpropagation algorithm.**Bibliography**:1) WATT, J.; BORHANI. R.; KATSAGGELOS, A. K., Machine Learning Refined. Foundations, Algorithms, and Applications. 2nd Ed. Cambridge: Cambridge University Press, 2020.

2) STRANG, G. Linear Algebra and Learning from Data. 1ª ed. Wellesley-Cambridge: Press,U.S., 2019.

3) HARRISON, M. Machine Learning Guia de Referencia Rapida: Trabalhando com Dados Estruturados em Python. 1ª ed; SÃO PAULO: Novatec Editora, 2019.

**Prerequisite**: MAT4161 and MAT4200. - MAT1406 - Scientific Computing and Differential Equations (
*SD*: MAT2447)**Syllabus**: Initial value problems: simple and multiple pass method, polynomial interpolation, stability and stiffness, linear and non-linear systems. Boundary value problems: finite difference method for linear problem and discretization. Methods for non-linear problems: shooting, projection, collocation, Garlekin and spline approximations. Explicit and implicit methods for elliptic, parabolic and hyperbolic equations. Fourier transforms. Discretization from integral form. Semi-discrete methods. Error and stability analysis.**Bibliography**:1) ISERLES, A., A first course in the Numerical Analysis of Differential Equations. CUP, 1996.

2) SMITH, G. D., Numerical Solution of Partial Differential Equations: Finite difference methods. Oxford : Clarendon Press, 1985.

3) GOLUB, G., ORTEGA, J., Scientific Computing and Differential Equations: an Introduction to Numerical Methods. Academic Press, 1992.

4) STRANG, G., Introduction to Applied Mathematics. Wellesley Cambridge Press, 1986.

**Additional Bibliography**:1) STRANG, G., Introduction to Applied Mathematics. Wellesley Cambridge Press, 1986

**Prerequisite**: MAT1202 or MAT1154 - MAT1907 - Introduction to Partial Differential Equation (
*SD*: MAT2907)**Syllabus**: Fourier series. Partial differential equation, heat equation, wave Laplace’s equation. Fourier method for initial value problems and boundary value problems.**Bibliography**:1) IÓRIO JÚNIOR, R. J., IÓRIO, V. B. de M., Equações Diferenciais Parciais. Rio de Janeiro: IMPA, 1988.

2) FIGUEIREDO, D. G., Análise de Fourier e Equações Diferenciais Parciais. Projeto Euclides, 1997.

3) FIGUEIREDO, D. G, NEVES, A. F., Equações Diferenciais Aplicadas. Coleção Matemática Universitária, 1997.

**Prerequisite**: (MAT1154 and MAT1604) or (MAT1154 and MAT1605) - MAT1414 - Introduction to Probability and Statistics (
*SD*: MAT2303)**Syllabus**: Axioms of Probability. Classical Probability Theory. Combinatorial Problems. Discrete and Continuous Random Variables. Independence. Expectation and Variance. The Law of Large Numbers. The Central Limit Theorem. Hypothesis testing. Parameter estimation. Regression.**Bibliography**:1) BARRY, R. J., Probabilidade: um Curso em Nível Intermediário. Rio de Janeiro: Projeto Euclides, 1981.

2) CHUNG, K. L., Elementary Probability Theory with Stochastic Processes. 3. ed. New York: Springer, 1979.

3) GNEDENKO, B. V., The Theory of Probability. Moscow: Mir Publ., 1969.

**Prerequisite**: MAT1604 or MAT1605 - MAT1502 - Introduction to Complex Analysis (
*SD*: MAT2502)**Syllabus**: Complex numbers. Definition and properties of elementary functions: power, exponential, logarithm and trigonometric complex functions. Analytic functions. Cauchy–Riemann equations. Integration. Cauchy formula. Maximum modulus principle. The fundamental theorem of algebra. Taylor and Laurent series. Classification of singularities. The residue theorem.**Bibliography**:1) SOARES, M., Cálculo em Uma Variável Complexa. Rio de Janeiro: Coleção Matemática Universitária (SBM), 2001.

2) CHURCHILL, R., Variável Complexa e suas Aplicações. McGraw Hill do Brasil, 1975.

3) ÁVILA, G., Funções de uma variável complexa. Rio de Janeiro. Livros Técnicos e Científicos, 1974.

**Prerequisite**: MAT1604 or MAT1605 - MAT1605 - Introduction to Analysis (
*SD*)**Syllabus**: Set and relation. Mathematical induction, proof by contradiction. Natural numbers. Cardinality and enumerability. Rational and real numbers. Limit, convergence of sequences and series. Topology of the real line: open, closed, compact, connected and dense sets. The Cantor ternary set. Continuous function: the Bolzano–Weierstrass theorem, the intermediate value theorem, uniform continuity.**Bibliography**:1) LIMA, E. L., Curso de Análise. Vol. I. Rio de Janeiro: Projeto Euclides, 1995.

2) BOAS, R. P., A Primer of real functions. 2. ed. Buffalo: Mathematical Association of America, 1972.

3) ÁVILA, G., Introdução à Análise Matemática. São Paulo: Ed. Edgard Blucher, 1999.

**Prerequisite**: None - MAT1606 - Real Analysis (
*SD*: MAT2620)**Syllabus**: Review of topology and continuity of real functions. The derivative, the Mean Value Theorem, L’Hôpital’s rule, Taylor approximants. Integration in the sense of Riemann. The Fundamental Theorem of Calculus. Sequences of functions. Power series and analytic functions. The Stone-Weierstrass theorem. The Theorem of Arzela-Ascoli. Introduction to harmonic analysis and Fourier series.**Bibliography**:1) LIMA, E. L., Curso de Análise. Vol. I. Rio de Janeiro: Projeto Euclides, 1995.

2) BOAS, R. P., A Primer of real functions. 2. ed. Buffalo: Mathematical Association of America, 1972.

3) ÁVILA, G., Introdução à Análise Matemática. São Paulo: Ed. Edgard Blucher, 1999.

