Professor: Wladimir Neves (IM-UFRJ)
Pré-requisitos: Análise no R^n, Álgebra Linear
Carga Horária: 48h. Início: 4 de Janeiro Término: A ser divulgado 
Horário e sala: Segundas, Terças e Sextas de 8:00 as 10:00 na Sala B-106 A 
Ementa: Sigma-álgebras; Medida positiva e medida exterior, medidas de Borel, regularidades e medidas de Radon; Construção de medidas; Medidas vetoriais, medidas absolutamente continuas, medidas singulares e medidas discretas; Classes importantes: Medidas de Lebesgue Stieltjes, medidas de Hausdorff, medidas de Haar. Funções mensuráveis e integral segundo Lebesgue; Conjuntos não mensuráveis a Lebesgue, o exemplo de Vitali; Teoremas de convergências; Os espaços L^p, completude e separabilidade. Duais e isometrias; Medidas-produto. Teorema de Fubini; Teorema de Lebesgue-Radon-Nikodym; Teorema de Representacao de Riesz para medidas de Radon vetoriais; Teorema de diferenciação de Lebesgue-Besicovitch.
Bibliografia: 
Wladimir Neves e Glauco Valle, Teoria da Medida Integração e Probabilidade, Editora UFRJ, 2015 
G. B. Folland, Real Analysis Modern Technics and Their Applications, 1999 
W. Rudin, Real and Complex Analysis, McGrawHill, New York, 1987. 
H.L.Royden, Real Analyses, Macmillan, New York, 1988.

Data da segunda Prova: 26/02/2016 - 8:00hrs

Professora: Bruno Scárdua (IM-UFRJ)
Pré-requisitos: Cálculo I e II ou equivalentes
Carga Horária: 48h, início: 4 de Janeiro, término: A ser divulgado
Horário e sala: Segundas, Quartas e Sextas de 10:00 as 12:00hs na sala D-120
Ementa: Conjuntos e funções. Conjuntos finitos, enumeráveis e não enumeráveis. Números reais. Sequências e séries de números reais. Topologia da reta. Limites de funções. Funções contínuas. Derivadas e Integrais.
Bibliografia: Elon Lima, Curso de “Análise

Data da segunda Prova: 26/02/2016 - 10:00hrs

Professor: Marina Silva Paez (IM-UFRJ)
Pré-requisitos: Cálculo I e II ou equivalentes
Carga Horária: 24h; Início: 4 de Janeiro. Término: A ser divulgado.
Horário e sala: Seg, Qua, Sex de 10h às 12h na sala C-119 (CT, Bloco C)
Ementa: Espacos amostrais e eventos. Probabilidade condicional. Variáveis aleatórias e distribuições de probabilidade. Valores esperados. Principais distribuições de probabilidade. Lei dos grandes números e teorema central do limite.
Bibliografia: DeGroot, MH; Schervish, MJ (2011), “Probability and Statistics”, Pearson (4a. ed.); Ross, S. (2012). A First Course in Probability (9a. ed.)

Prof. Rafael Labarca (Univerdad de Santiago de Chile - USACH)
Datas: de 1 a 4 fevereiro de 2016.
Horário: 10:00 às 12:00hs. Sala: C-116.
Ementa: A entropia topológica de uma aplicação contínua é um número (entre zero e infinito) que indica o grau de complexidade da aplicação. Este número é usado para dividir o espaço das aplicações contínuas em distintos níveis. Quando se trabalha com famílias parametrizadas, uma pergunta interessante é saber se estes níveis são conjuntos conexos ou não. Daremos a definição e discutiremos as propriedades básicas da entropia. Apresentaremos exemplos interessantes para exemplificar seu cálculo: o mundo lexicográfico e o mundo de Milnor-Thurston.
Bibliografia:
1) R.L. Adler, A.G. Konheim, M.H. Mc Andrew. Topological Entropy. TAMS vol. 144 (1965) pg. 309-319
2) R. Bowen. Entropy for groups endomorphisms and homogeneus spaces. TAMS vol. 153 (1971) pg. 404-414
3) Rafael Labarca. Notas do curso.
4) Rafael Labarca. La entropía topológica, propiedades generales y algunos cálculos en el caso del shift de milnor thurston. Ediciones IVIC. Venezuela 2011.
5) Karen Butt. An Introduction to topological entropy. REU 2014 Chicago. Webpage: http://math.uchicago.edu/~may/REU2014/REUPapers/Butt.pdf
6) Aaron Gelon So. Symbolic Dynamics.REU 2014 Chicago. W

