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MIMS Summer School: New Trends in Topology and Geometry
Jul 09, 2018 to Jul 12, 2018

Location : Tunis, Tunisia

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".

Organizing Commitee:
Sonia Ghorbal, Sadok Kallel, Paolo Salvatore, Hisham Sati,

Scientific Commitee:
Paolo Salvatore, Hisham Sati, Sadok Kallel

Sponsors:
MIMS, International Mathematical Union (IMU), Club of Innovation and Scientific Research (Dubai), GE2RI (Universite de Tunis).

 

Mohamed Abuzaid

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.

Joana Cirici

Weights on cohomology and application to formality

Given a differential graded algebra or any algebraic structure in cochain complexes, one may ask if it is quasi-isomorphic to its cohomology equipped with the zero differential. This property is called formality and has important consequences in algebraic topology. For instance, if the de Rham algebra of a manifold is formal, then certain higher operations in cohomology, called Massey products, are known to vanish. There are many examples of formal manifolds, like spheres, symmetric spaces or Lie groups. A very famous class of formal spaces is that of compact Kähler manifolds. Their formality was proven by Deligne, Griffiths, Morgan and Sullivan, using the fact that their cohomology with rational coefficients has a pure Hodge structure, a certain bigrading which is symmetric with respect to complex conjugation. 
 
For complex algebraic varieties, Deligne introduced mixed Hodge structures as a generalization of the structure enjoyed by the cohomology of Kähler manifolds. A key ingredient in mixed Hodge theory is the weight filtration which will occupy part of this course. The objective of this course is to explain the basics of mixed Hodge theory and its applications to homotopy theory. We will see how to compute the weight filtration in several simple but illustrating examples. Then, we will talk about purity and develop some applications to formality over the field of rational numbers. We will also discuss the theory of weights in étale cohomology as a tool for proving results of formality with torsion coefficients. 
 
These lectures include ideas from joint work with David Chataur, Francisco Guillén and Geoffroy Horel.

Paolo Salvatore

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.

Hisham Sati

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. 

Alex Suciu

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. 

In these introductory lectures, I will explain the algebraic notions underlying these constructions, and present some structural results.  I will illustrate the general theory with a number of examples, such as the computation of the resonance varieties of the Stanley-Reisner ring associated to a simplicial complex, or the Orlik-Solomon algebra associated to a hyperplane arrangement. I will conclude with a brief overview of some topics of current research, such as resonance varieties of CDGA models for spaces, the influence of formality, and the interplay 
between resonance and duality.

Craig Westerland

Arithmetic statistics and the homology of moduli spaces

Abstract: Many conjectures in number theory can be formulated in terms of questions about statistical or asymptotic distributions of invariants of arithmetic objects.  For instance, Malle has conjectured a formula on the growth rate of the set of number fields with specified Galois group as a function of discriminant.  Other conjectures concern the distribution of class groups (Cohen-Lenstra), elliptic curves as a function of conductor (Watkins), and invariants of elliptic curves, such as Selmer and Tate-Shafarevich groups (Bhargava-Kane-Lenstra-Poonen-Rains).  Very few of these conjectures currently seem accessible in full generality; for instance, Malle’s conjecture predicts a positive solution to the inverse Galois problem as a trivial corollary.  
 
Instead, one may reformulate these conjectures over function fields instead of number fields.  In this setting, they become questions of counting points on moduli spaces.  Through the machinery of the Weil conjectures, one may approach these problems by computing the homology of (the complex points of) these moduli spaces.  In many cases, this may be approached through familiar techniques in algebraic topology.
 
Our focus in the first lecture will be on formulating several of these number-theoretic conjectures, and defining all of the objects involved.  In the second lecture, we will explain how to transform such problems into questions about the homology of moduli spaces.  The third lecture will concern techniques of doing so based mainly of group homology for the braid group.

Adnene Chergui (Short communication), Universite Houari Boumedienne, Alger

On Levi-Civita's theorem for degenerate semi-riemannian manifolds

In semi-Riemannian manifolds the existence and uniqueness of the Levi-Civita connection is the consequence of the non-degeneracy of the metric. We propose an analogous of this theorem for the degenerate case.

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.

 

9-13 July  9h-10h20   10:50-12h10   14h-15h 15h10-16h10   16h45-17h45

Monday

 Cirici Coffe break   Suciu  lunch Westerland  Abouzaid Coffe break  Sati

Tuesday

  Abouzaid    Cirici  lunch  Abouzaid  Suciu    Salvatore

 Wed.

  Suciu   Westerland  lunch

Tunis visit:

Bardo Museum

 

Tunis visit:

Bardo Museum

 

  

19h30

Dinner Medina

Thursday

 Westerland    Cirici  lunch  Sati  Salvatore  

Short coms:

1. A. Chergui

2. N. Mehdi

 Friday

     Guided Tour of Tunis or short trip

   

List of participants to this conference
Jul 09, 2018 to Jul 12, 2018

Participant Institution
Mohammed Abouzaid Columbia University
Mouadh Akriche IPEIBizerte
Marwa Assili FST
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
Abdelkerim Chaabani FST
Esma Chelbi école normale supérieure
Adnene Chergui Universite Boumedienne Alger
Seifallah Cherif IPEIManar
Joana Cirici University of Barcelona
Abderraouf Dorai TPEIELMANAR
Moncef Ghazel IPEIMANAR
Sonia Ghorbal Faculté des sciences de Tunis
HASSINE Holia FST
saihi ines Faculté des sciences de Tunis
Nawal Irz Faculté des Sciences de Tunis
Hatem Issaoui IPEIN
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
HASNA RIAHI ENIT
Rhaiem Saber FST
Rebhi salem FST
Paolo Salvatore Roma Tor Vergata
Hisham Sati New York University, Abu Dhabi
Alexander Suciu Northwestern University
Walid Taamallah IPEIEM
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
Contact
secretary@mims-institut.org

MIMS Summer School: New Trends in Topology and Geometry