O Instituto de Ciências Matemáticas e de Computação (ICMC), da USP em São Carlos, promove no período de 11 de abril a 23 de maio o minicurso An introduction to geometric motivic integration, arc spaces and motivic zeta functions. O treinamento, a ser ministrado por Peter Konstantinov Petrov, será realizado às quartas-feiras, das 14h às 16h, sala 5-003 do ICMC.
O curso é gratuito e aberto a todos os interessados. Para se inscrever, é necessário preencher a ficha de inscrição e enviar para o e-mail eventos@icmc.usp.br até as 12h do dia 11 de abril.
O objetivo do minicurso é estudar espaços de arcos, anéis de Grothendieck de variedades, sua localização e completamento; promover estudo sobre medida e integração motívica e a regra de transformação birracional, teorema de Kontsevich para variedades suaves e sua generalização para variedades com no máximo singularidades canônicas de Gorenstein; estudar funções zeta motívicas e suas aplicações a singularidades e aritmética.
Detalhamento do curso (em inglês, informado pelo ministrante)
This course is planned to be an introduction to geometric motivic integration, arc spaces and motivic zeta functions. The aim is to explain and discuss the ideas and constructions in motivic integration to PhD students and post docs with only a very basic knowledge of algebraic geometry.
The course will start with some basic facts on arc spaces. Then the Grothendieck ring of varieties, its localization and completion will be defined as the ring of values of the integral. A motivic measure and integral will be introduced next, with the algebra of integrable sets, the equivalent of the Borel algebra in the case of Lesbesgue integral. Then will be formulated ``the user's friendly formula'', demonstrated on a few examples, and the basic theorem, the birational transformation rule. Using this the Kontsevich theorem, that smooth projective birational Calabi - Yau varieties have the same Hodge numbers will be obtained as an easy corrolary. It was the main motivation for inventing the motivic intgration by Kontsevich in 1995, developed over smooth varieties. A generalization of it for varieties with at worst Gorenstein canonical singularities (proposed by Denaf and Loeser) will be explained. Next, the motivic zeta functions, important for their applications to singularities and arithmetic, will be discussed briefly.
At the end of the course the arithmetic motivic integration, a powerful generalization of the geometric motivic integration, that has been developing in the last 10 years, will be introduced, and some of its applications to algebraic and Lie groups will be mentioned. Simple but informative examples will be frequently prefered before formal proofs.
Referências
1. Alistair Craw - An introduction to motivic integration, arXiv:math/9911179v2.
2. Julia Gordon, Yoav Yaffe - An overview of arithmetic motivic integration, 2008.
3. H.Kurke, A.Martin-Pizarro, A.Zheglov, I. Zhukov - Motivic integration: seminar talks, 2006.
4. Willem Veys - Arc spaces, motivic integration and stringy invariants, 2004.
Link para a ficha de inscrição: http://www.icmc.usp.br/~comunica/documentos/fichapetrov.doc
Informações:
Seção de Eventos do ICMC
Tel. (16) 3373-9146