Christopher Seaton (Rhodes College) e Hans-Christian Herbig (UFRJ)
Datas: 11 e 12 de janeiro de 2016.
Horário: de 10:00 as 12:00hs e de 15:00 as 17:00hs (duas aulas em ambos os dias). Sala: C100-B.
Ementa: Seja G um grupo de Lie compacto, e seja V uma G-represetação unitária. Então existe um mapa momento quadrático J com respeito ao qual V é uma variedade Hamiltoniana. Se Z denota a fibra zero do mapa momento, o quociente simplético correspondente é Z/G. Este quociente tem estrutura de espaço simplético estratificado e também de conjunto semi-algébrico, e vem equipado com uma álgebra de funções regulares, uma Poisson subálgebra da sua álgebra de funções suaves. Neste mini-curso introduziremos métodos para calcular a álgebra de funções regulares de um tal quociente simplético, baseados 'invariant theory' e geometria algébrica computacional. Além disso, explicarei como estes cálculos pode ser usadas para observer e verificar propriedades do quociente simplético. Tópicos incluem bases de Groebner para cálculo de polinômios invariantes, teoria de eliminação, e métodos para o cálculo de séries de Hilbert de álgebras de Cohen-Macaulay. Além disso, introduzirei os pacotes Mathematica e Macaulay2 usados para tais cálculos.

Vieri Benci (Universitá di Pisa)
Cronograma: O curso será dividido em 8 aulas, de 15 a 29 de fevereiro.
Horário e sala: a ser definido. Sala: a ser definida.
Ementa: Aula 1-2) Introdução à Teoria de Morse classica em dimensão finita. (Milnor: Morse Theory) Aula 3-4) Introdução à Teoria de Morse em variedades de Hilbert (em dimensão infinita), aplicações a problemas de EDP. (Chang: Infinite dimensional Morse Theory) Aula 5-6) Ilustração dos problemas que aparecem quando se tenta de estender a Teoria de Morse a variedades de Banach. Motivação para propor a matemática não Arquimediana neste contexto. Introdução às ultrafunções. (Notas de curso do Vieri Benci) Aula 7-8) Aplicação da Teoria das ultra funções à Teoria de Morse infinito dimensional. (Artigo: Benci-Isaia Nisoli)
Bibliografia:
1) Milnor, John; Morse Theory. Annals of Mathematics Studies 51, (1963). Princeton University Press
2) Chang, K.C.; Infinite Dimensional Theory and Multiple Solution Problems; Progress in Nonlinear Diferential Equations and Its Applications 4, (1993). Birkhauser
3) Notas de aula do Prof, Bencia
4) Benci, Vieri and Nisoli, Isaia; Towards a Morse theory on Banach spaces via ultrafunctions. Arab J. Math. Sci. (21), (2015), 144—158.

