SCHEDULE

**BATTISTON Giulia:**

**Galois descent for inseparable field extensions**The well known Galois descent allows us to descend algebraic objects along a field extension L/K when the latter is Galois, in particular separable. Using the theory of the automorphism group scheme and of descent along torsors, I will present a generalization of the classical Galois descent to the case where L/K is a possibly inseparable field extension, and give some examples of applications.

**BISWAS Indranil:**

**The fundamental group scheme of rationally chain connected varieties**(This is a joint work with M. Antei) Let k be an algebraically closed field and let X be a normal rationally connected variety over k. Kóllar and Chambert-Loir (independently) proved that the étale fundamental group of X at a point x is always finite and examples where it is not trivial are known. We prove that the local fundamental group scheme of X is finite and again the result is sharp.

**CADORET Anna:**

Geometric monodromy - semisimplicity and maximality

Let X be a connected scheme, smooth and separated over an algebraically closed field k of characteristic p, let f:Y-> X be a smooth proper morphism and x a geometric point on X. We show that the tensor invariants of bounded length d of the etale fundamental group \pi_1(X) acting on the \'etale cohomology groups H^*(Y_x,\F_l) are the reduction modulo-l of the tensor invariants of \pi_1(X,x) acting on H^*(Y_x,\Z_l) for l large enough depending on f:Y-> X, d. We use this result to discuss semisimplicity and maximality issues about the image of \pi_1(X,x) on H^*(Y_x,\Z_l).

**CA**SSOU-NOGUES Philippe:

Quadratic forms, twists and periods

Let $Y$ be a scheme in which $2$ is invertible and let $(V,q)$ be a rank $n$ quadratic form on $Y$. We consider the classifying topos of the orthogonal group ${\bf O}(q)$ whose cohomology ring $H^*(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})\simeq A_Y[HW_1(q),..., HW_n(q)]$ is a polynomial algebra over the \'etale cohomology ring $A_Y:=H^*(Y_{et},{\bf Z}/2{\bf Z})$ of the scheme $Y$. Here the $HW_i(q)$'s are Jardine's universal Hasse-Witt invariants. The ring $H^*(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})$ contains canonical classes $\mathrm{det}[q]$ and $[C_q]$ of degree $1$ and $2$ respectively, which are obtained from the determinant map and the Clifford group of $q$. The classical Hasse-Witt invariants $w_i(q)$ live in the ring $A_Y$. We state a theorem expressing $\mbox{det}[q]$ and $[C_{q}]$ as polynomials in $HW_{1}(q)$ and $HW_{2}(q)$ with coefficients in $A_Y$ written in terms of $w_1(q),w_2(q)\in A_Y$.

If a group-scheme $G$ acts on $q$ and $T$ is a $G$-torsor over $Y$ we may consider the twist $q_T$ of $q$ by $T$. The previous theorem then provides formulas relating the Hasse-Witt invariants of $q$ and $q_T$. This situation systematically arises whenever one is given an orthogonal object in a Tannakian category over a field of characteristic $\neq 2$. We apply this method to Nori's category of mixed motives over $\mathbb Q$. Given a smooth projective variety $V$ over $\mathbb Q$ of even dimension $d$, the motive $M=h^d(V)(d/2)$ has a canonical orthogonal structure. This yields formulas relating the torsor of periods of $M$ to the Hasse-Witt invariants of the de Rham quadratic form on $M_{dR}$ and the Betti quadratic form on $M_B$. This is a joint work with B. Morin.

**DEY Arijit:**Let X be a toric variety under the torus action T. T-equivariant vector bundles over X was first classified by Kaneyama and then Klyachko. In this talk we will discuss some of the results obtained in classifying T-equivariant principal G-bundles on X, where G is an affine algebraic group. This is a joint work with Indranil Biswas and Mainak Poddar.

Equivariant principal bundles on toric varieties

**EMSALEM Michel**:

On the fundamental group scheme over a Dedekind scheme

In this lecture we revisit the construction by Gasbarri of the fundamental group scheme of a connected faithfully flat S-scheme X, where S is a Dedekind scheme. We show that the hypothesis made in [2] that X is reduced is not sufficient, and we achieve the construction when X has reduced fibers or when X is normal ([1]). We will list some known properties of the fundamental group scheme, and address in particular the question of extending to the whole of $X$ a finite torsor over the generic fiber $X_\eta $.

[1] M. Antei, M. Emsalem, C. Gasbarri, Sur l'existence du schéma en groupes fondamental. Erratum à l'article [2]; http://arxiv.org/abs/1504.05082

[2] C. Gasbarri, Heights of vector bundles and the fundamental group scheme of a curve, Duke Math. J. 117, No.2, 287-311 (2003).

**HAI Phung Ho:**

**The structure of flat affine group schemes and applications**(This is a joint work with J. P. dos Santos and N.D. Duong).

We study affine group schemes over a discrete valuation ring R using two techniques: Néron blowups and Tannakian categories. We employ the theory developed to define and study differential Galois groups of integrable connections on a scheme over R. This throws light on how differential Galois groups of families degenerate.

