MM805 Tópicos de Análise I - 1° Semestre 2018

Polyfold theory

".. generalizes differential geometry and nonlinear Fredholm theory
to a class of spaces which are much more general than (Banach) manifolds.
These spaces may have (locally) varying dimensions and are described locally
by retracts on scale Banach spaces, replacing the open sets of Banach spaces
in the familiar local description of manifolds. .."

aulas: 4a/6a 10-12h sala 322 IMECC
primeiro encontro: 4a-f, 28 de Março (caráter organização das primeiras semanas)

Lecture Notes (AMS Open Math Notes) (to be extended as time permits)

Críterio de avaliação: Não haverá provas, nem notas, quem tiver frequência de pelo menos 75% de presença nas aulas será aprovado com o conceito 'S' -- assim o curso conta para os seus créditos sem afetar o CR (a média dos alunos).
Ementa: Chap. I of [HWZ]: M-polyfolds, sc-calculus, sc-Fredholm theory, orientations,
applications include Morse complex, Gromov-Witten theory, symplectic field theory (SFT).
Literatura principal:

[HWZ] Hofer, Wysocki, Zehnder "Polyfold and Fredholm Theory", (2017) 714pp, arXiv:1707.08941
[FFGW] Fabert et al, "Polyfolds: A first and second look", arXiv:1210.6670

Futuro: O curso disponibiliza uma parte do fundamento de varios novos projetos de pesquisa as quais começamos em 2018. Um colaborador, Urs Frauenfelder, vai visitar-nos no IMECC nas primeiras 2 semanas de Março.
Prerequisitos: Análise Funcional e Geometria Diferencial
Qualquer dúvida, por favor contata-me.

Some guide to some literature:

To get an overview of which deficits of classical approaches to infinite dimensional analysis, based on classical smoothness, the new theory is designed to overcome it is worthwhile to read the introductions of the following texts. In particular, the older papers are highly informative in this sense. Thereby one also gets an overview of the new notions, e.g. scale smoothness, of the new theory.

Hofer, Wysocki, Zehnder, Polyfolds and Fredholm Theory Part 1, (2005) 182pp
Overview of Leipzig Lectures (2005)

Concerning the new Fredholm theory:
Hofer, A general Fredholm theory and applications (2005) 77pp

Últimas notícias

Workshop on Polyfold Theory 25-31 August 2018 in Augsburg, a main Lecturer is Helmut Hofer

* arrived today (13/03/2018): The shift map on Floer trajectory spaces (arXiv:1803.03826 by Frauenfelder and myself)
So the break was fruitful and gave us one more manuscript we will follow in this lecture course.

The next class will take place friday 16 March. (Until then I have a visitor from abroad and it makes sense to fully focus on research with him. The topic relates to polyfold theory, so the course might eventually profit from that break anyway.)

Don't miss the talk: 5a-f, 1 de Março, 16h IMECC

Urs Frauenfelder (Augsburg)

What could Hamiltonian delay equations be?

Abstract: This is joint work with Peter Albers and Felix Schlenk. Although we do
not know what a Hamiltonian delay in general is, we describe a variational
approach to their periodic solutions. I will explain how periodic solutions of
some delayed Lotka-Volterra equations arise in this way. Finally I will discuss
how this leads to some new directions and challenges for Floer homology.

* arrived today (22/02/2018): First steps in Floer homology on scale Hilbert spaces (arXiv:1802.07445 by Albers, Frauenfelder, Schlenk)
   Urs Frauenfelder will visit us the first two weeks of this semester. Obviously attendance of his talk is highly recommended.
   See also arXiv:1802.07453 and arXiv:1802.07449.

* aplicação nova (Janeiro 2018): geometria simplética na dimensão infinita
   Crespo-Fabert "Hamiltonian partial differential equations and symplectic scale manifolds", arXiv:1801.05259

Descrição:
  Geralmente EDPs (chegando como gradientes de um funcionál) não são campos vetoriais
  nos espaços de Sobolev associados por causa do 'consumo de derivadas espaciais'.
  Salvação chegou há pouco (baseado no uso de "scale (sc) Banach spaces"): [HWZ]

  Nós queremos do início participar nesta teoría nova e aproveitar o poder dela.
  De um editor de Inventiones pode-se esperar contribuições perduráveis,
  então vale a pena aprender a nova ferramenta (e aplicar-la: em 2018 começamos
  um projeto de pesquisa sobre o "curve shortening flow" o qual queremos tratar
  com a teoria nova).

Joa Weber
sala 318
IMECC UNICAMP
e-mail: joa(at)ime.unicamp.br
fone: ++55 +19 352-16021
hora de atendemento em 2018-1: 6a-feira 12h-13h

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