Riemannian Geometry

Riemannian Geometry is a basic course for any graduate student in Mathematics who wants to study Geometry, Topology or Dynamic Systems, and is also a relevant course for students of Analysis and Applied Mathematics. Provide to the student the basic tools and some fundamental results of Riemannian Geometry. Institute of Mathematical and Computer Sciences (ICMC) São Carlos campus Program: Riemannian metrics; Connections; Completeness; Curvature; Isometric immersions; Variational calculus; Applications. Detailed program: (1) Riemannian metrics; Examples of Riemannian manifolds: the Euclidean space R^n, the sphere S^n, the real hyperbolic space H^n, product of Riemannian manifolds, conformal metrics, Riemannian coverings, flat tori, the Klein bottle, Riemannian submersions, the Hopf fibration and the complex projective space, quotient manifolds, Lie groups. (2) Connections; Parallel transport along a curve; Geodesics; Isometries and Killings vector fields; Induced connections. (3) Completeness; The Hopf-Rinow theorem; Cut locus, Examples. (4) The Riemann-Christoffel curvature tensor; The Ricci tensor and scalar curvature; Covariant derivative of tensors; Examples. (5) Isometric immersions; The second fundamental form; The fundamental equations. (6) Variational calculus; The energy functional; Jacobi vector fields; Conjugate points; Examples. (7) Space forms; The Synge theorem; The Bonnet-Myers theorem; Nonpositively curved manifolds. Fernando Manfio, Irene Ignazia Onnis 35 SMA5947 8 Two written tests. http://conteudo.icmc.usp.br/Portal/conteudo/1079/538/foreign-scholars

Computational techniques for Probability and Statistics I

Exploratory data analysis and computational techiques are fundamental to understand modern modern statistical methods. Introduce modern techniques of data analysis with concomitant using of computer. Using statistical packages. Institute of Mathematics ans Statistics (IME) São Paulo main campus 1. Exploratory data analysis (univariate and multivariate): position measurements, dispersion, asymmetry, robust measures, bivariate measures, association between variables, outliers identification, processing variables, graphics. 2. Linear Regression Models, Regression Tough and Smoothing Methods. 3. Stochastic Simulation: inversion methods, rejection, composition and resampling methods. 4. Numerical Optimization: Newton-Raphson, scoring, quasi-Newton. 5. EM Algorithm. 6. “Bootstrap” and “Jacknife”. 7. Monte Carlo methods and Gaussian quadratures Denise Aparecida Botter, Eduardo Jordao Neves, Anatoli Iambartsev 40 MAE5704 8 Tests, Lists of exercises and seminars https://www.ime.usp.br/en

Statistical Inference for Stochastic Processes

Stochastic processes are natural models for phenomena occurring in time and for spatial systems. Modeling natural phenomena using stochastic processes requires the knowledge of specific inferential and statistical model selection tools. Moreover, stochastic processes have also been used as computational tools in statistical inference, as exemplified by Monte-Carlo Markov chain algorithms for sampling probability distributions. To present basic notions of statistical inference for some important classes of stochastic processes. Institute of Mathematics ans Statistics (IME) São Paulo main campus 1) Statistical inference for Markov chains. Maximum likelihood estimation. Estimation of the order of the chain. 2) Statistical inference for stochastic chains with memory of variable length. The algorithm Context. 3) Context tree selection using the Bayesian Information Criterion (BIC). The algorithm_CTW. 4) Statistical inference for hidden Markov models. 5) Gibbs states. Interaction graph selection and maximum likelihood estimation for the_Ising_model. 6) Simulations using Monte-Carlo Markov chains (MCMC)._Glauber_dynamics, Gibbs sampler, Metropolis algorithm. 7) Perfect simulation algorithms. Jefferson Antonio Galves, Florencia Graciela Leonardi 40 MAE5741 8 Students will be evaluated through projects, seminars, exercise lists and write tests, https://www.ime.usp.br/en

Advanced Probability I

The general treatment of Probability Theory requires its formulation in abstract spaces, in the framework introduced by Kolmogorov. Introduce the basics of Probability Theory into abstract spaces, including the necessary elements of Measure Theory, in the framework formulated by Kolmogorov. Institute of Mathematics ans Statistics (IME) São Paulo main campus 1. Probability Spaces: (a) Lebesgue-Stieltjes Measure, Carath_dory Extension Theorem; (b) Measures of Probability, Random Variables; (c) Integration, Expectation, Convergence Theorems; (d) Product measures, Fubini’s theorem; (e) Independence; (f) Kolmogorov Extension Theorem; (g) Radon-Nikodym Theorem, Conditional Expectation. 2. Laws of Large Numbers: (a) Convergence in Probability and Almost Sure Convergence; (b) Weak Law of Large Numbers; (c) Borel-Cantelli lemmas; (d) Strong Law of Large Numbers. 3. Central Limit Theorem: (a) Convergence in Distribution; (b) Characteristic Functions; (c) TLC for Random Variables I.I.D; (d) TLC for Triangular Arrangements. Vladimir Belitsky, Miguel Natalio Abadi, Anatoli Iambartsev 50 MAE5811 8 Exam and exercises, with the possibility of collecting an article at the end of the course. https://www.ime.usp.br/en

Mathematica Statistics II

The formal study of the asymptotic properties of estimators and test statistics is fundamental to understand and propose modern statistical methods. To discuss formally asymptotic theory of statistical methods. Institute of Mathematics ans Statistics (IME) São Paulo main campus 1. Order of magnitudes and Taylor series. 2. Weak and strong convergence laws of the estimators. Univariate and multivariate cases. Slutsky’s Theorem. 3. Central limit Theorems _ Univariate, Multivariate and Martingales. Cram_r-Wold’s Theorem. Hajek-Sidak’ s Theorem and applications to regression models. Delta method and variance stabilizing transformations. 4. Asymptotic expansions. 5. Applications. Antonio Carlos Pedroso de Lima, Alexandre Galv_o Patriota 50 MAE5835 8 Exams, Worksheets and Seminars https://www.ime.usp.br/en