# Seminários

## HOJE

Fourier transform for locally compact abelian groups

**Expositor:**

*Oscar Guillermo Riano*- IMPA

**Colóquio dos Alunos**

**Resumo: **The Fourier transform has been widely studied for periodic $\mathbb{C}$-valued functions on $\mathbb{R}$, and functions defined on $\mathbb{R}^n$. This Fourier mapping and its characteristics are not only restricted to Euclidean spaces, they can be extended to certain mathematical spaces.

The main goal of this presentation is to briefly introduce the extension of the Fourier transform to locally compact abelian groups (LCA). Thus, we will start recalling some properties of these spaces. Later, we will obtain some properties of the Fourier transform that we have for granted on the real numbers.

Cohomology of Lie algebroids on schemes

**Expositor:**

*Ugo Bruzzo*- Sissa

**Teoria de Representações**

**Resumo: **I will consider Lie algebroids on noetherian separated schemes and will show how their cohomology can be described as a derived functor. I will also describe applications to the nonabelian extensionproblem for such Lie algebroids. (Partly in collaborationwith E. Aldrovandi and V. Rubtsov).

## PRÓXIMOS

Model order reduction for large-scale problems and applications to optimization with PDE constraints.

**Expositor:**

*Alessandro Alla*- PUC-Rio

**Matematica Aplicada e Computacional**

**Resumo: **Reduced order models (ROMs) plays a critically enabling role in large-scale scientific computing applications. The ROM architecture is being exploited in many simulation based physics and engineering systems in order to render tractable many high-dimensional simulations. Fundamentally, the ROM algorithmic structure is designed to construct low-dimensional subspaces, typically computed with SVD, where the evolution dynamics can be projected using a Galerkin method. Thus, instead of solving a high-dimensional system of differential equations (e.g. millions or billions of degrees of freedom), a rank* *$l$ model can be constructed in a principled way.

In this talk we will introduce some order reduction techniques such as Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) to show the effectiveness of these methods to approximate numerically nonlinear PDEs. We will also show examples of model reduction to solve optimal control problems (e.g. feedback control) for PDEs. Joint works with M. Falcone, N. Kutz, V. Simoncini and S. Volkwein.

Solution of piecewise smooth equation systems by successive abs-linearization

**Expositor:**

*Andreas Griewank*- Yachaytech (Equador)

**Otimização**

**Resumo: **

Most nonsmooth vector functions of practical interest are piecewise smooth and many can be easily expressed in abs-smooth form, i.e. as the composition of smooth elementals and the abs. max, or min function. Linearizing the smooth elementals at a reference point one obtains a piecewise smooth local approximation with a second order approximation error. Many properties of this abs-linearization, which can be obtained by minor extensions of AD tools. are closely related to that of the underlying nonsmooth system. We discuss these relations and the local convergence properties of the natural successive abs-Iinear equation solving (SALES) generalization of Newton's method in the smooth case. Of course, there are many connections to other nonsmooth equation solving algorithms and theorems.

Introduction to turbulence and blow up

**Expositor:**

*Uriel Frisch*- Laboratoire Lagrange, Observatoire and Universite Cote d'Azur Nice, France

**Palestra Especial**

**Resumo: **This lecture is intended to give a rough idea of some of questions arising in turbulence: Reynolds number and instabilities, chaos and the butterfly effect, fully developed turbulence as a spatial variant of Brownian motion, fractals in Nature and in turbulence, simple models leading to finite time explosion (blow-up), nonlinear depletion and possible avoidance of blow-up.

The lecture will be elementary and accessible to students whose background is in mathematics, or physics, or engineering or numerical analysis.

Burgers turbulence and the spurious $k^{-3}$ energy spectrum debunked

**Expositor:**

*Uriel Frisch*- Laboratoire Lagrange, Observatoire and Universite Cote d'Azur Nice, France

**Matematica Aplicada e Computacional**

**Resumo: **Formal manipulations of the one-dimensional inviscid and unforced Burgers equation suggest that the energy spectrum $E(k)$ might follow a $|k|^{-3}$ rather than the $|k|^{-2}$, generally associated with shocks. The use of the Fourier-Lagrangian solution, introduced by Fournier and Frisch (L'equation de Burgers deterministe et statistique. 1983. J. Mec. Theor. Appl, vol. 2, pp. 699-750.) shows that the spurious $|k|^{-3}$ solutions stem from an improper use of the multi-stream behaviour associated with the Burgers-Vlasov equation. This will be a blackboard lecture intended for advanced students and specialists in PDEs, field theory and/or catastrophy theory.

Color Lie (super)algebras and Z2 × Z2 symmetries in the Levi-Leblond equation.

**Expositor:**

*Zhanna Kuznetsova*- UFABC

**Teoria de Representações**

**Resumo: **We investigate systems with color Lie (super) algebra symmetries.In the first part of the talk I give a brief introduction in color Liealgebras and superalgebras focusing on the Z2*Z2 particular case.In the second part I present an analysis of the symmetry operators of theLevy-Leblond equation which is a nonrelativistic wave equation of a spin 1/2 particle (nonrelativistic analog of the Dirac equation). It is shown that the equation has two kinds of symmetries. One is given by the super Schroedinger algebra and the other one by a Z2×Z2 graded Lie superalgebra. The realizationof the Z2*Z2 superalgebra is presented in terms of matrix differential operators.