**Prerequisite**: MAT1605 or MAT1604 - MAT1612 - Analysis on the R
^{n}Space (*SD*: MAT2624)**Syllabus**: Topology of N-dimensional euclidean spaces: metric structures, topological structures and the notion of completeness. Scalar fields, continuity, and the notion of derivative of a scalar valued function in R^N. The contraction principle, the Inverse Function Theorem and the Implicit Function theorem. The rank theorem and normal forms for mappings between euclidean spaces. Taylor’s formula for the approximant. Jordan measurable sets. The integral in the sense of Riemann, and the notion of integrable function. Fubini’s theorem and the change of variable formula for N-dimensional domains.**Bibliography**:1) LIMA, E. L., Curso de Análise. Vol. 2. Rio de Janeiro: Projeto Euclides, 2000.

2) LANG, S., Undergraduate Analysis. New York: Springer, 1997.

3) SPIVAK, M., Calculus on manifolds: a modern approach to classical theorems of advanced. Menlo Park, Calif. : W. A. Benjamin 1965.

**Prerequisite**: MAT1606 or MAT1610 - MAT1620 - Introduction to Fourier Analysis (
*SD*: MAT2640)**Syllabus**: The genesis of Fourier analysis. Basic properties of Fourier series: introduction to problems and examples; uniqueness of the Fourier series; convolutions and their properties; summability kernels. Convergence of Fourier series: review of elementary notions of vector spaces and inner product; convergence in the mean-square sense. Point convergence; a local result and an example of a continuous function with divergent Fourier series. The Fourier transform on the line: definition of the Fourier transform in the space of functions of moderate decrease; the Schwartz space; the Fourier inversion formula; Plancherel's theorem; The Poisson summation formula; the Heisenberg uncertainty principle; applications to EDPs.**Bibliography**:1) STEIN, E. M., SHAKARCHI, R., Fourier Analysis: An Introduction, Vol. I; Princeton: Princeton Lectures in Analysis, 2003.

2) GRAFAKOS, L., Classical Fourier analysis, Vol. 2; New York: Springer, 2008.

3) FIGUEIREDO, D. G. de, Análise de Fourier e Equações Diferenciais Parciais. IMPA; Rio de Janeiro: Coleção Projeto Euclides, 2018.

**Prerequisite**: MAT1223 and MAT1612 and MAT1502 (or MAT1510). - MAT1702 - Introduction to Topology (
*SD*: MAT2714)**Syllabus**: Metric spaces, Topological spaces. Continuity. Connected and compact spaces. Fundamental group. Covering spaces. Classification of surfaces.**Bibliography**:1) LIMA, E. L., Grupo Fundamental e Espaço de Recobrimento. Rio de Janeiro: Projeto Euclides, 1993.

2) MASSEY, W., A Basic Course in Algebraic Topology. New York: Graduate Texts in Mathematics (Springer), 1991.

3) LIMA, E. L., Elementos de Topologia Geral. IMPA, 1976.

**Prerequisite**: (MAT1153 and MAT1605) or (MAT1153 and MAT1604) or (MAT1173 and MAT1605) or (MAT1173 and MAT1604) or (MAT1163 and MAT1605) or (MAT1183 and MAT1605) or (MAT1604 and MAT1605) - MAT1811 - Differential Geometry (
*SD*: MAT2812)**Syllabus**: Planar and spatial curves. Frenet frame and applications. Surfaces.in Euclidean space. Calculus on surfaces: areas, isometries, conformal mappings. Orientation. Gauss normal map, curvatures, special lines (curvature lines, asymptotic lines, geodesics). Gauss egregium theorem. Gauss-Bonnet theorem and applications.**Bibliography**:1) DO CARMO, M., Geometria Diferencial de Curvas e Superfícies. Rio de Janeiro: Coleção Matemática Universitária (SBM), 2005.

2) MONTIEL, S., ROS, A., Curves and surfaces. Providence, RI: American Mathematical Society, 2005.

3) KÜHNEL, W., Differential Geometry. Rio de Janeiro: American Mathematical Society, 2002. SPIVAK, M., A comprehensive Introduction to differential geometry. Volume III. Publish or Perish, 1970.

**Prerequisite**: (MAT1153 and MAT1605) or (MAT1153 and MAT1604) or (MAT1173 and MAT1605) or (MAT1173 and MAT1604) or (MAT1163 and MAT1605) or (MAT1183 and MAT1605) or (MAT4163 and MAT1605) - MAT1901 - Ordinary Differential Equations (
*SD*: MAT2905)**Syllabus**: Differential equations of first order. Reduction of high order equations to first order systems. Existence and uniqueness of solutions. Dependence on the initial conditions. Extension of solutions. Linear systems with constant coefficients. Non-homogeneous linear equations and non-autonomous linear equations. Poincaré-Bendixson theorem.**Bibliography**:1) SOTOMAYOR, J., Lições de Equações Diferenciais Ordinárias. Rio de Janeiro: 2) Projeto Euclides, 1979.

2) HIRSH, M., SMALE, S., Differential Equations, Dynamical Systems and Linear Algebra. New York: Academic Press, 1974.

3) CODDINGTON, E. A., LEVINSON, N., Theory of ordinary differential equations. New York: McGraw-Hill, 1955.

**Prerequisite**: (MAT1154, MAT1223 and MAT1606) or (MAT1154, MAT1220 and MAT1606) or (MAT1154, MAT1220 and MAT1610) or (MAT1154, MAT1223 and MAT1610)

The Graduate Program Core Courses are indicated by “** DF**”; these courses have special rules - see the Graduate Program Regulations.

Courses with same content in the undergraduate and graduate programs are indicated by "** SD**" in each syllabus.

- MAT2215 - Linear Algebra (
*SD: MAT1223*)**Syllabus**: Fields, vector spaces, bases, dimension, matrix algebra, linear operators. n-dimensional real and complex vector spaces as normed spaces. Gaussian elimination, determinants. Invertible matrices. Eigenvalues, eigenvectors, invariant subspaces. Characteristic polynomial. Diagonalization of operators. Real and complex Jordan forms. Inner product. Orthogonal bases. Singular value decomposition. Self-adjoint operators, symmetric matrices. Spectral theorem.**Basic Bibliography**:1) Lima, E. L., Álgebra Linear, Coleção Matemática Universitária (SBM), 1999.

2) Hoffman, K.; Kunze, R., Álgebra Linear, LTC, 1979.