Jose Yallouz (Technion - Israel Institute of Technology, Bell Labs Israel)
Cronograma: O curso será divido em 4 aulas de 90 minutos, de 25 a 29 de janeiro.
Horário e sala: a ser definido. Sala: a ser definida.
Ementa: Coping with network failures has been recognized as an issue of major importance in terms of social security, stability and prosperity. It has become clear that current networking standards fall short of coping with the complex challenge of surviving failures. The need to address this challenge has become a focal point of networking research. In particular, the concept of tunable survivability offers major performance improvements over traditional approaches. Indeed, while the traditional approach is to provide full (100%) protection against network failures through disjoint paths, it was realized that this requirement is too restrictive in practice. Tunable survivability provides a quantitative measure for specifying the desired level (0%-100%) of survivability and offers flexibility in the choice of the routing paths. In this mini-course, we will focus on several aspects of this novel metric under two important transmission methods, namely unicast and broadcast. We will establish efficient algorithmic schemes for optimizing the level of survivability under two classes of QoS requirements, namely ``bottleneck'', e.g. bandwidth, and ``additive'', e.g. delay and cost. For the unicast method we aim to find a pair of paths such that maximizes the survivability level under the desirable QoS requirement. First, we establish some (in part, counter-intuitive) properties of the optimal solution. Then, we establish efficient algorithmic schemes for optimizing the level of survivability under additive end-to-end QoS bounds. Subsequently, through extensive simulations, we show that, at the price of negligible reduction in the level of survivability, a major improvement (up to a factor of 2) is obtained in terms of end-to-end QoS performance. Finally, we exploit the above findings in the context of a network design problem, in which we need to best invest a given ``budget'' for improving the performance of the network links. For the brodcast method we aim to find a set of spanning trees such that maximizes the survivability level under the desirable QoS requirement. We establish efficient algorithmic schemes for optimizing the level of survivability under various QoS requirements. In addition, we derive theoretical bounds on the number of required trees for maximum survivability. Finally, through extensive simulations, we demonstrate the effectiveness of the tunable survivability concept in the construction of spanning trees. Most notably, we show that, typically, negligible reduction in the level of survivability results in major improvement in the QoS performance of the resulting spanning trees. The minicourse we will include topics from the following fields: graph theory, approximation algorithm and convex optimization.
Bibliografia:
1) Ron Banner and Ariel Orda. The power of tuning: A novel approach for the efficient design of survivable networks. 15(4):737 --749, 2007.
2) Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
3) J.Tapolcai, Pin-Han Ho, P.Babarczi, and L.Rónyai. Internet Optical Infrastructure - Issues on Monitoring and Failure Restoration. Springer, 2014.
4) VijayV. Vazirani. Approximation Algorithms. Springer-Verlag New York, Inc., New York, NY, USA, 2001.
5) Jose Yallouz and Ariel Orda. Tunable qos-aware network survivability. Proceedings of the {IEEE} {INFOCOM} 2013, Turin, Italy, April 14-19, 2013, pages 944--952, 2013.
6) Jose Yallouz, Ori Rottenstreich, and Ariel Orda. Tunable survivable spanning trees. In {\em {ACM} {SIGMETRICS} / International Conference on Measurement and Modeling of Computer Systems, {SIGMETRICS} '14, Austin, TX, {USA} – June 16 - 20, 2014}, pages 315-- 327, 2014.

Bernard Dacorogna (Ecole Polytechnique Fédérale Lausanne, Suiça)
Cronograma:
Primeira aula: dia 15/01/2016 (sexta-feira). Título: A Dirichlet problem involving the divergence operator.
Segunda aula: dia 18/01/2016 (segunda-feira). Título: Symplectic decomposition, Darboux theorem and ellipticity.
Terceira aula: dia 21/01/2016 (sexta-feira). Título: Differential inclusions, isometric imbeddings and origami.
Horário e sala: Todas as aulas serão na sala C-116 tendo início as 10:30hs.
Bibliografia:
Implicit Partial Differential Equations. (Progress in Nonlinear Differential Equations and Their Applications), Bernard Dacorogna and Paolo Marcellini.

