
Realisation of birational maps
Anna Bot (Basel)
Abstract
Given a matrix $F$ in $\mathrm{GL}_n(\mathbb{Z})$, we investigate under which conditions it can be realized as a birational map $f$ of $\mathbb{P}^2$,
meaning if there exists a blowup to a rational surface $X$ on which $f$ lifts to an automorphism whose action on the Picard group of $X$ is precisely $F$.
Based on ideas by McMullen and Uehara, we give sufficient conditions on points on a cuspidal cubic, and indicate how this can be used to prove
that the ordinal of all dynamical degrees of all birational maps of $\mathbb{P}^2$ is $\omega^\omega$.

Instantons on Fano threefolds
Gaia Comaschi (IMB Dijon)
Abstract
The notion of mathematical instanton was originally introduced by Atiyah, Drin
feld, Hitchin and Manin to define the stable rank 2 vector bundles $E$ on $\mathbb{P}^3$ having
$c_1(E) = 0$ and such that $H^1(E(−2)) = 0$. Thereafter, the notion of instanton had
been generalized to rank 2 vector bundles on other Fano varieties besides $\mathbb{P}^3$. The
aim of this talk is to present an even further generalization that extends the notion
of instanton to non necessarily locally free sheaves of arbitrary rank on Fano 3folds
of Picard rank one. I will present the main features of these sheaves focusing in
particular on the properties of instantons, and of their moduli, on Fano 3folds of
index 2.

On the existence of smooth orbital varieties
Lucas Fresse (IECL Nancy)
Abstract
Let $G$ be a complexe reductive group. The orbital varieties are certain Lagrangian subvarieties of the nilpotent $G$orbits.
These varieties are in general singular. In this talk we show that, in the case where $G$ is classical, every nilpotent $G$orbit contains at least one
smooth orbital variety. We also point out that this property can fail in the case where $G$ is of exceptional type.

Free $\mathbb{G}_a$Actions of Winkelmann type
Gene Freudenburg (Western Michigan)
Abstract
Let $k$ be a field of characteristic zero and $B$ an affine $k$domain. A regular action
of $\mathrm{SL}_2(k)$ on $B$ corresponds to a fundamental pair for $B$, that is, a pair of locally nilpotent
derivations $(D, U )$ of $B$ which satisfies $[D, [D, U ]] = −2D$, $[U, [D, U ]] = 2U$, and
$B=\bigoplus_{d \in \mathbb{Z}} B_d$ is a $\mathbb{Z}$grading where $B_d = ker ([D, U ] − dI)$.
Let $A \subset B$ be the kernel of $D$. Classical techniques show that $A$ is affine. The Structure Theorem
describes $A$ as an $\mathbb{N}$graded ring, as well as the degree modules and image ideals of $D$. The Extension
Theorem describes the extension of $D$ to $B[t]$ by an $\mathrm{SL}_2(k)$invariant function, using the degree
modules of $D$ to give an explicit set of generators for the kernel of the extended derivation on $B[t]$.
Winkelmann constructed a free $\mathbb{C}_+$action on $\mathbb{C}^4$ with singular algebraic quotient, and a locally
trivial $\mathbb{C}_+$action on $\mathbb{C}^5$ with smooth algebraic quotient which is not globally trivial. Finston and
Jaradat used similar methods to construct a locally trivial $\mathbb{C}_+$action on $\mathbb{C}^5$ with singular algebraic
quotient. We generalize these constructions by recognizing and exploiting the role of the underlying
$\mathrm{SL}_2(\mathbb{C})$module, showing in particular that the quotient morphism of such an action cannot be
surjective. The Extension Theorem gives a simple way to confirm the results of Finston and Jaradat,
who used the van den Essen algorithm (and Singular) to find generators for their ring of invariants.
This was the first example of a locally trivial $\mathbb{C}_+$action on an affine space having a singular algebraic
quotient. We discuss families of examples, including a simpler example of a locally trivial $\mathbb{C}_+$action
on $\mathbb{C}^5$ with singular algebraic quotient.

On multiplicities of fibers of Fano fibrations
Chuyu Zhou (EPFL)
Abstract
In this talk, I will present how to reduce various conjectures in birational geometry, including MckernanShokurov
conjecture and bounded conjecture of rationally connected CalabiYau varieties, to a conjecture on multiplicities of
fibers of a special kind of Fano fibrations. This is based on a recent joint work with Guodu Chen.