- MAT2220 - Algebraic Structures I (
*DF*;*SD*: MAT1224)**Syllabus**: Rings, polynomial rings, Ideals. Quotient rings. Homomorphisms. Field of fractions of an integral domain. Euclidian domains. Irreducibility of polynomials. Groups. Permutation groups. Matrix groups. Abelian groups. Homomorphisms and quotient groups. Group actions.**Basic Bibliography**:1) Garcia, A.; Lequain, Y., Elementos de Álgebra, Projeto Euclides, 2002.

2) Artin, M., Álgebra, Prentice-Hall, 1991.

- MAT2221 - Algebraic Structures II (
*SD: MAT1225*)**Syllabus**: Fields and Field extensions. Algebraic number fields. Finite fields. Characteristic of a field. Constructions by ruler and compass. Galois Theory. Examples of low degree. Resolution of polynomials equations of degree 3 and 4 in one variable. Solvable groups, resolution by radicals. Examples of equations that cannot be solved by radicals.**Basic Bibliography**:1) Edwards, H. M., Galois Theory, Graduate Texts in Mathematics, Springer, 1997.

2) Chambert-Loir, A., A Field Guide to Algebra, Undergraduate Texts in Mathematics, Springer, 2004.

- MAT2225 - Commutative Algebra
**Syllabus**: Ideals in commutative rings. Spectrum of a ring. Zariski topology. Radicals. Modules. Tensor product. Localization. Noetherian and Artinian rings. Primary decomposition. Support. Algebraic extensions, Noether's normalization theorem and Hilbert Nullstellensatz. Integral extensions, "going-up" and "going down" theorems. Discrete valutation rings. Invertible Ideals. Completion, Artin-Rees' lemma, Krull's theorem, Hensel's theorem. Dimension theory.**Basic Bibliography**:1) Atiyah, M.F.; Macdonald, I.G.Introduction to Commutative Algebra, Addison-Wesley, 1994.

2) Matsumura, H. Commutative Algebra, Benjamin-Cummings Pub Co, 1980.

3) Peskine, C. An algebraic introduction to Complex Projective Geometry: 1. Commutative Algebra, Cambridge University Press, 2009.

- MAT2229 - Computational Linear Algebra (
*DF; SD: MAT1231*)**Syllabus**: Vector and matrix norms, orthogonal projections. Matrix algebra algorithms with rounding error analysis. System of linear equations: LU decomposition, positive definite systems, band symmetric, bloc and sparse matrices. QR and SVD decompositions with applications. Iterative methods, Krylov subspace methods, conjugate gradient and related methods. Algorithms for eigenvalue decomposition.**Basic Bibliography**:1) Demmel, J., Applied Numerical Linear Algebra, SIAM, 1997.

2) Golub, G.; Van Loan, C., Matrix Computations, Johns Hopkins University Press, 1989.

- MAT2255 - Algebraic Geometry
**Syllabus**: Affine space. Closed algebraic subsets. Zariski topology. Regular functions. Sheaves. Algebraic Varieties. Morphisms. Projective Varieties. Properness theorem. Irreducible components. Rational functions. Finite morphisms. Dimension: Krull dimension, transcendence degree, Zariski tangent space. Krull's lemma. Local properties. Smooth points. Rational maps. Blow-up. Normalization. Dimension of the fibers. Bertini's theorem. Vector bundles. Canonical line bundle. Adjunction formula. Divisors, inversible sheaves, canonical divisor. Linear systems. Ample line bundles, immersions in the projective space. Coherent sheaves. Riemann-Roch for curves. Applications. Intersection numbers. Hodge index theorem. Birational maps of surfaces.**Basic Bibliography**:1) Shafarevich, I.R. Basic Algebraic Geometry, Springer-Verlag, 1994.

2) Kempf, G. Algebraic Varieties, Cambridge University Press, 1993.

3) Smith, K.E., Kahampää, L., Kekäläinen, P., Traves, W. An Invitation to Algebraic Geometry, Springer-Verlag, 2000.

- MAT2256 - Introduction to Complex Geometry
**Syllabus**: Holomorphic functions in several variables. Complex varieties. Kähler metrics. Blow-up. Complex vector bundles, connections, curvature, Chern classes. Sheaves and cohomology. Harmonic forms, Hodge theorem and applications. Kähler identities, Hodge decomposition. Serre-Kodaira duality. Lefschetz decomposition. Divisors and line bundles. Bertini theorem. Adjunction formula. Kodaira's vanishing theorem. Lefschetz's theorem on hyperplane sections. Lefschetz's theorem on (1,1) classes. Algebraic varieties. Chow's theorem. Kodaira's immersion theorem. Picard and Albanese varieties. Applications.**Basic Bibliography**:1) Griffiths, P.; Harris, J. Principles of Algebraic Geometry, Wiley-Interscience, 1994.

2) Voisin, C. Théorie de Hodge et Géometrie Algébrique Complexe, Societe Mathematique de France, 2002.

3) Huybrechts, D. Complex Geometry, Springer-Verlag, 2004.

- MAT2303 - Introduction to Probability (
*DF; SD: MAT1414*)**Syllabus**: Kolmogorov’s axioms of probability. Discrete random variables. Counting problems and probability as the relative frequency of events. Continuous random variables. Mean and variance. Conditional mean and variance. Generating functions and characteristic functions. The deMoivre-Laplace limit theorem. The Poisson limit theorem. The law of large numbers. The basic central limit theorem. Introduction to random walks, markov chains and probability on graphs. The Monte Carlo method.**Basic Bibliography**:1) James, B.R. Probabilidade: um Curso em Nível Intermediário, Projeto Euclides, 1981.

2) Feller, W., Introdução à Teoria das Probabilidades e suas Aplicações, Edgard Blucher, 1976.

3) Grimmett, G., Stirzaker, D. Probability and Random Processes, 3rd ed, Oxford, 2001.

- MAT2304 - Probability Theory I
**Syllabus**: Probability spaces, basic properties. Construction of probability measures in R and Rn. Random variables and vectors. Distributions of probability and distribution functions in Rn. Independence and product measures. Expectation of random variables: basic properties and inequalities. Types of convergence of random variables. Law of large numbers: weak convergence and Borel-Cantelli lemmas. Strong law of large numbers. Kolmogorov's three-series Theorem. Characteristic functions and convergence in distribution in Rn. The Theorem of Lindeberg-Feller. Applications. Further topics: Kolmogorov's extension theorem (existence of sequences of i.i.d. random variables).**Basic Bibliography**:1) Jacod, J.; Protter, P. Probability Essentials, Springer, 2004.