David Dritschel (University of St Andrews, Reino Unido)
Cronograma: O curso será dividido em 3 aulas, de 20 a 27 de janeiro.
Horário e sala: a ser definido. Sala: a ser definida.
Ementa: This course will discuss both the mathematics and the numerical methods enabling the study of inviscid incompressible fluid flow on two-dimensional surfaces, with an emphasis on closed surfaces. Examples of closed surfaces include the sphere and the ellipsoid, both practically relevant to planetary atmospheric dynamics. Despite these applications, relatively little is known about fluid flows on such surfaces, compared to flow in the two-dimensional plane. We begin with the simplest fluid model, namely the motion of singular point vortices, a model first introduced by Kirchhoff in the 1800s. We shall see that surface geometry has a profound effect on the dynamics of such vortices. In particular, surface curvature tends to be destabilising, especially when the curvature varies across the surface. We then discuss an accurate numerical method for studying vortex motion on general surfaces of revolution. Students will be given the opportunity to use this method (together with associated visualisation tools) to explore various aspects of vortex motion. We then generalise the analysis to finite distributions of vorticity on closed surfaces - vortex patches. We discuss the `inversion problem', namely how to obtain the velocity field from the instantaneous distribution of vorticity. We show how this simplifies for surfaces of revolution and then discuss a hybrid Lagrangian-Eulerian numerical method for accurate flow simulation. The method, called the Contour-Advective Semi-Lagrangian (CASL) algorithm, blends the most accurate and efficient parts of a grid-free contour dynamics method with a grid-based conventional method. Again, students will have the opportunity to use this method to explore the complex vortex dynamics which may occur on closed surfaces. The schedule for the course is as follows: Lecture 1: Mathematical aspects of fluid flow on closed surfaces. Point vortex dynamics. Lecture 2: Numerical methods for point vortices on closed surfaces. Hands-on exploration of the dynamics. Lecture 3: Generalisation to vortex patches. The inversion problem and simplifications for surfaces of revolution. The CASL algorithm. Hands-on exploration of the dynamics.

Wai-Sun Don (Qingdao, China)
Datas: 27 a 29 de Janeiro.
Horário: 13:00 as 15:00hs. Sala: LIG1.
Ementa: High order linear methods — such as spectral methods, optimized compact and central finite difference schemes, and discontinuous Galerkin methods — are widely used for solving PDEs with smooth solutions because of their efficient resolution and their ability to compute both large and fine scale structures including those existing in turbulence. However, they usually perform poorly in the presence of shocks and/or high gradients due to the well-known Gibbs phenomenon. In contrast, high order nonlinear shock capturing finite difference schemes — such as the weighted essentially non-oscillatory scheme (WENO) — resolve such discontinuities and high gradients in an essentially non-oscillatory manner while also computing smooth structures with acceptable high resolution. In this short course, I will describe the basic ideas of the WENO reconstruction procedure and give an overview of the essential building blocks in the numerical solution of hyperbolic conservation laws such as the linear scalar wave equation, nonlinear scalar Burgers equation and nonlinear system of Euler equations. If time allows, I will also discuss hybrid schemes that conjugate a linear scheme such as the compact scheme with the nonlinear WENO scheme along with high order shock detection algorithms such as Multi-resolution analysis and conjugate Fourier analysis. The short course consists of 1.5-2 hours lectures and, immediately following the lectures, a 1.5-2 hours practice programming session over the course of three days. The Matlab programming session will provide students with firsthand experience conceptualizing the material discussed during lectures, visualizing the performance of the algorithms by programming and analyzing results. Some background on basic finite difference schemes for solving wave type PDEs and familiarity with Matlab programming would be helpful in appreciating the beauty behind the ideas of nonlinear adaptive algorithms in solving once intractable classes of physical problems.
Bibliografia:
Harten, B. Engquist, S. Osher and S. Chakravarthy, Uniformly high order essentially non-oscillatory schemes, III, Journal of Computational Physics, 71:231-303, 1987.
G. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 126:202-228, 1996.
X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, Journal of Computational Physics, 115:200-212, 1994.
C.-W. Shu, High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM Review, 51:82-126, 2009.
Eleuterio F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics : A Practical Introduction.
Randell, LeVeque, Numerical Methods for Conservation Laws

