TO GET PREPARED ..


Part I. Stacks and fundamental group

I.1. orbifolds, stacks and groupoids and dictionaries between them, gerbes
I.2. algebraic fundamental group of schemes and stacks; fundamental group scheme
I.3. algebraic patching
I.4. further aspects of the fundamental group


Part II. Moduli spaces and stacks of coverings (Hurwitz spaces)

II.1. models of curves
II.2. geometry of Hurwitz spaces (including construction)
II.3. topological, group and combinatorial aspects
II.4. reduction modulo p and compactification
II.5. connected Hurwitz space components and inverse Galois theory


Part III. Grothendieck-Teichmueller (GT) tower and Galois actions

III.1.  the action of the arithmetic Galois group on the geometric fundamental group of the thrice punctured projective line
III.2. profinite braid groups and the genus 0 GT group.
III.3. McLane relations, discrete complexes of curves and how to pass to strictly positive genus.
III.4  profinite complexes of curves and another geometric view of the GT group.
III.5. the contractibility conjecture and its consequences.


Part IV. Modular Towers (MT)

IV.1  construction and the main conjectures
IV.2. cusp and component structure on MTs
IV.3. modular curve towers as MTs, and alternating group towers
IV.4. Inverse Galois Theory, Abelian Varieties and the MT program
IV.5. MTs with a spire and generalizing Serre's Open Image Theorem

