### Summer Program in Mathematics 2018

The Summer Program in Mathematics of IMECC is destined mainly to

- Graduate students in mathematics and/or related areas at any stage;
- Undergraduate students in mathematics, physics, engineering and/or related areas at any stage;
- Candidates applying for Master's degree program in Mathematics at IMECC;
- Graduate students incoming in the first semester of 2018 who wish to advance their credits and studies;
- Professors of the second and third cycles who wish to recycle.

#### Period of courses and activities:** January 04 to February 20, 2018.**

#### Application period: **September 01 to October 31, 2017.**

** **

#### Registration procedures

It is only needed to provide the following documents:

- Application form and a brief Academic Transcript that must be sent via the link.

#### Financial Support

We expect to have scholarships for students who are still depending on the decision of external funding agencies.We encourage students not yet graduated and at the end of undergraduation to apply for the summer program.

#### Courses

During the months of January-February 2017, the Summer Program in Mathematics at IMECC will offer 6 (six) courses. These belong to the framework of the Graduate/Undergraduate Courses of Mathematics of IMECC and are listed below.

** **

**MM202: Introduction to Analysis** (Undergraduate/Specialization level)

Prof. Maicon José Benvenutti (UFSC-Blumenau, SC)

__Program:__ Real numbers.Numerical sequences and series. Some topological notions on the real line. Real functions. Limit of functions. Continuous functions. Uniform continuity. Derivable functions. Taylor's formula. Aplications of the derivative of a function.

__Bibliography:__ (1) Lima, E. L., Curso de Análise, Vol. 1, Projeto Euclides, 14 ed., IMPA, 2012. (2) de Figueiredo, D. G., Análise I, 2 ed., 1996. (3) Rudin, W., Principles of Mathematical Analysis, Mc Graw Hill, 1976.

**MM719: Linear Algebra** (Master's level)

Prof. Dessislava H. Kochloukova (Unicamp, SP)

__Program:__ Vector spaces, basis and coordinates, linear transforms and matrices, rank, nullity, inner product, normal and self-adjoint operators, diagonalization. Dual space and the transpose, Cayley-Hamilton theorem, minimum polynomial of a linear endomorphism, Jordan form, real Jordan form, rational form. Multilinear transform, alternating function, determinant, tensor product of vector spaces, tensor algebra, symmetric tensor algebra. Grassmann's algebra, Clifford's algebra, structure of bilinear and quadratic forms, orthogonal and symplectic transforms.

__Bibliography:__ (1) K. Hoffman and R. Kunze, Linear Algebra (2nd edition), Prentice Hall (1971). (2) A. Kostrikin and Y. Manin, Linear algebra and geometry, Gordon and Breach (1989). (3) D. Northcott, Multilinear Algebra, Cambridge Univ. Press (1964). (capítulos 1 e 2). (4) Outras Referências: R. J. Santos, Álgebra Linear e Aplicações, disponível em versão eletrônica (pdf) em http://www.mat.ufmg.br/~regi/ S. Axler, Down with determinants, Springer (1967). K. Ikramov, Linear algebra: Problems book, Mir (1983).

**MM425: Functional Analysis I** (Ph.D. level)

Prof. Nestor Felipe Castañeda Centurión (UESC, BA)

__Program:__ Normed spaces and Banach spaces. Holder and Minkowski inequalities. Banach spaces of sequences and Banach spaces of functions. Subspace and quotient space. Finite-dimensional normed spaces and the Riesz Theorem. Hahn-Banach's theorem and its applications. Representation of linear functionals in the spaces l_p and L_p. Riesz's Representation Theorem. Theorem of Lax-Milgram. Duality. Reflexive Banach spaces. Uniform boundedness principle. The open mapping theorem and the closed-graph theorem. Spaces endowed with inner product and Hilbert spaces. Orthogonal projections. Orthonormal sets. Bessel inequality and Parseval identity. Linear and continuous operators. Compact operators in Banach spaces. Spectral theorem for self-adjoint compact operators in Hilbert spaces. Weak topology and weak-star topology. The Banach-Alaoglu Theorem.