Dr Alfonso Artigue (Universidad de la República, Uruguay) & Dr. José Vieitez (Universidad de la República, Uruguay)
Datas: de 15 a 19 de fevereiro de 2016.
Horário: 10:00 às 12:00hs. Sala: C-116.
Ementa: Basic results: uniform expansiveness, Lyapunov functions, topology of stable sets: case of a manifold. Classification of expansive homeomorphisms on surfaces. Some aspects on greater dimension. Robust expansiveness, quasi-Anosov diffeomorphisms. Fathi's metric for expansive homeomorphisms: topological entropy of expansive homeomorphisms. Variations on the concept of expansiveness: cw-exp, h-exp, measure-exp, N-exp, countable-exp, hyper-exp. 2- expansiveness on surfaces. Robust N-expansiveness on C^r topology. Local conectedness on cwexp of surfaces.
Bibliografia:
Notes to be prepared by Artigue and Vieitez.
N. Aoki and K. Hiraide, Topological theory of dynamical systems, North-Holland, 1994.
A. Artigue, J. Brum, and R. Potrie, Local product structure for expansive homeomorphisms, Topology Appl. 156 (2009).
A. Artigue, Hyper-expansive homeomorphisms, Publicaciones Matemáticas del Uruguay 14 (2013).
A. Artigue, M. J. Pacífico, and J. L. Vieitez, N-expansive homeomorphisms on surfaces, Communications in Contemporary Mathematics (2014).
R. Bowen, Entropy-expansive maps, Trans. of the AMS 164 (1972).
A. Fathi, Expansiveness, hyperbolicity and Hausdor dimension, Commun. Math. Phys. 126 (1989).
K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math. 27 (1990).
H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993).
J. Lewowicz, Lyapunov Functions and Topological Stability, J. Di. Eq. 38 (1980).
J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Bras. Mat. 20 (1989).
R. Mañé, Expansive diffeomorphisms, Dynamical SystemsWarwick 1974, 1975.
R. Mañé, Expansive homeomorphisms and topological dimension, Trans. of the AMS 252 (1979).
C. A. Morales and V. F. Sirvent, Expansive measures, 29o Colóq. Bras. Mat., IMPA, 2013.
C. A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst. 32 (2012).
M. J. Pacifico, E. R. Pujals, M. Sambarino, and J. L.Vieitez, Robustly expansive codimensionone homoclinic classes are hyperbolic, Ergodic Theory Dynam. Systems 29 (2009).
M. J. Pacifico, E. R. Pujals, and J. L. Vieitez, Robustly expansive homoclinic classes, Ergodic Theory Dynam. Systems 25 (2005).
M. J. Pacifico and J. L. Vieitez, Entropy expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut. 21 (2008), no. 2.
W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), no. 6.
J. L. Vieitez, Expansive homeomorphisms and hyperbolic dieomorphisms on 3-manifolds, Ergod Theory Dynam. Systems 16 (1996).
J. L. Vieitez, Lyapunov functions and expansive diffeomorphisms on 3D-manifolds, Ergod Theory Dynam. Systems 22 (2002),

Dominik Kwietniak, Universidad Jagiellonian, Cracóvia, Polónia
Datas: de 25 a 28 de janeiro de 2016.
Horário: 13:00 às 15:00hs. Sala: C-116.
Ementa: Topological dynamics, as the name suggests, studies properties of dynamical systems from the topological viewpoint. This general approach allows extra degree of flexibility and is complementary to the more geometric setting encountered in smooth dynamics. Shift spaces arising via symbolic representations can be considered as perfect examples of purely topological actions. The goal of this minicourse would be to present an introduction to the field and may be complementary to minicourses on expansive dynamics and entropy.
Bibliografia:
D.Lind B.Marcus An introduction to symbolic dynamics and coding
P.Kurka Topological and symbolic dynamics
J.de Vries Topological dynamical systems