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On the enveloping skew field of a Lie algebra
Jacques Alev
Abstract
In this talk we will start with a survey of results on the theme of the structure of the skew field of fractions of the enveloping algebra of a Lie algebra
in various situations (quantum, super, positive characteristic) considered since the statement of the Gelfand-Kirillov Conjecture in 1966.
In the second part of the talk we will present some recent results about an analogous case not covered by the conjecture.
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Dynamical degrees of automorphisms of the affine space
Jérémy Blanc
Abstract
The automorphisms of the affine space of dimension n are still mysterious so far. We can define the (first) dynamical degree as the exponential
degree growth of the iterates, and ask if this one is an algebraic integer, and of which degree. It seems that we can only have algebraic integers of degree $< n$.
I will give examples and partial classification for very simple automorphisms, namely the ones obtained by composing an affine with a triangular map.
Joint work with Immanuel Van Santen.
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Bounded endomorphisms of projective algebraic varieties
Michel Brion
Abstract
Let $X$ be a projective algebraic variety over an algebraically closed field. The monoid $\mathrm{End}(X)$ of endomorphisms of $X$ is known to be “locally algebraic”: it has at most countably many connected components, andall of these are of finite type. We say that an endomorphism $f$ is bounded if the closure in $\mathrm{End}(X)$ of the set of positive powers of $f$ is of finite type; equivalently, this set meets only finitely many components. The talk will present a structure result for bounded endomorphisms, and discuss some questions of birational geometry motivated by this result.
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Structure of the tree at infinity of a polynomial in two variables
Pierrette Cassou-Nogues
Abstract
Let $f:\mathbb{C}^2\rightarrow \mathbb{C}$ be a primitive polynomial. Extend $f$ to a morphism $\Phi:X\rightarrow \mathbb{P}^1$ where $X$ is a non singular projective surface that contains $\mathbb{C}^2$ as an open set and where $D=X\setminus \mathbb{C}^2$ is an SNC-divisor of $X$. The dual graph of $D$ is a tree. We consider substructures of the tree, that we call ”combs” such that a tree can be decomposed in a finite number of combs. One of our main results is that the number of combs in a tree is less or equal to $1 + 2g$ where $g$ is the genus of the generic curve of the polynomial. In particular, if $f$ is a rational polynomial, we get one comb. If time permits, we shall discuss dicriticals of degree $1$ and recover the case of simplerational polynomials. This is a joint work with Daniel Daigle.
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Generating properties of polynomial automorphisms
Éric Edo
Abstract
All is over a field of characteristic zero. About a special polynomial automorphism $f$, we can consider the two following ”generating proper-ties”:
1) The normal subgroup generated by $f$ is the normal closure of the special linear group in the special automorphism group.
2) Together with the affine subgroup, it generates the entire tame subgroup.
We survey and compare some results in this two situations with the idea of measure the complexity of a polynomial automorphism.
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On the Finite Generation of Certain Graded Factorial Domains
Gene Freudenburg
Abstract
Our main result implies that, if $D$ is a locally nilpotent derivation of $\mathbb{C}[n]$ which is homogenous for a non-degenerate positive $\mathbb{N}^{n−2}$-grading, then the kernel of $D$ is finitely generated, and in case $n= 4$, we show further that such a kernel is generated by at most four elements. Our main result is as follows: Let $k$ be an algebraically closed field, and let $B$ be a unirational factorial $k$-domain of finite transcendence degree $n$ over $k$. If $B$ admits a non-degenerate positive $\mathbb{N}^{n−1}$-grading over $k$, then $B$ is rational and affine over $k$. We further give trinomial relations satisfied by generators of $B$. From this, we recover Mori’s description of factorial affine surfaces with an elliptic torus action (1977) and Ishida’s description of factorial affine threefolds with an elliptic torus action of complexity one (1977), and we generalize the description of Hausen, Herppich and Suss of factorial affine varieties with an elliptic torus action of complexity one (2011) to (algebraically closed) fields of any characteristic. This talk represents joint work with Takanori Nagamine.
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On Borel subgroups of $\mathrm{Bir}(\mathbb{P}^2)$
Jean-Philippe Furter
Abstract
A Borel subgroup of a linear algebraic group is defined as a maximal closed connected solvable subgroup. It is well-known that all Borel subgroups are conjugate. We will address this issue on the Cremona group $\mathrm{Bir}(\mathbb{P}^2)$. This is a joint work with Isac Hedén.
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Nilpotent $B$-orbits associated to sets of orthogonal roots
Jacopo Gandini
Abstract
Let $G$ be a semisimple algebraic group, with a fixed Borel subgroup $B$ and maximal torus $T$ in $B$. To any set of pairwise strongly orthogonal roots I will attach a nilpotent $B$-orbit in the Lie algebra of $G$, and will explain how the combinatorics of the involutions in the affine Weyl group of $G$ is related to the geometry of such $B$-orbits. The talk is based on joint works with A. Maffei, P. Moseneder Frajria and P. Papi.
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Perpetuants---a lost treasure
Hanspeter Kraft
Abstract
This talk is about the following beautiful result conjectured by MacMahon in 1884 and proved by Emil Stroh in 1890.
Theorem The dimension of the space of perpetuants of degree $k>2$ and weight $g$ is the coefficient of $x^g$ in
$$
{\frac {x^{2^{k-1}-1}}{(1-x^{2})(1-x^{3})\cdots (1-x^{k})}}.
$$
For $k=1$ there is just one perpetuant, of weight 0, and for $k=2$ the number is given by the coefficient of $x^g$ in $x^2/(1-x^2)$.
We will explain the notion since it has some mathematical interest, and also Stroh's proof which is quite remarkable and in a way very modern.