2) Billingsley, P. Probability and Measure, 3rd ed. Wiley, 1995.

3) Chung, K.L. A course in Probability Theory, third ed., Associated Press, 2001.

**Additional Bibliography**:1) Shyriaev, A.N.; Boas, R.P. Probability, Springer, 1995.

2) Durrett, R. Probability: Theory and Examples, fourth ed. Cambridge, 2010.

3) Varadhan, S.R.S. Probability Theory. Courant Lecture Notes. AMS, 2001.

- MAT2305 - Probability Theory II
**Syllabus**: Stable laws and infinitely divisible laws. Expectation and conditional probability; properties, existence theorems and regularizations. Discrete time Martingales: Doob's decomposition theorem, Doob's inequalities, stopping times, optional stopping theorem, crossing number inequality, Martingale convergence theorem. Markov chains; random walks in countable spaces, transient and recurrent behavior. Birkhoff's ergodic theorem.**Basic Bibliography**:1) Jacod, J.; Protter, P. Probability Essentials, Springer, 2004.

2) Billingsley, P. Probability and Measure, 3rd ed. Wiley, 1995.

3) Chung, K.L. A course in Probability Theory, third ed., Associated Press, 2001.

**Additional Bibliography**:1) Shyriaev, A.N.; Boas, R.P. Probability, Springer, 1995.

2) Durrett, R. Probability: Theory and Examples, fourth ed. Cambridge, 2010.

3) Varadhan, S.R.S. Probability Theory. Courant Lecture Notes. AMS, 2001.

- MAT2447 - Scientific Computing and Differential Equations (
*DF; SD: MAT1406*)**Syllabus**: Initial value problems: simple and multiple pass method, polynomial interpolation, stability and stiffness, linear and non-linear systems. Boundary value problems: finite difference method for linear problem and discretization. Methods for non-linear problems: shooting, projection, collocation, Garlekin and spline approximations. Explicit and implicit methods for elliptic, parabolic and hyperbolic equations. Fourier transforms. Discretization from integral form. Semi-discrete methods. Error and stability analysis.**Basic Bibliography**:1) Iserles, A. A first course in the Numerical Analysis of Differential Equations. Cambridge Unive. Press, 1996.

2) Smith, G. D. Numerical Solution of Partial Differential Equations: Finite Difference Methods. 2nd ed. Oxford Univ. Press, 1985.

3) Golub, G.; Ortega, J. Scientific Computing and Differential Equations: an Introduction to Numerical Methods, Academic Press, 1991.

- MAT2451 - Analysis and approximation of physical phenomena simulations
**Syllabus**: Fourier analysis and wavelets for discrete PDE's. Approximation spaces, finite elements. Viscosity solutions for PDE's. Variational formulations. Invariant approximations. Physical invariants and discrete exterior calculus. Simulation with reduced sampling and particle methods.**Basic Bibliography**:1) Mallat, S. A wavelet tour of signal processing. Academic Press, 1999.

2) Hughes, T.J.R. The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Dover, 2000.

3) Stern, A.; Desbrun, M. Discrete Geometric Mechanics for Variational Integrators. Proceedings SIGGRAPH 2006.

**Additional Bibliography**:1) Gawlik, E.; Mullen, P.; Pavlov, D.; Marsden, J.E.; Desbrun, M. Geometric, Variational Discretization of Continuum Theories. Physica D: Nonlinear Phenomena, 240(21), 1724-1760, 2011.

2) Paiva, A.; Petronetto, F.; Lewiner, T.; Tavares, G. Simulação de fluidos sem malha, uma introdução do método SPH. 27o. Colóquio Brasileiro de Matemática, IMPA, 2009.

- MAT2461 - Mathematical Elements for Computer Graphics (
*SD: MAT1303*)**Syllabus**: Geometric transformations; 3d interface; arcball and quaternions; curve drawing; sampling; basis of geometric data structure; rendering and shading; graphs of 2d and 3d functions; programming notions in C/C++ or python; openGL basics.**Basic Bibliography**:1) Pesco, S., Lopes, H. Notas de elementos matemáticos para computação gráfica. Departamento de Matemática, PUC-Rio.

2) Angel, E. Interactive Computer Graphics: A Top-Down Approach with OpenGL, 4th Edition, Addison-Wesley 2006.

3) Angel, E. OpenGL: A Primer. 4th Edition, Addison-Wesley 2007.

- MAT2462 - Introduction to Geometric Modeling (
*SD: MAT1305*)**Syllabus**: Splines; geometric interpolation; Delaunay triangulations; mesh data structure; parametric and implicit surfaces; boolean operations.**Basic Bibliography**:1) Farin, G. Curves and Surfaces for CAGD. Academic Press, 1993.

2) Mortenson, M.E. Geometric Modeling, Wiley, 1997.

3) Botsch, M., Pauly, M., Kobbelt, L., Alliez, P., Levy, B., Bischoff, S., Rössl, C. Geometric Modeling Based on Polygonal Meshes. SIGGRAPH Course Notes, 2006.

**Additional Bibliography**:1) Boissonnat, J.-D., Yvinec, M. Algorithmic Geometry, Cambridge University Press, 1998.

2) Figueiredo, L.H. Geometria Computacional, 18.o Colóquio Brasileiro de Matemática, IMPA, 1991.

3) Sorkine, O. Laplacian Mesh Processing, Eurographics STAR 2005.

- MAT2463 - Mathematical Fundamentals for Image Processing
**Syllabus**: Image models, discrete convolution, smoothing, linear filters, scale spaces, colour histograms manipulations; gaussian mixture models and edition, statistical learning, colorization, snakes, watershed, distance transform, image foresting transform, min cut segmentation, mathematical morphology, features, SIFT, tracking, openCV.**Basic Bibliography**:1) Teixeira, R. C. Introdução aos Espaços de Escala (EDPs em Processamento de Imagens). 23.o Colóquio Brasileiro de Matemática, IMPA, 1991.