__Bibliography:__ (1) Conway, J. B., A course in functional analysis. Second edition. Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990. (2) Honig, C. S., Análise Funcional e Aplicações, IME-USP. (3) Kreyszig, E., Introductory Functional Analysis with Applications, John Wiley. (4). Taylor, A. E., Lay, D. C., Introduction to functional analysis, Second edition, John Wiley &Sons, New York-Chichester-Brisbane, 1980. (5) Brezis, H., Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. (6) Bachman, G., Narici, L., Functional analysis, Dover Publications, Inc., Mineola, NY, 2000. (7) Lax, P. D., Functional analysis, Pure and Applied Mathematics, Wiley-Interscience, New York, 2002.

** **

**MM456-Ordinary Differential Equations **(Master's level)

Prof.: Douglas D. Novaes (Unicamp, SP)

__Program:__ Existence and Uniqueness theory. Successive approximation method for the existence and uniqueness of solutions. Peano's existence theorem. Maximal solutions, flows. Linear systems and their maximal solutions. Differentiable dependence of solutions on parameters and initial conditions. Differential of flow. Tubular Flow Theorem. Complete fields. Lie bracket of vector fields. Phase space. Classification of orbits. Hartman-Grobmann's theorem. Lyapunov Stability, Lyapunov functions and Lyapunov exponents. Poincaré-Bendixon Theorem. Conservative fields. Recurrence and Poincaré recurrence theorem.

Prerequisites: 1. Differential Calculus of Several Variables (or in Banach spaces). 2. General topology or topology of metric spaces.

__Bibliography:__ (1) Sotomayor, J. Lições de EDO. Priojeto Euclides. 1979. (2) Hartman, Philip, Ordinary Differential Equations, 2nd Ed., Society for Industrial & Applied Math, 2002. (3) Coddington, E.A. and Levinson, N. Theory of ordinary differential equations. New York: McGraw-Hill, 1955. (4) Hale, J.K. Ordinary differential equations. New York: Wiley-Interscience, 1969. (5) Hirsch, M.N. & Smale, S. Differential equations, dynamical systems and linear algebra. New York: Academic Press, 1974.

**MM634- Harmonic Analysis** (Master's and Ph.D. level)

Prof.: Mahendra P. Panthee (Unicamp, SP)

__Program:__ Fourier series and integrals. Hilbert transform. H(p)-spaces. Singular integrals. Interpolation theorems. Hardy-Littlewood maximal function. Calderon-Zygmund theory. Littlewood-Paley theory and multiplier operators. Hardy and BMO spaces. Applications.

__Bibliography:__ 1. Duoandikoetxea, J., Fourier Analysis, Graduate Studies in Mathematics, 29, AMS, Providence, RI, 2001. 2. Stein, E., Harmonic Analysis, Princeton University Press, Princeton, New Jersey, 1993. 3. Stein, E., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. 4. L. Grafakos, Classical and modern Fourier analysis. Pearson Education, Inc., Upper Saddle River, NJ, 2004.5. Sadosky, Cora Interpolation of operators and singular integrals. An introduction to harmonic analysis. Monographs and Textbooks in Pure and Applied Math., 53. Marcel Dekker, Inc., New York, 1979.

**MM813:** **Topics in Geometry I (An Introduction to Foliations Theory)** (Master's and Ph.D. level)

Prof.: Gabriel Ponce (Unicamp, SP)

__Program:__ Differentiable manifolds, Foliations, Topology of leaves, Holonomy and stability theorems, foliations and fibered bundles, Novikov's Theorem.

The main goal of this course is to study problems and tools from Foliations Theory. Foliations theory developed in an accelerated manner in the 70's and created a wide range of tools which are nowadays used in divers areas of mathematics as stochastic dynamics, dynamical systems, ergodic theory, group actions etc. In this course the objective will be to present an introduction to foliations from a more geometrical perspective, focusing on the topological, differential and dynamical behavior of foliations.

Prerequisites: Though we will give a quick revision of some of the required prerequisites in the first week, it is recommended that the student has basic knowledges in differential manifolds and multivariable calculus.

__Bibliography:__ 1. Cesar Camacho and Alcides Lins Neto. Geometric theory of foliations. Birkhäuser Boston, Inc., Boston, MA, 1985. 2. A. Candel and L. Conlon. Foliations I, vol. 23 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2000.