Title: Superfícies especiais e soluções de equações diferenciais parciais
Abstract: The basic concepts of surfaces and the class of Weingarten linear surfaces will be reviewed. This special class of surfaces
contains those of constant Gaussian curvature or constant mean curvature.
The partial differential equations whose solutions correspond to these surfaces will be considered in this talk.
Given such a surface and the solution of the corresponding equation, geometric transformations (by Bäcklund and Ribaucour) that allow to obtain
new surfaces of the same type and therefore new solutions of the same differential equation.
The composition of such transformations allows to obtain a multitude of new solutions algebraically. In particular, for surfaces of Gaussian
curvature K = -1, which correspond to the solutions of the sine-Gordon equation, we will answer the question: composition of two Bäcklund
transformations is equivalent to one transformation of Ribaucour?
Several examples of surfaces and solutions of differential equations will be displayed.
Email:email@example.com Lattes CNPq:Clique aqui Minibio: She has a degree in Mathematics from the Universidade Federal do Rio de Janeiro (1967), a
master's degree in Mathematics from the University of Michigan (1969), a doctorate in Mathematics from the
Instituto de Matemática Pura e Aplicada (1972) and a post-doctorate from the University of California, Berkeley
(1975-1978). She is currently Professor Emeritus at the Universidade de Brasília, CNPq productivity scholarship,
member of the CNPq Mathematical/Statistics Advisory Committee, member of the Advisory Board of FAPDF and
president of the Fundação de Estudos Em Ciências Matemáticas (since 1997) She was president of the Sociedade
Brasileira de Matemática (SBM), representative of the area of Mathematics, Probability and Statistics at CAPES
(twice) and representative of Exact Sciences at the Scientific Technical Council of CAPES. She was Editor of the
magazine Matemática Contemporânea published by SBM, for 20 years. She was Coordinator of the Graduate Program
in Mathematics at UnB for 10 years, Head of the Mathematics Department at UnB and a member of the Science and
Technology Committee of the Distrito Federal. She was elected a full member of the Academia Brasileira de Ciências
and The World Academy of Sciences (TWAS). She was distinguished by the Presidency of the Republic with the
National Order of Scientific Merit in the Comendador Class and in the Grã-Cruz Class. She received a Moção de Louvor
from the Legislative Chamber of the Distrito Federal and the title of Honorary Associate of SBM. As a researcher
in Mathematics she works with an emphasis on Differential Geometry. It is mainly dedicated to the study of the
geometry of varieties and the interaction between differential geometry and differential equations. She
supervised 25 masters and 27 doctors. She was a visiting professor at several Brazilian and foreign universities,
including being a guest speaker at a considerable number of conferences at universities and national and
international scientific congresses. She has several publications, including books and research articles
published in specialized international journals.
The following post-graduate courses will be offered in XLIX Summer School. They will occur from January 4th to February 12th, 2021.
Attention: The courses will be taught in Portuguese.
Email:firstname.lastname@example.org Personal Page:Click here Minibio: Born in goiânia-GO, where he studied physics at PUC - Goiás, he has a doctorate from Universidade
de Brasília (2014-2016). His formation revolves around Ergodic Theory and Statistical Mechanics. Currently, his
favorite research topics are, Thermodynamic Formalism Via Transfer Operators and Random Dynamical Systems.
Functions of one complex variable.
Cauchy-Goursat theorem, Cauchy's integral formula.
Taylor and Laurent series.
Poles and residues.
Conformal mappings, Riemann's theorem.
Entire and meromorphic functions.
1. John Conway, Functions of One Complex Variable, Springer-Verlag, 1978;
2. Ahlfors L.; Complex Analysis, MC Graw-Hill/New York, 1972;
3. Hille E.; Analytic Function Theory, Adisson Wesley, 1971;
4. Rudin W.; Real and Complex Analysis, Graw-Hill/New York, 1968.
Email:email@example.com Personal Page:Clique aqui Minibio: Alex obtained his degree in Mathematics (2007) from Universidade Estadual de Londrina (UEL).
He obtained his master’s degree in Mathematics (2011)
from the Universidade Estadual de Maringá (UEM) and his PhD in Mathematics (2016) from the Universidade de Brasília. He defended
his thesis in Algebra, more specifically, in Group Theory. From 2016 to 2018, he worked as an assistant professor at the Universidade
Tecnológica Federal do Paraná (UTFPR), Guarapuava campus. Since 2018, he is an adjunct professor at the Universidade de Brasília.
His research is focused mainly in groups generated by automatons, groups closed by state and groups with finite orbits via automorphism.
