This is a four day school at the PhD level. It will consist of lecture series by Mohamed Abouzaid (Columbia), , Joana Cirici (Barcelona), Alexander Suciu (Northeastern University), Craig Westerland (University of Minnesota), Hisham Sati (NYU-Abu Dhabi) and Paolo Salvatore (Roma Tor Vergata).
The topics of the school will center around Topology and Geometry.
The first MIMS school was held in 2012 under the title "Operads and Configuration Spaces".
Sonia Ghorbal, Sadok Kallel, Paolo Salvatore, Hisham Sati,
Paolo Salvatore, Hisham Sati, Sadok Kallel
MIMS, International Mathematical Union (IMU), Club of Innovation and Scientific Research (Dubai), GE2RI (Universite de Tunis).
Flow categories for symplectic topology
Cohen-Jones-Segal introduced the notion of flow categoriesto construct homotopy types associated to Morse functions. In particular, they proved that the stable homotopy type of a manifold may be recovered from what they call a "framed flow category." I will revisit their construction, adapting it to the needs of symplectic topology, which requires considering Morse theory in the infinite dimensional setting. This is joint work with Andrew Blumberg.
Weights on cohomology and application to formality
Formality in Algebra and Topology
An algebraic structure (commutative algebra, operad..) with a differential is called formal if it is equivalent to its homology in the derived sense. Two related famous formality results are due to Kontsevich: the formality conjecture in deformation quantization and the formality of the little discs operad. We review the obstruction theory to formality in a general setting, and focus particularly on algebraic structures originating from topological spaces. Then we report on some recent non-formality results related to euclidean configuration spaces.
Twisted generalized cohomology and applications
Twisted forms of various generalized cohomology theories have been gaining prominence in recent years, both for mathematics as well as for applications in physics. We will survey this area, starting with twisted de Rham cohomology and twisted K-theory, and then generalizing to more recently constructed theories such as twisted elliptic cohomology, twisted Morava K-theory and E-theory, and twisted iterated algebraic K-theory of the topological K-theory spectrum. I will describe the construction of the latter theories and then give geometric/differential refinements of a few and present computational techniques, which will be illustrated with examples. We will also highlight connections to twisted higher tangential structures, such as String and higher structures. We end with applications (to physics), including T-duality as an isomorphism of twisted cohomology theories, fields as Chern characters of elements of such theories, and charges of branes as pushforwards.
Geometry and topology of cohomology jump loci
The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. The geometry of these varieties, and the interplay between them sheds new light on the topology of the original space and that of its abelian covers.
Arithmetic statistics and the homology of moduli spaces
Adnene Chergui (Short communication), Universite Houari Boumedienne, Alger
On Levi-Civita's theorem for degenerate semi-riemannian manifolds
Mehdi Nabil, Universite Cadi Ayyad, Marrakech.
Cohomology of coinvariant forms
Let M be a differentiable manifold and $\Gamma$ a group acting on M by diffeomorphisms. We call a $\Gamma$-coinvariant form on M any linear combination of differential forms of the type $\omega-\gamma^*\omega$ with $\omega\in\Omega^*(M)$ and $\gamma\in\Gamma$. The space of such forms is denoted $\Omega^*(M)_\Gamma$, it is a subcomplex of $(\Omega^*(M),d)$ we therefore write $H^p(\Omega^*(M)_\Gamma)$ for its $p$-th cohomology group. We study the action of the group $\Gamma$ on the manifold M in various situations by observing the relationship of this newly introduced complex with the complex of $\Gamma$-invariant forms on M; $\Omega^*(M)^\Gamma$ and the complex of diffenrential forms $\Omega^*(M)$ and illustrating the interplay between their respective cohomologies by means of direct sum decompositions or exact sequences, depending on the case of study. This eventually leads to some cohomological obstructions for the existence of certain group actions (Isometric actions or properly discontinuous actions). Travail en collaboration avec Abdelhak Abouqateb et Mohamed Boucetta.
|Cirici||Coffe break||Suciu||lunch||Westerland||Abouzaid||Coffe break||Sati|
1. A. Chergui
2. N. Mehdi
|Guided Tour of Tunis or short trip|
|Mohammed Abouzaid||Columbia University|
|Naoufel BATTIKH||Faculté des sciences de Tunis|
|Aziz Ben Ouali||Institut préparatoire aux études d'ingénieurs de Monastir|
|Marwa Bouali||Tunis El Manar University|
|Mohamed Amine Boubatra||Faculty of science of Tunis|
|Moez Bouzouita||institut préparatoire du kairouan|
|Esma Chelbi||école normale supérieure|
|Adnene Chergui||Universite Boumedienne Alger|
|Joana Cirici||University of Barcelona|
|Sonia Ghorbal||Faculté des sciences de Tunis|
|saihi ines||Faculté des sciences de Tunis|
|Nawal Irz||Faculté des Sciences de Tunis|
|Fatma Kadi||ENS Kouba-Algeria|
|Sadok Kallel||American University of Sharjah|
|soula maroi||facuté de science de sfax|
|Mohammed El Amine Mekki||Université Mustapha Stambouli|
|Mehdi Nabil||Cadi Ayyad University, Marrakesh|
|Roberto Pagaria||SNS Pisa|
|Andrea Pizzi||Roma Tor Vergata|
|Alessio Ranallo||University Roma II, Tor Vergata|
|Jammazi Refki||FST El Manar|
|Chaabane REJEB||Institut préparatoire aux études d'ingénieurs El Manar|
|Paolo Salvatore||Roma Tor Vergata|
|Hisham Sati||New York University, Abu Dhabi|
|Alexander Suciu||Northwestern University|
|Oumaima Tibssirte||Cadi Ayyad University, Marrakesh|
|Marwa Troudi||Faculté des sciences de Tunis|
|Craig Westerland||University of Minnesota|
|Mhamdi Zeinab||Faculte de science Sfax|
|safa zouari||institut préparatoire à l'étude scientifique et technologique|