2) Bradski, G., Kaehler, A. Learning OpenCV: computer vision with the OpenCV library. O' Reilly, 2008.

3) da Fontoura Costa, L., Marcondes Cesar Jr., M. Shape Analysis and Classification. CRC, 2000.

**Additional Bibliography**:1) Brigham, E.O., The Fast Fourier Transform and its Applications. Prentice Hall, 1988.

2) Serra, J. Image Analysis and Mathematical Morphology. Academic Press, New York, 1982.

- MAT2464 - Introduction to Computational Topology
**Syllabus**: Revision of cell complexes and their topological properties, definition of discrete topological invariants as Euler characteristic, Betti numbers, fundamental homology cycles. Morse, Morse-Smale, Witten-Morse theories and their discretization through piecewise-linear, finite elements and Forman approaches with applications to Computer Graphics. Applications of those techniques to geometry processing and shape edition.**Basic Bibliography**:1) Zomorodian, A.J. Topology for computing, Cambridge Univ. Press, 2005.

2) Mäntylä, M. Computational topology: a study of topological manipulations and interrogations in computer graphics and geometric modeling, Finnish Academy of Technical Sciences, 1983.

3) Edelsbrunner, H. Geometry and topology for mesh generation, Cambridge Univ. Press, 2001.

- MAT2465 - Discrete and Computational Geometry
**Syllabus**: Triangulations and simplicial complexes, Voronoï and Laguerre diagrams, Delaunay and regular triangulations, interpolations with paraboloids, medial axes and alpha-shapes, sampling on surfaces, geodesic and discrete curvature computation, discrete Laplace operators and minimal surfaces, Laplacian surface deformation.**Basic Bibliography**:1) Boissonnat, J.-D.; Yvinec, M. Algorithmic Geometry. Cambridge University Press, 1998.

2) de Berg, M.; van Kreveld, M.; Overmars, M.; Schwarzkopf, O. Computational Geometry, Algorithms and Applications. Springer, 1997.

3) Figueiredo, L.H. Geometria Computacional, 18.o Colóquio Brasileiro de Matemática, IMPA, 1991.

**Additional Bibliography**:1) Sorkine, O. Laplacian Mesh Processing. State-of-The-Art Report, Eurographics 2005.

- MAT2502 - Analysis in one complex variable (
*DF; SD: MAT1502*)**Syllabus**: Complex derivative; Cauchy-Riemann equations. Power series; analytic functions. Complex line integrals. Index of a curve, homotopy. Cauchy’s theorem; Cauchy’s integral formula. Homologous curves. Morera and Goursat’s theorems. Poles. Laurent series. Residuals. Riemann sphere; meromorphic functions. Maximum modulus theorem. Schwartz lemma. Möbius mappings; cross-ratio. Normal families, Montel’s theorem. Riemann mapping theorem. Additional topics.**Basic Bibliography**:1) Gamelin, T.W. Complex Analysis. Springer, 2001.

2) Conway, J.B. Functions of One Complex Variable I. Springer; 2nd edition (1978)

3) Needham, T. Visual Complex Analysis. Oxford, 1999.

- MAT2620 - Real Analysis (
*SD: MAT1606*)**Syllabus**: Review of topology and continuity of real functions. The derivative, the Mean Value Theorem, L’Hôpital’s rule, Taylor approximants. Integration in the sense of Riemann. The Fundamental Theorem of Calculus. Sequences of functions. Power series and analytic functions. The Stone-Weierstrass theorem. The Theorem of Arzela-Ascoli. Introduction to harmonic analysis and Fourier series.**Basic Bibliography**:1) Lima, E. L., Curso de Análise, vol. I, Projeto Euclides, 1995.

2) Ávila, G. Introdução à Análise Matemática, Ed. Edgard Blucher, 1999.

3) Abbott, S. Understanding Analysis. Springer, 2001.

- MAT2621 - Measure Theory (
*DF*)**Syllabus**: Measure spaces. Exterior measure; Carathéodory extension theorem. Completion. Examples: Lebesgue measure. Measurable functions. Integral. Convergence theorems. Borel measures; regularity. Riesz representation theorem in the space of continuous functions. L^p spaces. Product measures; Fubini-Tonelli Theorem. Signed measures. Hahn decomposition theorem, absolute continuity, Radon-Nikodym theorem, Lebesgue decomposition. Differentiation of monotonic functions, functions of bounded variation, differentiation of an indefinite integral, Lebesgue points of density, absolute continuity.**Basic Bibliography**:1) Fernandez, P. J. Medida e Integração. Projeto Euclides, IMPA, 1976.

2) Rudin, W. Real and Complex Analysis, 3rd ed. Mc-Graw Hill, 1976.

3) Bartle, R.G. The Elements of Integration and Lebesgue Measure. Wiley, 1995.

- MAT2622 - Functional Analysis (
*DF*)**Syllabus**: Normed vector spaces and Banach spaces. Dual spaces. Zorn's lemma and the Hahn-Banach theorem - analytic and geometric forms. Baire's theorem and the Banach-Steinhaus theorem. Open mapping and closed graph theorems. Weak topologies, reflexive spaces. Separable spaces. Hilbert spaces. The Lax-Milgram and Stampacchia´s theorems. Spectral theory in Hilbert spaces. Lebesgue-measurable functions and L^p spaces. Sobolev spaces. Applications to boundary value problems for partial differential equations.**Basic Bibliography**:1) Brezis, H. Functional Analysis, Sobolev spaces, and Partial Differential Equations, Springer 2010.

2) Rudin, W. Functional Analysis, McGraw-Hill, 1991.

3) Schechter, M. Principles of Functional Analysis, American Mathematical Society 2001.

- MAT2624 - Analysis on the RN space (
*DF; SD: MAT1612*)**Syllabus**: Topology of N-dimensional euclidean spaces: metric structures, topological structures and the notion of completeness. Scalar fields, continuity, and the notion of derivative of a scalar valued function in R^N. The contraction principle, the Inverse Function Theorem and the Implicit Function theorem. The rank theorem and normal forms for mappings between euclidean spaces. Taylor’s formula for the approximant. Jordan measurable sets. The integral in the sense of Riemann, and the notion of integrable function. Fubini’s theorem and the change of variable formula for N-dimensional domains.**Basic Bibliography**:1) Lima, E. L., Curso de Análise, vol. 2, Projeto Euclides, 2000.