SPEAKERS
JOSE BERTIN (lecture)
Stacks  and Gerbs  with a view toward  curves and covers
- Stacks as sheaves in groupoids
- Stacks as quotients under a groupoid action
- punctual stacks,  BG
- morphisms, substacks
- Sheaves, line bundles
- Moduli of stables curves
- Cohomology, if times permits
MUSTAFA KORKMAZ (lecture)
Mapping Class Groups
- mapping class groups of surfaces : preliminaries
- generators, relations, some known algebraic structures
- role of the mapping class group in the the theory of 3- and 4-manifolds.
RAZVAN LITCANU (lecture)
Intersection Theory on Algebraic Stacks
- intersection theory on algebraic stacks
- geometric and arithmetic aspects ; examples ("Shimura varieties-like" stacks)
- some K-theory
SINAN UNVER (lecture)
Multi-Zeta Values and the Grothendieck-Teichmuller Group
- different realizations of the motivic fundamental group ; de Rham-Betti comparison ; multi-zeta values ; including survey of~:
- paper by Deligne : Le groupe fondamental de la droite projective moins trois points  
- paper by Deligne and Goncharov : Groupes fondamentaux motiviques de Tate mixte.
MATTHIEU ROMAGNY (lecture) 
Models of curves.  The main topic of these lectures is to present the stable reduction theorem with the point of view of Deligne and Mumford. We introduce the basic material needed to manipulate models of curves, including desingularization, intersection theory on regular arithmetic surfaces, and the structure of the jacobian of a singular curve. The proof of stable reduction in characteristic 0 is given, while the proof in the general case is explained and important parts are proved. We give applications to the moduli of curves and covers of curves.
1. Models of curves
1.1 Definitions : normal, regular, (semi)stable models
1.2 Existence of minimal regular models
1.3 Intersection theory on regular arithmetic surfaces
1.4 Blow-up, blow-down, contraction
2. Stable reduction
2.1 Stable reduction <=> semistable reduction
2.2 Proof of semistable reduction in char. 0
2.3 Relation with semistable reduction of abelian varieties
3. Application to moduli of curves
3.1 Valuative criterion for \bar{M}_g
3.2 The finite monodromy
3.3 Reduction of Galois covers of order prime to char(k).
ANNA CADORET (lecture)
 Galois Categories
0. Short intro.
I. Galois categories
I-1 Definition;
I-2 Main theorem;
I-2 Comparison functor/fundamental groups;
I-3 First examples
a- topological covers
b- spectrum of fields;
c- normal schemes;
d- geometrically connected schemes of finite type over a field ("canonical short exact sequence");
e- abelian varieties over the complex;
f- RET;
In this last example, I plan
(i) to define the analytic space associated with a scheme of finite type over the complex;
(ii) to state the theorem about the equivalence of category between etale analytic and algebraic covers;
(iii) to give the consequence about the computation of fundamental groups (curves, some configuration spaces etc).
STEFAN WEWERS (lecture)
Algebraic patching and covers of curves
1. Lecture:  Grothendieck's Existence Theorem
 * motivation: deformation of curves and the smoothness of M_g
 * formal schemes and GET
 * sketch of a proof of GET for P^1
2. Lecture:  Formal and rigid patching
 * motivation: deformation of covers + algebraic construction of Hurwitz spaces
 * (formal) deformation of tamely ramified covers
 * the rigid point of view
 * degenerating branch points
3. Lecture:  Applications
 * the Grothendieck-Beckmann-Theorem
 * p-adic points on Hurwitz spaces, HM-tupels
 * construction of tamely ramified rational functions  
MICHAEL FRIED (lecture)
Riemann produced many tools that allow generalizing Abel's production of modular curves. A missing ingredient was how to generalize Galois' introduction and analysis of the higher level curves we call X^0(p^{k+1}), k��� 0. While Shimura's projects are related, they don't capture the most useful part of Riemann's program. Using moduli spaces of covers requires deducing properties of the spaces from their cusps. New techniques for identifying cusps of Hurwitz spaces combine with connectedness results to identify advantageous cusps. Our applications here show modular curve-like properties for Modular Towers over alternating group Hurwitz spaces.
1. Dihedral Groups: MT view of Modular curve cusps
  http://www.math.uci.edu/~mfried/talklist-mt/London1-ModCurves.html
2. Updating an Abel-Gauss-Riemann Program: Cusp and component structure on MTs (including the introduction of spires)
  http://www.math.uci.edu/~mfried/talklist-mt/ucicoll05-22-08.html
3. Conway-Fried-Parker-Voelklein connectedness results and presentations of the absolute group of Q.
  http://www.math.uci.edu/~mfried/talklist-mt/CFPVIstanbul.html
MICHAEL EMSALEM (lecture)
On the Fundamental Groupoid Scheme
1. Fundamental Group Scheme
2. Group schemes, groupoids, torsors.
3. Statement of the tannakian duality theorem ; representions of a stack, of a gerbe, of a groupoid.
4. Fundamental groupoid scheme
5. Link with the etale fundamental group and the Grothendieck's short exact sequence.
Prerequisites: definition of stacks, descent theory, Grothendieck topologies (?) (in the talk of J.Bertin)  fundamental groups, Galois categories (in the talk of A.Cadoret)
PIERRE DEBES (lecture)
1. Foundations of Modular Towers
1.1. p-universal Frattini cover
1.2. Characteristic quotients and lifting lemma
1.3. Towers of moduli spaces
1.4. The dihedral group example
2. The Modular Tower Conjecture
2.1. The Main Conjecture
2.2. The dihedral group example
2.3. The Fried-Kopeliovich theorem
2.4. Moduli space and stack versions of the MT conjecture
2.5. Original and reduced forms of the MT conjecture
3. Galois Covers, Abelian Varieties and Modular Towers
3.1. Central Results
3.2. Torsion of abelian varieties
3.3. Application to the MT conjecture
3.4. l-adic points on Harbater-Mumford modular towers
3.5. Generalization of the central theorem
Debes' Lecture Notes (In particular, see ``Revetements topologiques'' and ``Arithmetique des revetements de la droite'')
SEYFI TURKELLI (research talk)
Connected Components of Hurwitz Schemes and Malle's Conjecture. Let k be a global field, G be a transitive subgroup of Sym(n), and H<G be a one-point stabilizer. Define the counting function Z_G(k,X) to be the number of G-extensions L/k  with the norm of  discriminant  Norm[ D(L^H / k)] <X where L^H is the fixed field of H. Malle's conjecture states that Z_G(k,X) is asymptotic to X^a(logX)^b for some constants a and b. Recently, Klueners gave a counterexample to Malle's conjecture. In this talk, we will "fix" this conjecture by using Hurwitz schemes of G-N covers.
MARCO A. GARUTI (research talk)
P-adic representations of the fundamental group scheme. The fundamental group scheme of a scheme S over a base B classifies torsors over S under finite flat B-group schemes, thus  generalizing the algebraic fundamental group. In this talk we will introduce the notion of p-adic representation of the fundamental group scheme of S/B, generalizing the Katz correspondence for p-adic local systems. We will show that these represenations are closely related to the maximal nilpotent quotient of the fundamental group scheme.
DAJANO TOSSICI (research talk)
Weak and strong extension of torsors. Let R be a discrete valuation ring of unequal characteristic with fraction field K which contains a primitive p^2-th root of unity. Let  X be a faithfully flat R-scheme and let G be a finite abstract group.  Let us consider a G-torsor Y_K->X_K and let Y be the normalization of X in Y_K. There are two possible questions which arise naturally. Is it  possible to extend this torsor to a G'-torsor Y->X over R for some R-model G' of G (Strong extension)? Is it  possible to extend this torsor to a G'-torsor Y'->X with Y' a  (possibly not normal) model of Y_K over R and G' an R-model of G (Weak extension)? We will give a positive answer to the  weak extension  for G abelian and X any scheme with mild hypothesis. While  we will give a  criterion to determine when a torsor is strongly extendible for  G = Z/p^n with n <3 and X a local normal scheme.  The explicit classification of finite and flat R-group schemes isomorphic to Z/p^n (n<3) over K  plays a crucial role. For n = 1 this classification was already known and the problem of extension of Z/p Z-torsors has already been studied (for instance by Raynaud, Green-Matignon, Henrio, Saidi) when X is a formal curve. The problem of extension of torsors arises naturally in the local study of group actions on an R-scheme.
MARCO ANTEI (research talk) 
Galois closure for towers of torsors