Schedule:

Monday to Friday, 9 am to 11:30 am

Room:

ICC ANF 12

The official language of the course is Portuguese.

Professor:

André Caldas de Souza

Tutors:

Henrique Souza
Mateus Fleury
Nowras Ali

Tutoring schedule: Monday, Wednesday and Friday, from 11 am to 12:30 pm.
Room: ICC ANF 12

Abstract:

For more information about the course, exercises lists and others, see
http://andrec.mat.unb.br/ensino/topologia/
The professor recommends that before the first day of class, everyone has read chapters 1 and 2 of the class notes.
1. Conceitos básicos de topologia geral
2. Topologias construídas a partir de outras topologias
3. Conexidade
4. Compacidade
5. Axiomas de separação

References:

1. J. L. Kelley, General Topology, Graduate Texts in Mathematics 27, Springer, 1955
2. J. R. Munkres, Topology, Prentice Hall, 2ª edição
3. Notas de Aula

Schedule:

Monday, Tuesday, Thursday and Friday, 2pm to 5 pm

Room:

Department of Mathematics, -Room B 427/10

The official language of the course is Portuguese.

Professor:

Sandra Imaculada Moreira (UnB e UEMA)

Abstract:

1. Espaços normados;
2. Teoremas de Hahn-Banach, Baire, Aplicação aberta, Gráfico fechado e Banach-Steinhaus;
3. Formas geométricas do Teorema de Hahn-Banch;
4. Topologias fracas; reflexividade e separabilidade;
5. Espaços L^p;
6. Espaços de Hilbert;
7. Teoria espectral de operadores lineares compactos e auto-adjuntos;
8. Espaços de Sobolev sobre a reta e problemas variacionais.

References:

1. Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira, Fundamentos de Análise Funcional, SBM, Coleção Textos Universitários, Rio de Jan_eneiro, 2014;
2. Brézis H.: Functional Analysis, Sobolev spaces and partial differential equations, universitext. Srpinger, New York, 2011;
3. Conway, J.B.: A course in Functional Analysis. Second edition. Graduate Texts in Mathematics, 96. Spinger-Verlag, New York, 1990;
4. Folland, G.B.: Real Analysis. Modern techniques and their applications . Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., Ney York, 1999;
5. Rudin W.: Functional Analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991;
6. Kesavan, S.: Topins in Functional Analysis ans applications. John Wiley & Sons, Inc., New York, 1989;
7. Yosida K.; Functional Analysis, Springer-Verlag, New York, 1974;
8. Kreyszig, E.: Introdutory Functional Analysis with applications. John Wiley & Sons, Inc., New York, 1989;
9. Oliveira, C.: Introdução à Análise Funcional, IMPA;
10. Stein, E.M.; Shakarchi, R.: Functional Analysis. Introduction to further topics in Analysis. Princeton Lectures in Analysis, 4. Princeton University Press, Princeton, New Jersey, 2011.

Schedule:

Monday to Friday, 2pm to 4:30 pm

Room:

ICC ANF 12

The official language of the course is Portuguese.

Professor:

Alexei Krassilnikov

Abstract:

1. Sistemas de equações lineares.
2. Espaços vetoriais.
3. Polinômios.
4. Decomposição primária.
5. Forma canônica de Jordan.
6. Teorema espectral.
7. Produto interno.
8. Formas multilineares – tensores.

References:

1. Hoffman/Kunze; Álgebra Linear; Ed. Livros Técnicos e Científicos, 1979;
2. Lang, Serge; Álgebra Linear; Ed. Ciência Moderna, 2003;
3. Halmos, P. Espaços Vetoriais de Dimensão Finita; Ed. Campus, 1978;
4. Lipschutz, S; Álgebra Linear; Ed. McGraw-Hill Makron Books 1994.

Schedule:

February 3rd to 7th, 2020, 10am - 12noon

Room:

Department of Mathematics, Room A - Miniauditorium

Professor:

Guram Donadze (Institute of Cybernetics of Georgian Technical University and Universidade de Brasília)

Abstract:

We will study the theory of Hilbert polynomials and prove that for regular local rings the Krull, Chevalley
and global dimensions are the same.

References:

1. Atiyah M. Macdonald I. G. Introduction to commutative algebra. Massachusetts: AddisonWesley Publishing, 1969.
2. Nagata M. Local rings. Interscience Publishers a division of John Wiley and Sons, New York-London, 1962.
3. Serre J. P. Local algebra. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000.