1. Systems of linear equations.
2. Vector spaces.
4. Primary decomposition.
5. Canonical form of Jordan.
6. Spectral theorem.
7. Inner product.
8. Multilinear forms – tensors.
Minibio: Edgard got his degree in Mathematics at the Instituto Superior Técnico (Portugal, 2013)
under the direction of Diogo Gomes. Since January 2017, Edgard has worked as an Assistant Professor in the
Department of Mathematics at PUC-Rio. Researcher of the National Council of Science and Technology and Jovem
Cientista do Estado do Rio de Janeiro (FAPERJ-Brasil), Edgard is Junior Associate Fellow of the International
Centre for Theoretical Physics (ICTP-Trieste) and grantee of the Instituto Serrapilheira. In December 2019,
Prof. Pimentel was elected Affiliated Member of the Brazilian Academy of Sciences. Edgard likes to think about
Partial Differential Equations, mostly at the intersection of regularity theory and geometric analysis
A natural question in the analysis of PDEs concerns the regularity of the solutions to a given problem. That is,
the impact of the equation's structure on the smoothness degree of its solutions. A typical instance where it can
be observed is available in basic courses on mathematical analysis: twice-differentiable functions that happen to
be harmonic are, indeed, analytic. In parallel to regularity properties of the solutions, one finds the analysis of
geometric properties intrinsic to the equation, e.g. the local regularity of the zero-level sets of solutions. We
are interested in the intersection of those two variants: the structure of the equation under analysis changes
(discontinuously) with respect to properties of the solutions. To be more precise, we start our discussion with a
non-convex Hamilton-Jacobi equation in the presence of gradient constraints; in this setting we detail the optimal
regularity of solutions and derive a free boundary condition. In the sequel we focus on a fully nonlinear problem
whose diffusion structure depends on the sign of solutions; in this setting we talk about optimal regularity and
geometric properties of the free interface. Finally, we consider degenerate fully nonlinear problems; in this
setting the sign of the solutions affects the degeneracy rates of the problem, yielding a two-flavors free boundary
problem. We cover the basics of viscosity solutions, discuss a few strategies in regularity theory and put forward
elementary material on free boundary analysis.
Automata, languages and groups of automorphisms of rooted trees
The minicourse will be taught in English.
January 18th, 19h and 21th of 2021, from 10am to noon
Minibio: Marialaura Noce studied mathematics at the University of Salerno (Italy), where she obtained her degree in 2016. After that, she earned a PhD in mathematics in a cotutelle agreement between the
University of the Basque Country (Bilbao) and the University of Salerno, in December 2019. From January
2020 to July 2020, she was a Researcher Associate at the University of Bath (U.K.) and since September
2020, she is a Research Assistant at the University of Göttingen (Germany). Her research activity is
mainly based on groups of automorphisms of rooted trees and branch groups, Engel conditions and
algorithmic problems in groups. She has authored 10 papers in international journals and delivered
invited and contributed talks at international conferences.
In this course we will give an introduction to automata groups, explaining their connections with groups of
automorphisms of rooted trees, and formal languages. Then, we will discuss remarkable examples, important
recent developments of this theory, and open problems.
Minibio: Kaye Silva is a professor at the Institute of Mathematics and Statistics at Universidade
Federal de Goiás. He obtained his bachelor degree in Mathematics from Unieversidade Estadual de Goiás (2006) and
his master's degree in mathematics from Universidade Federal de Goiás (2012). He obained his Ph.D. in Mathematics from Universidade Federal de Goiás in 2015. His current research interests are focused on variational methods, in particular, the applicability of the Nehari manifold method in some rough circumstances
We introduce the extremal parameters and show its relation with topological changes of the Nehari set. As a
consequence we deduce existence, non-existence and multiplicity of solutions to a large class of variational
Minibio: Pedro Gaspar has a degree in Mathematics from the Institute of Mathematical and
Computer Sciences, Universidade de São Paulo (2013) and a PhD in Mathematics from the
Instituto Nacional de Matemática Pura e Aplicada (2018), under the guidance of Professor Fernando Codá
Marques. He is currently L. E. Dickson Instructor at the University of Chicago, United States. He has
experience in Geometric Analysis, working mainly on minimal surfaces, phase transitions and Allen-Cahn
equation. In 2019, he received the Professor Carlos Teobaldo Gutierrez Vidalon Award, for the best thesis
in mathematics defended in Brazil in the year prior to the award, considering originality and quality.
Minimal surfaces are critical points of the area function and are among the most studied objects in
Differential Geometry and Geometric Analysis, with deep connections with different areas, such as Partial
Differential Equations (PDEs), Calculation of Variations and Mathematical Physics. In this minicourse, we will
discuss the relationships between such surfaces and the theory of phase transitions - in particular, the
Allen-Cahn equation. These relationships have been explored since the 1970s to predict and prove results on
minimal hypersurfaces using classical techniques of Calculation of Variations and PDEs and, conversely, to
obtain information on certain semilinear equations through knowledge of the geometry of such hypersurfaces.
With a focus on geometric aspects of the Allen-Cahn equation, we will talk about some approaches to prove the
existence of solutions and study their qualitative properties and discuss how the solutions of this equation
(and its sets of zeros) approach minimal hypersurfaces, providing a useful approximation of the functional area.
Analysis on higher dimensions and basic knowledge of Differential Geometry and Partial Differential Equations.
Some familiarity with notions of Riemannian Geometry (Riemannian metrics, geodesics, submanifolds) is recommended.