**PARAMESWARAN****A. J.****:**

**On a theorem of Deligne and the slice theorem**The aim of this talk is to prove a structure theorem for doubly saturated affine schemes of finite type in GL(V ) over algebraically closed fields. This is built on recent results of Deligne and the results of Serre. We obtain Luna’s étale slice theorem in positive characteristics as an application. This is a Joint work with V. Balaji.

**PEPIN Cédric:**Torsors under commutative finite group schemes over a local field of characteristic p

We will define and study some coarse moduli spaces for such torsors. In particular, we will establish a duality between the spaces associated to a group and to its Cartier dual.

**PIEROPAN Marta:**

**Torsors, Cox rings and Manin's conjecture**

A conjecture of Manin dating back to 1989 predicts an asympotic formula for the distribution of rational points on Fano varieties over number fields. The conjecture has been verified for some families of varieties using various methods. One approach consists of parameterizing the set of rational points via integral points on some affine spaces in order to reduce to a lattice point counting problem. This has been done mostly for varieties over the field of rational numbers Q. The talk presents a systematic way to produce the parameterization over arbitrary number fields using torsors under quasitori and related Cox rings. This procedure gives a geometric interpretation of previous ad hoc parameterizations over Q and led to prove Manin's conjecture for new varieties over number fields beyond Q.

**RAYNAUD Michel**:

Espaces algébriques et schémas en groupes

Soit G un S-espace algébrique en groupes, lisse, séparé, à fibres connexes où S est un schéma normal noethérien. Lorsque S est de dimension ≥ 2, on n'a pas d'énoncé général qui assure que G soit un schéma. On obtient néanmoins une réponse positive lorsque G ne dégénère pas trop. Par exemple lorsque les fibres de G sont semi-abéliennes (extension d'une variété abélienne par un tore) ou lorsque G a des fibres affines et possède, localement pour la topologie étale, des tores maximaux qui sont leur propre centralisateur (les sous-groupes de Cartan sont des tores).

**VISTOLI Angelo**:

Fundamental group schemes

This is joint work with Niels Borne. I will discuss a class of generalizations of the fundamental group scheme and the unipotent fundamental group scheme defined by M. Nori.

**ZUO Kang:**

**p-adic Simpson's correspondence via Higgs-De Rham flow**In my talk I shall first explain briefly the notion of Higgs-de Rham flow and p-adic Simpson's correspondence between the category of crystalline representations of algebraic fundamental group of a smooth quasiprojective scheme over W(k) and the category of periodic Higgs bundles via this flow. As an application we construct an absolut irreducible rank-2 crystalline representation of algebraic fundamental group of a generic p-adic hyperbolic curve via the uniformization Higgs bundle, which should be regarded as a p-adic analogue of Hitchin-Simpson's uniformization theorem on complex hyperbolic curves via uniformization Higgs bundles. This talk is based on my joint papers with Lan, Sheng and Yang.

TUESDAY
19/01/2016

11h45-12h15 Welcome!

A welcome packet will be given

(lunch)

14h00 BISWAS Indranil

15h20 BATTISTON Giulia

WEDNESDAY 20/01/2016

9h45 HAI Phung Ho

11h15 EMSALEM Michel

On the fundamental group scheme over a Dedekind scheme

(lunch)

14h00 RAYNAUD Michel

Espaces algébriques et schémas en groupes

15h20 CADORET Anna

THURSDAY 21/01/2016

9h45 VISTOLI Angelo

Fundamental group schemes

11h15 DEY Arijit

(lunch)

14h00 CASSOU-NOGUES Philippe

Quadratic forms, twists and periods

15h20 A. J. PARAMESWARAN

SOCIAL DINNER

FRIDAY 22/01/2016

9h00 PEPIN Cédric

10h30 PIEROPAN Marta

11h40 ZUO Kang

(lunch)

11h45-12h15 Welcome!

A welcome packet will be given

(lunch)

14h00 BISWAS Indranil

**The fundamental group scheme of rationally chain connected varieties**

15h20 BATTISTON Giulia

**Galois descent for inseparable field extensions**WEDNESDAY 20/01/2016

9h45 HAI Phung Ho

**The structure of flat affine group schemes and applications**11h15 EMSALEM Michel

On the fundamental group scheme over a Dedekind scheme

(lunch)

14h00 RAYNAUD Michel

Espaces algébriques et schémas en groupes

15h20 CADORET Anna

**Geometric monodromy - semisimplicity and maximality**THURSDAY 21/01/2016

9h45 VISTOLI Angelo

Fundamental group schemes

11h15 DEY Arijit

**Equivariant principal bundles on toric varieties**(lunch)

14h00 CASSOU-NOGUES Philippe

Quadratic forms, twists and periods

15h20 A. J. PARAMESWARAN

**On a theorem of Deligne and the slice theorem**SOCIAL DINNER

FRIDAY 22/01/2016

9h00 PEPIN Cédric

**Torsors under commutative finite group schemes over a local field of characteristic p**10h30 PIEROPAN Marta

**Torsors, Cox rings and Manin's conjecture**11h40 ZUO Kang

**p-adic Simpson's correspondence via Higgs-De Rham flow**(lunch)

ABSTRACTS OF THE TALKS