2) Lang, S., Undergraduate Analysis, Springer, 1997.

3) Pugh, C. Real Mathematical Analysis. Springer, 2010.

- MAT2640 - Introduction to Fourier Analysis (
*DF; SD: MAT1620*)**Syllabus**: The genesis of Fourier analysis. Basic properties of Fourier series: introduction to problems and examples; uniqueness of the Fourier series; convolutions and their properties; summability kernels. Convergence of Fourier series: review of elementary notions of vector spaces and inner product; convergence in the mean-square sense. Point convergence; a local result and an example of a continuous function with divergent Fourier series. The Fourier transform on the line: definition of the Fourier transform in the space of functions of moderate decrease; the Schwartz space; the Fourier inversion formula; Plancherel's theorem; The Poisson summation formula; the Heisenberg uncertainty principle; applications to EDPs.**Basic Bibliography**:1) STEIN, E. M., SHAKARCHI, R., Fourier Analysis: An Introduction, Vol. I; Princeton: Princeton Lectures in Analysis, 2003.

2) GRAFAKOS, L., Classical Fourier analysis, Vol. 2; New York: Springer, 2008.

3) FIGUEIREDO, D. G. de, Análise de Fourier e Equações Diferenciais Parciais. IMPA; Rio de Janeiro: Coleção Projeto Euclides, 2018.

- MAT2712 - Algebraic Topology I (
*DF*)**Syllabus**: Simplicial and singular homology. Excision. Mayer-Vietoris sequence. Singular and De Rham cohomology. Orientation and duality in manifolds.**Basic Bibliography**:1) Hatcher, A. Algebraic Topology. Cambridge University Press, 2002.

2) Massey, W. A Basic Course in Algebraic Topology. Graduate Texts in Mathematics. Springer, 1991.

- MAT2713 - Algebraic Topology II
**Syllabus**: Higher order homotopy groups. Fibrations and fiber bundles; homotopy exact sequence. Universal fibrations. Elementary calculation of some homotopy groups of the classical groups.**Basic Bibliography**:1) Hatcher, A. Algebraic Topology. Cambridge University Press, 2002.

2) Spanier, E.H. Algebraic Topology. Revised edition. Springer, 1994.

- MAT2714 - Introduction to Topology (
*SD: MAT1702*)**Syllabus**: Metric spaces, Topological spaces. Continuity. Connected and compact spaces. Fundamental group. Covering spaces. Classification of surfaces.**Basic Bibliography**:1) Munkres, J.M. Topology, 2nd edition. Prentice-Hall, 2000.

2) Lima, E.L. Grupo Fundamental e Espaço de Recobrimento, Projeto Euclides, 1993.

3) Massey, W. A Basic Course in Algebraic Topology. Graduate Texts in Mathematics. Springer, 1991.

- MAT2715 - Differential Topology
**Syllabus**: Sard Theorem. Transversality. Intersection theory mod 2: intersection number, degree, winding number. Oriented intersection theory. Lefschetz fixed point theorem. Euler characteristic. Vector fields; Poincaré-Hopf Theorem. Introduction to Morse theory. Classification of compact surfaces.**Basic Bibliography**:1) Hirsch, M. W. Differential Topology. Graduate Texts in Mathematics. Springer, 1976.

2) Milnor, J.W. Topology from the Differentiable Viewpoint. Revised edition. Princeton University Press, 1997.

3) Guillemin, V.; Pollack, A. Differential Topology. Prentice Hall, 1974.

- MAT2812 - Differential Geometry on R3 (
*DF; SD: MAT1811*)**Syllabus**: Planar and spatial curves. Frenet frame and applications. Surfaces.in Euclidean space. Calculus on surfaces: areas, isometries, conformal mappings. Orientation. Gauss normal map, curvatures, special lines (curvature lines, asymptotic lines, geodesics). Gauss egregium theorem. Gauss-Bonnet theorem and applications.**Basic Bibliography**:1) Do Carmo, M., Geometria Diferencial de Curvas e Superfícies, Coleção Matemática Universitária (SBM), 2005.

2) Kühnel, W., Differential Geometry, American Mathematical Society, 2002.

- MAT2815 - Riemann Surfaces I
**Syllabus**: Definition of Riemann surface. Holomorphic maps and their properties. Isothermal parameters. Construction of Riemann surfaces. Riemann surface of an algebraic equation. Conformal structures. Branched coverings. Hurwitz formula. Riemann relation. Analytic continuation. Uniformization theorem, proof and examples: the unit disc as the universal covering of the sphere minus three points. Riemann surfaces as quotient of its universal covering surface, Koebe-Poincaré theorem. Conformal structures on the tori. Weierstrass P function and other elliptic functions. Conformal structures on the annuli. Great Picard theorem.**Basic Bibliography**:1) Ahlfors, L. Conformal Invariants. McGraw-Hill 1973.

2) Sá Earp, R.; Toubiana, E. Introduction à la géométrie hyperbolique et aux surfaces de Riemann. Cassini, 2009.

3) da Costa, C.J. Funções elípticas, algébricas e superfícies mínimas. 18.o Colóquio Brasileiro de Matemática, IMPA, 1991.

- MAT2816 - Riemann Surfaces II
**Syllabus**: Sheaves. Algebraic Functions. Fundamental group and singular (co)homology of compact Riemann surfaces. Monodromy. Algebraic Curves. Divisors, line bundles, canonical line bundle. Linear systems, maps to the projective space. Sheaves cohomology, finiteness theorems. Dolbeault's theorem. Serre duality. Riemann-Roch theorem. Harmonic forms. Vanishing of the cohomology, ample line bundles, immersion into the projective space. Hyperellitpic curves. Picard group. Jacobian. Abel's theorem. Jacobi's theorem. Applications to algebraic curves and their jacobians.**Basic Bibliography**:1) Foster, O. Lectures on Riemann Surfaces, Springer-Verlag, 1981

2) Narasimhan, R. Compact Riemann Surfaces, Birkhäuser, 1996.

3) Miranda, R. Algebraic Curves and Riemann Surfaces, American Mathematical Society, 1995.

