Seminários

Lagrange and Markov spectra

Resumo: The Lagrange spectrum and Markov spectrum are subsets of the real line with complicated fractal properties that appear naturally in the study of Diophantine approximations. These objects give a rich connection between Number theory and Dynamical systems, which has had an impressive development in recent years Perron gave a symbolic dynamic characterization of those sets. Symbolic dynamics is a powerful tool to encode the trajectory of a point by a sequence, where each symbol of a sequence corresponds to a piece of the space.

In this talk we will give the definition of the spectra, the continuity of its Hausdorff dimension, as well as explaining its connection with combinatorial objects such as the Cohn tree. Most of the definitions are elementary, so the talk can be attended by a wide audience.

The asymptotic description of the wave propagation on the lattice

Resumo: We consider the equation of wave propagation on the lattice with the governing equation similar to the wave equation but with the Laplacian replaced by its difference approximation. We can present the difference derivative with the help of the shifting operator which leads to some continuous equation. This equation is in such a form that for its investigation the semi-classical method (WKB method) of asymptotic analysis can be applied. We pose the Cauchy problem with localized initial conditions and want to study the semi-classical asymptotics while the parameter of localization tends to zero. We have two small parameters in this problem: the lattice step and localization parameter and we will discuss different situations of the ratio of these parameters and the form of the solution in these situations. In many cases the asymptotic can be presented with the help of Airy functions, but in some cases the asymptotic can be presented also with the help of Jacobi theta-function.

On Natural Language Processing (NLP): 1) its connection to high quality images via Stable Diffusion, and 2) its unifying app ChatGPT that organizes all the existing text functions.

Resumo: In the last year, Natural Language Processing (NLP) was fundamental in the creation of images of high quality via appropriate text prompts (the inverse of image captions). One such program was DALL-E, of OPENAI. Unfortunately, it is not open source. But later (in August of 2022) StabilityAI created Stable Diffusion which also creates high quality images and it is open source, so we can understand it and extend it. On item 2) we observe that it uses the already existing Large Language Model GPT3.5 which is already used for well known text functions, like summarization, translation, text generation, sentiment analysis, and others. But it adds value in the sense you do not have to program those text functions, or integrate them. Also, it is very powerful in programing; it can create a homepage for you, including all the technical items, connection to internet, CSS, HTML, database creation and handling. In other areas of knowledge, it is not very reliable, but partial information could be useful if one has a critical eye.

O problema de Burnside

Resumo: A proposta original do problema de Burnside era: um grupo de torção finitamente gerado é necessariamente finito? Construiremos um contraexemplo encontrado por Golod e Shafarevich em 1964. Além disso, faremos uma breve exposição de alguns refinamentos dessa questão: os chamados problema de Burnside limitado e problema de Burnside restrito.