Seminários

Resumo: Minimal surfaces with finite total curvature in three-dimensional spaces have been widely studied in the recent decades. A celebrated result in this subject states that, if $\Sigma \subset \mathbb{R}^3$ is a complete immersed minimal surface of finite total curvature, then it has finite conformal type. Moreover, its Weierstrass data can be extended meromorphically to the punctures and its total curvature is an integral multiple of $4 \pi$.

In this talk, the goal is to present some theorems concerning minimal surfaces in $\mathbb{M}^2 \times \mathbb{R}$ having finite total curvature, where $\mathbb{M}^2$ is a Hadamard manifold. We obtain analogous versions of classical results in Euclidean three-dimensional spaces. The main result gives a formula to compute the total curvature in terms of topological, geometrical and conformal data of the minimal surface. In particular, we prove the total curvature is an integral multiple of $2\pi$.

Resumo: Dada uma distribuição numa variedade projetiva, estudamos o problema de achar uma folheação por curvas tangente com o mesmo corpo de integrais primeiras racionais.

Resumo: In the seminar we discuss isospectral symmetries of matrix Schrödinger operators with periodic potentials (with a possible generalization to the quasi-periodic case, and to the stochastic case). Spectral theory of the (matrix) Schrödinger operator is an important mathematical tool in such areas as functional analysis, dynamical systems, integrable models etc. Following A.P. Veselov and A.B. Shabat “...the KdV theory is exactly the theory of isospectral symmetries of type (1) of the Schrodinger operator and, therefore, could have arisen independently within the framework of spectral theory”. We will study the dressing chain of associated matrix Riccati equations. Our main finding in this case is a special (local) symmetry between diagonal operators and operators with a non-trivial interaction. We also consider possible applications of these results in the theory of integrable systems, analysis of stability of mechanical systems, design of periodic structures with desired spin/polarization transport (1-D anisotropic photonic crystals).

Resumo: In this talk I will present experiments on how to integrate the power and flexibility of a general purpose programming language (Python, running on Jupyter) with the stability and extensive sound and effects library of a modern digital audio workstation (Logic Pro X) to compose an electronic dance music track. I will introduce and use simple music theory concepts to generate chord progression, bass line, arpeggios and other musical elements, which will be recorded in Logic via MIDI, using Logic's MIDI Clock messages for synchronization. The main advantage of using Python is the fact that it is popular and general-purpose -- there's no need to learn a specific music-related programming language. The main advantage of sending algorithmically-generated music to Logic is that this environment holds a comprehensive library of sounds and effects that allow fully producing a music piece into a professional-sounding track. I will present one such track, composed specially for this talk using some of the algorithms to be discussed.