Marion JeanninPostdoc

• 09.2023 to 12.2023: PhD course on algebraic groups with Seidon Alsaody, at Uppsala University. Online notes should be available soon!
• 01.2023 to 03.2023: Teaching Assistant for the course of Linear algebra II at Uppsala University,
• 10.2020-09.2021: ATER, University of Burgundy, (Dijon, France).
  Teaching assistant for various Bachelor's courses for students from different majors (mathematics, computer sciences, physics, chemistery, biology, economy).
• 09.2016-09.2020: Teaching assistant and then Demi-ATER, Claude Bernard University (Lyon, France).
  Teaching assistant and tutor for various Bachelor's courses for students from different majors (mathematics, computer sciences, physics, economy) and oral examiner for students in preparatory school (for engenering schools).
• 02.2013-06.2015: Teaching assistant, EPFL (Lausanne, Switzerland).

Group schemes are mathematical objects that encode the symmetries of algebraic and algebro-geometric objects. Examples include the general linear group of invertible matrices of size n, but also orthogonal and symplectic groups, which consist, respectively, of isometries with respect to a quadratic or symplectic form. These are examples of reductive groups, which have a combinatorial structure theory in terms of root data.

Let G be a group scheme acting on a scheme X. Defining a “reasonable” structure for the quotient of this action is a complicated task in algebraic geometry: for instance let us consider the action of the filed of complex numbers on the affine space of dimension n by scalar multiplication. A naive approach would be to define the quotient for this action as the variety that parametrizes the closed orbits. Unfortunately, this turns out to provide a rather “poor” object: in the previous example the unique closed orbit is the origin, while the orbit closures are the lines through the origin. A more satisfying definition of a quotient Y for the G-action on X is formalised with the notion of geometric quotient. When X is affine, such a quotient verifies that:

1. the space Y identifies with the orbit space X/G equipped with the quotient topology,
2. the quotient Y is endowed with a uniquely defined structure of variety. In particular, when X is irreducible the field of rational functions of Y is the field of G-invariant rational functions of X.

In the above example, ageometric quotient does not exist, but if ones restricts the action to the affine space without the origin, a geometric quotient exists (as a projective variety though). Actually, by a theorem of M. Rosenlicht, any irreducible reduced variety X acted on by an algebraic group G contains a non-empty G-stable open subset, its semi-stable locus, that admits a geometric quotient for the induced action. The Hilbert–Mumford criterion provides a combinatorial characterisation of this semi-stable locus by mean of destabilizing cocharacters. One may furthermore aim to construct quotients of more sophisticated geometric objects such as vector bundles over a curve (a geometrically connected scheme of dimension 1) defined over a field k. The previous notions extend to this setting: (semi)-stable vector bundles are (semi)-stable points of moduli spaces of bundles (and, as such, can be characterised by the Hilbert–Mumford criterion). They admit a combinatorial characterisation which directly involves their degree. The lack of semi-stability of a given vector bundle V defined over X is encoded by its Harder–Narasimhan filtration which is such that the successive quotients are semi-stable and of strictly decreasing slopes, where the slope of V is the ratio of its degree by its rank. The stabiliser of the filtration for the action of GL(V ) is a parabolic subgroup.

A torsor of a reductive group G is a scheme E on which G acts on a free, transitive and “locally trivial” way (for a given topology, usually for the fppf topology in the context of my research). For instance, torsors of the general linear group (defined over the curve X) can be thought as rank n vector bundles over X. The semistability of torsors of reductive groups over X is ruled by that of the Lie algebra of the twisted group (which is a vector bundle over X) and is encoded by a canonical parabolic subgroup of this twisted group. Semi-stability of G-torsors raises several questions among them is the integration of such Lie algebras. Namely: the Lie algebras bundle of a group can be seen as its tangent bundle at identity. In this way one often says that such Lie algebras bundles “derive” from the starting group. On the other hand, one can wonder whether, given a Lie algebra l, there exists a group L such that l = Lie(L). One often requires L to be connected (this is anodyne as one can always restrict to the connected component of the identity of L, which has the same Lie algebra as L) and smooth (which is automatically satisfied for affine algebraic groups in characteristic zero but is rather restrictive in positive characteristic). Because of both the existence of the exponential and a theorem of Cartier that implies that any affin group scheme defined over a field of characteristic zero is smooth, integration is a well-known settled question in characteristic zero. In positive characteristic, this is a different story...

Preprints:

• Analogues of Morozov Theorem in characteristic p>0
• Harder-Narasimhan games, with Huayi Chen