With our method we shall in fact exhibit a basis of perpetuants, the main new result.
Let us shortly explain what a perpetuant is. In the classical invariant theory of binary forms $R_{n}:=\Bbbk[x,y]_{n}$, the $U$-invariants (sometimes called semi-invariants) $S(n)={\mathcal O}(R_{n})^{U}$ play a central role.
So far, the indecomposable $U$-invariants, i.e. the generators of the algebra of $U$-invariants $S(n)$ are only known for small $n$, and there seems to be no hope to find the Hilbert-series nor the dimensions of the spaces of indecomposables $U$-invariants in general.
However, there is a natural embedding $S(n) \subset S(n+1)$, i.e. every $U$-invariant of binary forms of degree $n$ is also a $U$-invariant of binary forms of degree $n+1$. (This was efficiently used by the classics, in particular in their computations.)
It is not difficult do see that an indecomposable $U$-invariant of $R_{n}$ might become decomposable as a $U$-invariant of $R_{m}$ for some $m>n$. This leads to the definition of a perpetuant.
Definition A perpetuant is an indecomposable element of $S(n)$ which remains indecomposable in all $S(m), \ m\geq n$.
Setting $I_{n}\subset S(n)$ to be the homogeneous maximal ideal, then a perpetuant gives an element of $I_n/I_n^2$ which {\em lives forever}, that is it remains nonzero in all $I_m/I_m^2$ for $m\geq n$. In this sense it is {\em perpetuant}.
Although there is no hope to describe $I_{n}/I_{n}^{2}$ in general, we have the beautiful formula above for the perpetuants $P_{n} \subseteq I_{n}/I_{n}^{2}$.
Joint with Claudio Procesi.
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Generators of proper normal subgroups of the Cremona group
Anne Lonjou
Abstract
We say that an element $g$ of a group $G$ generates a proper normal subgroup if the smallest normal subgroup containing $g$ is a proper
subgroup of $G$. We are interested in finding such elements (up to some power) in the group of birational transformations of the projective plane over
any algebraically closed field. In this talk, we give a complete classification of Cremona transformations of infinite order. It is known that some
loxodromic elements satisfy this property, we prove that it is also the case of Halphen twists. This is a joint work with Serge Cantat and Vincent Guirardel.
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Real rational models of surfaces and the real Cremona group
Frédéric Mangolte
Abstract
We survey results on real rational surfaces of the last decade focusing on the links between their topology and their birational geometry.
In particular, we will see that stereographic projections can be used to « separate » infinitely near points and that the infinite transitivity
of some subgroup of the real Cremona group can serve to prove that there is exactly one rational algebraic model for each nonorientable topological surface.
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Smooth horospherical projective varieties with small Picard number
Boris Pasquier
Abstract
In birational geometry, projective varieties with small Picard number (which is at least 1) appear to be the ”smallest” varieties. In particular, one of the principle of the Minimal Model Program is to decrease the Picard number of the variety, and sometimes it terminates with a fibration whose general fiber is of Picard number 1. To know projective varieties with small Picard number is then essential in this theory. In this talk, I will explain how to classify and study projective varieties with small Picard number in the family of horospherical varieties. This family is a subfamily of spherical varieties, which contains flag varieties and toric varieties. The first main idea to such classification is to construct many automorphisms of the variety.
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$\mathbb{A}^1$-contractible varieties
Sabrina Pauli
Abstract
The affine line $\mathbb{A}^1$ is the only smooth, complex, $\mathbb{A}^1$-contractible curve and it is conjecturedthat $\mathbb{A}^2$ is the only smooth, complex, $\mathbb{A}^1$-contractible surface. However, there exist nontrivial examples of smooth, complex, $\mathbb{A}^1$-contractible varieties in dimension 3 and higher. One of them is the Koras-Russell cubic $\{x^2y+z^2+t^3+x= 0\}\subset \mathbb{A}^4$ which sticks out since it is one of the few affine examples. Notrivial affine $\mathbb{A}^1$-contractible varieties serve as potential counter examples to the Zaraski cancellation problem which is still unsolved in dimension greater or equal to 3 and characteristic 0.
In my talk I will give a short survey on $\mathbb{A}^1$-homotopy theory and $\mathbb{A}^1$-contractible varieties and highlight the interplay between $\mathbb{A}^1$-homotopy theory and affine algebraic geometry.
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Uniformly Rational Varieties and equivariant version
Charlie Petitjean
Abstract
An algebraic variety is uniformly rational if every point of this variety admits an open neighborhood isomorphic to an
open subset of the affine space. Such a variety is necessarily smooth and rational,but the converse is an open question.
In this talk, we will consider varieties with group actions and introduce several equivariant defnitions of uniform rationality. Part of the results are joint work with A.Liendo (University of Talca).
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Embedding some open Riemann surfaces into the complex plane
Pierre-Marie Poloni
Abstract
Forster conjectured in 1967 that every Stein manifold of dimensionnadmits a proper holomorphic embedding into $\mathbb{C}^N$ with $N= [3n/2] + 1$.
Eliashberg, Gromov (1992) and Schurmann (1997) proved that this is true for all $n\geq 2$. On the other hand the question
whether every one-dimensional Stein manifold (i.e. every open Riemann surface) embeds into the complex plane remains wide open.
In this talk, we enlarge the class of examples for which a positive answer is known. More precisely, we will show that the Riemann sphere,
with a nonempty countable closed subset containing at most two accumulation points removed, as well as any compact Riemann surface of genus one,
with a nonempty closed discrete subset containing at most one accumulation point removed, are all embeddable into $\mathbb{C}^2$.
Our construction is inspired by a result of Sathaye stating that every smooth affine algebraic curve of genus one is a plane algebraic curve.
This is joint work with Frank Kutzschebauch.
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