Christina Brech: Um espaço de Tsirelson não-separável

Data: quinta-feira, 28 de setembro de 2017, às 10h

Sala: Auditório Antonio Gilioli 

Palestrante: Christina Brech (IME-USP)

Título: Um espaço de Tsirelson não separável 

Resumo:  Apresentaremos elementos da construção obtida conjuntamente com J. Lopez-Abad e S. Todorcevic de um espaço de Tsirelson não separável: um espaço de Banach reflexivo de densidade não enumerável que não possui cópias dos espaços $\ell_p$. A construção envolve famílias de subconjuntos finitos de um conjunto de índices não enumerável similares às famílias de Schreier. 

Noé de Rancourt: Ramsey theory, games, and Banach-space dichotomies, part 3

Data: quinta-feira, 24 de agosto de 2017, às 13h30h

Sala: Sala B-05 (Bloco B do IME)

Palestrante: Noé de Rancourt (Université Paris 7)

Título: Ramsey theory, games, and Banach-space dichotomies, part 3

Resumo:  In this talk, which is a sequel of the two preceding ones, I will explain the difficulties that make it impossible to have a real analogue of Mathias-Silver theorem in a general setting, and discuss how these
difficulties can be solved. Then, in the abstract setting defined in the last talk, I will present a general Ramsey result, where the possible conclusions will be formulated in terms of the existence of winning strategies in two-players games with perfect information. I will then present Gowers' Ramsey-type theorem for block-sequences, which is an approximate version of the last result in the setting of Banach spaces with a basis, and use it to end the proof of Gowers' first dichotomy. If time remains, I will present a generalisation of the abstract Ramsey theorem which also implies Borel determinacy of games on the integers, and hence is much stronger than an usual Ramsey theorem, in a matemathical way.
Apoio: Projeto USP-Cofecub "Geometria dos espaços de Banach"

Noé de Rancourt: Ramsey theory, games, and Banach-space dichotomies, part 2

Data: quinta-feira, 17 de agosto de 2017, às 13h30h

Sala: Auditório Antonio Gilioli (Bloco A do IME)

Palestrante: Noé de Rancourt (Université Paris 7)

Título: Ramsey theory, games, and Banach-space dichotomies, part 2

Resumo: In this talk, I will give an introduction to Ramsey theory and discuss how this theory can apply to give a proof of Gowers' first dichotomy (which I introduced in the last talk). This theory consists in a collection of results ensuring that given a partition of a structure, at least one of the pieces of the partition contains a sufficiently large substructure. I will begin with introducing Mathias-Silver theorem, which is a Ramsey result in the contexts of sets without structure, and explain what difficulties can happen when we try to adapt this result to other contexts, like Banach spaces with bases. Then, in an abstract and purely combinatorial formalism, I will discuss how these difficulties can be solved and I will present a general Ramsey result, where the possible conclusions will be formulated in terms of the existence of winning strategies in two-players games with perfect information. This theorem, in the context of Banach spaces, is a form of the Ramsey-type theorem Gowers used to prove his dichotomies.

Apoio: Projeto USP-Cofecub "Geometria dos espaços de Banach"

Noé de Rancourt: Ramsey theory, games, and Banach-space dichotomies, part 1

Data: quinta-feira, 10 de agosto de 2017, às 13h30h

Sala: Auditório Antonio Gilioli (Bloco A do IME)

Palestrante: Noé de Rancourt (Université Paris 7)

Título: Ramsey theory, games, and Banach-space dichotomies, part 1

Resumo: This is the first of a series of three talks, during which I will present some combinatorial tools that can be used in the study of Banach-space dichotomies. These dichotomies are one of the fundations of Gowers' loose classification project, which aims to build a list of "well-understood" classes of separable Banach spaces such that every space has a subspace in at least one class. In this first talk, I will present in more details Gowers' project, its motivations and some of the dichotomies Gowers obtained, and then I will give an introduction to Ramsey theory, the combinatorial tool the most used in the proof of dichotomies. This theory consists in a collection of results ensuring that given a partition of a structure, at least one of the pieces of the partition contains a sufficiently large substructure. I will in particular introduce Mathias-Silver theorem (one of the main results in Ramsey theory) and begin to explain how it can be adapted to the setting of Banach spaces.

Apoio: Projeto USP-Cofecub "Geometria dos espaços de Banach"