**Additional Bibliography**:1) Gunning, R.C. Lectures on Riemann Surfaces, Princeton University Press, 1966.

- MAT2817 - Representation Theory
**Syllabus**: Representations of finite groups. Schur's lemma. Characters. Class functions, irreducible representations and conjugacy classes. The regular representation. Induced and restricted representations. Frobenius reciprocity. Group Algebra. Applications. Elements of Lie Groups and Lie algebras. Lie theorems. Killing form. Semi-simple Lie algebras. Cartan subalgebras. Maximal tori. Roots, weight spaces. Weyl group. Unitary trick for compact Lie groups. Representations of compact Lie groups. Applications. Irreducible representations of SL(n,C) and GL(n,C).**Basic Bibliography**:1) Simon, B. Representations of Finite and Compact Groups, AMS, 1995.

2) Kirilov Jr., A. An Introduction to Lie Groups and Lie Algebras, Cambridge University Press, 2008.

3) Humphreys, J.E. Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1973.

**Additional Bibliography**:1) Fulton, W.; Harris, J.Representation Theory, Springer-Verlag, 1991.

2) Bump, D. Lie Groups, Springer-Verlag, 2011.

3) Miller, W. Symmetry Groups and their Applications, Elsevier, 1972.

- MAT2821 - Differentiable Manifolds
**Syllabus**: Differentiable manifolds. Examples. Submanifolds. Tangent space. Differentiable maps. Embeddings and immersions. Partitions of unity. Orientations. Manifolds with boundary. Differential forms. Exterior derivative. Frobenius theorem. Integration of forms. Stokes Theorem. Applications.**Basic Bibliography**:1) Tu, L.W. An Introduction to Manifolds, 2nd edition. Universitext. Springer, 2010.

2) Lee, J.M. Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Springer, 2002.

- MAT2824 - Introduction to Lie Groups and Groups of Transformations
**Syllabus**: Topological groups, the classical groups, Lie groups, homomorphisms of Lie groups, subgroups, coverings, Lie algebra associated to a Lie group, simply connected Lie groups, exponential mapping, closed subgroups, elementary representation theory, adjoint representation, maximal tori, group actions, orbits and orbit spaces. Homogeneous spaces, fixed points, actions on coverings.**Basic Bibliography**:1) Chevalley, C. Theory of Lie Groups. Princeton Univ. Press, 1999.

2) Warner, F. W. Foundations of Differentiable Manifolds and Lie Groups. Springer, 1983.

3) Bredon, G.E. Introduction to Compact Transformation Groups. Academic Press, 1972.

- MAT2826 - Riemannian Geometry I
**Syllabus**: Riemannian metrics. Riemannian connection. Geodesics. Curvatures. Jacobi fields. Complete Riemannian manifolds. Isometric immersions. Spaces of constant curvature. Variations of energy. Rauch comparison theorem. Morse index theorem. Homogeneous spaces.**Basic Bibliography**:1) do Carmo, M.P. Geometria Riemanniana. Projeto Euclides, IMPA, 1988.

2) Cheeger, J.; Ebin, D.G. Comparison theorems in Riemannian geometry. AMS, 2008.

3) Petersen, P. Riemannian geometry, Springer, 2006.

**Additional Bibliography**:1) Hicks, N.J. Notes on differential geometry, Van Nostrand, 1965.

2) Warner, F.W. Foundations of differentiable manifolds and Lie groups. Springer, 1983.

- MAT2828 - Minimal Surfaces in R3 I
**Syllabus**: Several equivalent definitions of minimal surfaces in Euclidean space. Classic examples and their geometric characterizations. The Weierstrass representation. Curvature estimates and Bernstein's theorem. Schwarz reflection principle. Conjugate and associate minimal surfaces. Complete minimal surfaces of finite total curvature. The maximum principle, Rado theorem and the half-space theorem. Douglas-Rado solution to the Plateau problem.**Basic Bibliography**:1) Dierkes, U.; Hildebrandt, S.; Sauvigny, F. Minimal surfaces. Springer, 2010.

2) Blaine Lawson, H. Lectures on minimal submanifolds. Publish or Perish, 1980.

3) Osserman, R. A survey of minimal surfaces. Dover, 1986.

**Additional Bibliography**:1) Osserman (Ed.), R. Geometry V. Encyclopaedia of Mathematical Sciences, volume 90, Springer, 1997.

2) Colding, T.B.; Minicozzi II, W. A course in minimal surfaces, AMS, 2011.

- MAT2905 - Ordinary Differential Equations (
*DF; SD: MAT1901*)**Syllabus**: Differential equations of first order. Reduction of high order equations to first order systems. Existence and uniqueness of solutions. Dependence on the initial conditions. Extension of solutions. Linear systems with constant coefficients. Non-homogeneous linear equations and non-autonomous linear equations. Poincaré-Bendixson theorem.**Basic Bibliography**:1) Sotomayor, J., Lições de Equações Diferenciais Ordinárias, Projeto Euclides, 1979.

2) Hirsch, M; Smale, S., Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, 1974.

- MAT2907 - Partial Differential Equations I (
*DF; SD: MAT1907*)**Syllabus**: Introduction. Classic methods for solving PDE. First order equations. Cauchy problem. Caucky-Kowalevskaya theorem. Classical second-order PDEs and boundary value problems. Well-posed problems. Generalizations to systems of PDE and higher order equations.**Basic Bibliography**:1) John, F. Partial differential equations. Vol. 1. Springer, 1981.

2) Smirnov, M. M. Second-order partial differential equations. Noordhoff, 1966.

- MAT2915 - Modern theory of second-order elliptic PDE - classical and strong solutions
**Syllabus**: Maximum principles for second order linear elliptic equations. Schauder a priori estimates. Compactness principles for sequences of solutions of elliptic PDE. Existence theorem for the classical Dirichlet problem for second-order linear elliptic PDE - Perron’s method (sub- and super-solutions method). Alexandrov-Bakelman-Pucci maximum principle. Applications to the theory of solvablity and regularity of solutions of general PDEs - Krylov-Safonov theory. First eigenvalue of an elliptic operator, maximum principle. Applications to the qualitative theory - Alexandrov’s moving planes method, symmetry of positive solutions of elliptic PDE.**Basic Bibliography**:1) Gilbarg, D.; Trudinger, N.S. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001.