Schedule:

January 27 to 31, 2020, 8am - 10am

Room:

Department of Mathematics, Room A - Miniauditorium

Professor:

João Vitor da Silva (Universidade de Brasília)

Abstract:

In this series of Lectures we will address some recent progresses in regularity theory of nonlinear elliptic/p´arabolic PDEs and Free Boundary Problems, which were developed in the last years. The insights behind the expositions consist of explaining their mathematical relevance, intrinsic difficulties in being overcome and applications in other classes of problems. In the end of the Lectures we will present new directions and some mathematical expectations for the next years in such subjects of research.

References:

1. M.D. Amaral, J.V. da Silva, G.C. Ricarte & R. Teymurazyan, Sharp regularity estimates for quasilinear evolution equations. Israel J. Math. 231 (2019), no. 1, 25-45.
2. J.V. da Silva, P. Ochoa & A. Silva, Regularity for degenerate evolution equations with strong absorption. J. Differential Equations 264 (2018), no. 12, 7270-7293.
3. J.V. da Silva, J. Rossi & A. Salort, Regularity properties for p−dead core problems and their asymptotic limit as p → ∞. J. London Math. Soc. (2) 99 (2019) 69-96.
4. J.V. da Silva & A. Salort, Sharp regularity estimates for quasi-linear elliptic dead core problems and applications. Calc. Var. Partial Differential Equations 57 (2018), no. 3, 57: 83.
5. J.V. da Silva & E.V. Teixeira, Sharp regularity estimates for second order fully nonlinear parabolic equations. Math. Ann. 369 (2017), no. 3-4, 1623-1648.

The course will be taught in Portuguese.

Schedule:

February 4th to 7th, 2020, 10am - 12noon.

Room:

Department of Mathematics, Room B - 427/10

Professor:

Augusto Ponce (Université catholique de Louvain)

Abstract:

O objetivo do curso é desenvolver e aplicar em detalhe técnicas que permitem demonstrar a existência e regularidade de soluções do problema de Dirichlet associado ao operador de Schrödinger:
−Δu + V u = μ ∈ Ω,
u = 0 on ∂ Ω, onde V é um potencial positivo e µ é uma medida boreliana finita. Os pré-requisitos são conceitos básicos de cursos introdutórios em teoria da medida, equações diferenciais parciais e espaços de Sobolev.

References:

1. Class notes
2. Augusto C. Ponce, Elliptic PDEs, Measures and Capacities. From the Poisson equation to Nonlinear Thomas-Fermi problems. EMS Tracts in Mathematics. European Mathematical Society, Zürich, 2016. EMS Monograph Award in 2014.

The course will be taught in Portuguese.

Schedule:

January 20 to 24, 2020, 10am - 12noon.

Room:

Department of Mathematics, Room A - Miniauditorium

Professor:

Felippe Guimarães (Universidade de Brasília)

Prerequisites:

Conceitos básicos de variedades diferenciáveis, geometria Riemanniana, tensores e grupos.

Abstract:

Dado um fibrado vetorial, podemos definir o conceito de transporte paralelo ao longo de curvas da base com relação a uma conexão sobre este fibrado. Ao considerar curvas fechadas em um ponto, o transporte paralelo nos fornece um isomorfismo natural da fibra sobre este ponto nela mesmo. O conjunto de isomorfismos obtidos por esta construção formam um grupo chamado Grupo de Holonomia.
O objetivo deste minicurso é introduzir o Grupo de Holonomia de uma variedade Riemanniana, associado a conexão de Levi-Civita e o fibrado tangente, e apresentar a sua relação natural com a geometria intrinsica de uma variedade. Para no final motivar o famoso Teorema de Berger, sobre classificação de grupos de holonomia em variedades Riemannianas.

References:

1. Genaro, R., Grupo de holonomia e o teorema de Berger, Dissertação, UNICAMP. Disponível em: repositorio.unicamp.br/handle/REPOSIP/306399
2. Clarke, A., Santoro, B., Holonomy groups in riemannian geometry. Publicações Matemáticas do IMPA, PM-39. Disponível em: https://arxiv.org/pdf/1206.3170.pdf

Schedule:

January 27 to 31, 2020, 10am - 12noon.

Room:

Department of Mathematics, Room A - Miniauditorium

Professor:

Stefano Buccheri (Universidade de Brasília)

Abstract:

The study of Elliptic Partial Differential Equation (EPDE) is a classical, huge and still really active research field in Mathematics. The attempt of giving a reasonably good description of the development of just one of its branch require a full-course of at least one semester. The much more modest aim of this small-course is to give a flavor, and maybe awake some interest, on some strategy and tools you can use when the data of your problem get tough.