2) Evans, L.C. Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.

3) Han, Qing; Lin, Fanghua. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.

- MAT2916 - Modern theory of second-order elliptic PDE - weak solutions
**Syllabus**: Sobolev spaces. Weak solutions in Sobolev spaces of elliptic PDE in divergence form. Solvability of linear elliptic equations in divergence form and regularity of the weak solutions. “Bootstrap” methods for regularity of the weak solutions of nonlinear equations. Variational characterization of the eigenvalues of a self-adjoint elliptic operator of second order. Variational formulation of solutions of divergence-form PDE. Methods for searching critical points - direct minimization, “mountain-pass” and “linking” type theorems. Notion of weak viscosity solution of an elliptic PDE. Existence and regularity of the viscosity solutions of general nonlinear elliptic PDE.**Basic Bibliography**:1) Gilbarg, D.; Trudinger, N.S. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001.

2) Evans, L.C. Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.

3) Han, Qing; Lin, Fanghua. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.

**Additional Bibliography**:1) Willem, M. Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996.

2) Struwe, M. Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2010.

3) Caffarelli, L.A.; Cabré, X. Fully nonlinear elliptic equations. American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, 1995.

4) Crandall, M.G.; Ishii, H.i; Lions, P.-L. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67.

- MAT2917 - Methods of nonlinear analysis and applications
**Syllabus**: Schauder’s fixed point theorem for compact maps. Nonlinear Leray-Schauder alternative. Leray-Schauder fixed point theorem. Existence of solutions of quasilinear elliptic equations following from a priori estimates for the solutions and their gradient. Maximum principles for nonlinear elliptic equations of second order. A priori estimates in the C^1-norm for solutions of constant mean curvature equations in various settings. Applications to the Dirichlet problem in bounded domains with smooth boundary data. Perron’s method, applications to Dirichlet problems in unbounded domains with continuous boundary data.**Basic Bibliography**:1) Gilbard, D.; Trudinger, N. Elliptic partial differential equations of second order. Springer, 2001.

2) Granas, A.; Dugundji, J. Fixed point theory. Springer, 2010.

3) Agarwal, R.; Meegan, M.; O'Regan, D. Fixed point theory and applications. Cambridge University Press, 2004.

**Additional Bibliography**:1) Barbosa, J.L.M.; Sá Earp, R. Prescribed mean curvature hypersurfaces in Hn+1 with convex planar boundary II. Seminaire de théorie spectrale et géométrie de Grenoble, volume 16, 43-79 (1998).

- MAT2920 - Introduction to Dynamical Systems
**Syllabus**: Basic notions of dynamics: periodicity, recurrence, transitivity, minimality. Fundamental examples: contractions, linear maps, rotations, gradient flows, Morse-Smale functions. Circle dynamics: rotation number, Denjoy example and theorem, Poincaré´s classification. Expanding maps, Symbolic dynamics, topological mixing, shifts of finite type, Smale horseshoe, toral automorphisms, geodesic and horocyclic flows on surfaces, kneeding theory.**Basic Bibliography**:1) Brin, M.; Stuck, G. Introduction to Dynamical Systems. Cambridge University Press, 2002.

2) Hasselblatt, B.; Katok, A. A First Course in Dynamics: with a Panorama of Recent Developments. Cambridge University Press, 2003.

3) Devaney, R. An Introduction to Chaotic Dynamical Systems. 2nd ed. Westview Press, 2003.

- MAT2921 - Hyperbolic Dynamics
**Syllabus**: Local stability theory for hyperbolic periodic points of diffeomorphisms and closed orbits of flows. Hartman-Grobman Theorem, and existence of invariant submanifolds. Morse-Smale diffeomorphisms, Hyperbolic sets, examples: Anosov linear systems, Plykin atractor, solenoid. Lambda-Lemma, transversal homoclinic points and horseshoes, symbolic dynamics. Stable manifold theorem for Anosov systems, expansiveness and shadowing property, Anosov closing lemma and stability of hyperbolic sets. Structural stability of Morse-Smale systems, cycles and filtrations, omega stability for Axiom A systems without cycles, omega explosions.**Basic Bibliography**:1) Brin, M.; Stuck, G. Introduction to Dynamical Systems. Cambridge University Press, 2002.

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

- MAT2922 - Ergodic Theory
**Syllabus**: Invariant measures; the weak topology and existence of invariant probabilities for continuous maps. Examples. Recurrence and ergodicity: Poincaré recurrence theorem, topological and metric versions. Birkhoff theorem. Ergodicity, unique ergodicity, mixing. Examples: shifts, rotations, expanding maps of the circle, toral automorphisms, Furstenberg example, geodesic and horocyclic flows on surfaces. Ergodic decomposition. Topological and metric entropies: generating partitions, Kolmogorov-Sinai theorem.**Basic Bibliography**:1) Mañé, R. Teoria Ergódica. Projeto Euclides, IMPA, 1983.

2) Petersen, K. Ergodic Theory. Cambridge University Press, 1990.

3) Walters, P. An Introduction to Ergodic Theory. Springer, 2007.

- MAT2923 - Introduction to conservative systems
**Syllabus**: Linear symplectic spaces and their geometry. Darboux’s theorem for symplectic forms and contact forms. The nonlinear theory of symplectic spaces and basic notions of symplectic topology. Lagrangean geometry. Weinstein’s Lagrangean neighborhood theorem. The theorem of Givental on germs of Lagrangean varieties. Weinstein’s trick of the Lagrangian graph. Momentum maps and reduction theory. An introduction to one dimensional variational calculus. The geometry of Hamiltonian and Lagrangian systems. Applications to classical mechanics, geometric optics and integrable systems.**Basic Bibliography**:1) Moser, J., Zehnder, E.J.: Notes on dynamical systems. Courant Lecture Notes in Mathematics, 12 (2005).

2) Arnold, V.I.: Mathematical Methods of classical mechanics. Graduate Texts in Mathematics, 60, Second edition, Springer-Verlag (1989).

3) Cannas, A.: Lectures on symplectic geometry. Lecture Notes in Mathematics, 1764, Springer-Verlag (2001).

**Additional Bibliography**:1) Paternain, G.: Geodesic flows. Progress in Mathematics, 180. Birkhäuser (1999).

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