We gratefully acknowledge support from:
ANR FIBALGA
Institut de Mathématiques de Bourgogne
Swiss Academy of Sciences
Swiss mathematical society
University of Basel
30 January - 3 Feburary 2023, in Charmey
(near Gruyères, Fribourg, Switzerland)
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Xavier Caruso (Université de Bordeaux) |
Random p-adic polynomials. |
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Nguyen-Bac Dang (Université Paris-Saclay) |
Degree growth of iterates of rational mappings: a functional analytic viewpoint. |
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Victoria Hoskins (Radboud University Nijmegen) |
Moduli problems and geometric invariant theory. |
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Monday January 30 |
Tuesday January 31 |
Wednesday February 1 |
Thursday February 2 |
Friday February 3 |
| 12h30 welcome |
breakfast 8h45-9h45 mini-course 1 10h15-11h15 mini-course 2 11h45-12h45 mini-course 3 |
breakfast 8h45-9h45 mini-course 1 10h15-11h15 mini-course 2 11h45-12h45 mini-course 3 |
breakfast 8h45-9h45 mini-course 1 10h15-11h15 mini-course 2 11h45-12h45 mini-course 3 |
breakfast 8h45-9h45 mini-course 1 10h-11h mini-course 2 11h15-12h15 mini-course 3 |
| lunch | lunch | lunch | lunch | bus at 12h42 |
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14h30-15h30 mini-course 1 16h00-17h00 mini-course 2 17h30-18h30 mini-course 3 19h dinner |
time for discussion / enjoying the mountain side 17h20-18h10 Bouillet 18h30-19h20 Jaramillo Puentes 19h30 dinner |
time for discussion / enjoying the mountain side 17h20-18h10 Cangini 18h30-19h20 Abboud 19h30 dinner |
time for discussion / enjoying the mountain side 17h20-18h10 Benozzo 18h30-19h20 Dujella 19h30 dinner |
VIVA GRUYERE Charmey, Rte des Arses 4, 1637 Charmey
The journey to Charmey is 2h10 from Geneva, 2h30 from Basel/Zürich, 1h30 from Lausanne.
See timetables on www.cff.ch, the bus stop is "Charmey (Gruyère), Le Chêne". The place is very close to the bus stop.
Here is a hike map for the area.
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In this mini-course based on a joint work with Charles Favre,
I will show how one can endow the space of b-divisors or b-classes
(divisors or cycle classes that live in all birational compactifications
of a given projective variety X) to understand certain problems arising
in algebraic dynamics. Precisely, we will study the asymptotic growth of
the sequence of algebraic degrees of iterates of a given rational map f.
We will show how the pullback action by f on the space acts on the space
of b-divisors as a continuous operator and show that under certain
generic conditions, this operator has a unique eigenvector for the
largest eigenvalue (a spectral gap). I will then explain how one can
construct this eigenvector and then show that the uniqueness follows
from a generalization of the Hodge index theorem to b-divisors.
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Many moduli spaces are constructed as group quotients using geometric invariant theory (GIT). First, we will review GIT for reductive groups and outline some of the moduli spaces one can construct using reductive GIT. The heart of the course will focus on explaining a recent extension of this theory to non-reductive groups and describing applications to the construction of new moduli spaces, including moduli spaces of hypersurfaces in weighted projective spaces and moduli spaces of unstable objects.
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Random polynomials with real coefficients is a classical object
of study in probability, which has outstanding applications to
different areas in mathematics and physics. Recently, the p-adic
counterpart of this question has attracted some attention, and
several progress towards the distribution of roots of a random
p-adic polynomial have been achieved.
The aim of this lecture is to give a fair introduction of this
family of results and techniques.
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Let f be a dominant polynomial transformation of the complex affine plane. The dynamical degree lambda_1 of f is defined as the limit of the n-th root of the degree of the n-th iterate of f. In 2007, Favre and Jonsson showed that the dynamical degree of any polynomial endomorphism of the affine plane is a quadratic integer. For any affine surface S0, there is a definition of the dynamical degree that generalizes the one on the affine plane. We show that the result still holds for any affine surface: the dynamical degree of an endomorphism of any affine surface is a quadratic integer. The proof uses the space of valuations centered at infinity V. The endomorphism f defines a transformation on V and studying the dynamics of f on V gives information about the dynamics of f on S0. The main result is that under certain hypothesis, f admits an attracting fixed point in V that we call an eigenvaluation. This implies that one can find a good compactification S of S0 such that f admits an attracting fixed point p at infinity and f has a normal form at p; the result on the dynamical degree follows from the normal form.
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An important problem in birational geometry is trying to relate in a meaningful way the canonical bundles of the source and the base of a fibration.
Classically, results in this direction have been established only thanks to Hodge theory, which is a tool we cannot use over fields of positive characteristic. However, recently the problem has been approached using ideas from the minimal model program.
We can take inspiration from these methods to prove a canonical bundle formula result in positive characteristic.
This is work in progress.
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Over fields of characteristic p, the Lie algebra of an algebraic group does not carry as much
information as it does in characteristic 0. Still, the presence of the "p-mapping" on Lie(G)
allows to reconstruct at least the Frobenius kernel of G. In this talk, we describe the restricted
locus of the universal Lie algebra (i.e. the locus where it admits a p-mapping) and the moduli space
of p-Lie algebras over the flattening stratification of the center. Then we revisit the classical example
of the moduli space of Lie algebras of rank 3, showing it is representable by a finitely presented
scheme over the ring of intergers. Using the beautiful linkage theory, we will show that it has two
irreductible components, it is flat, and with reduced Cohen-Macaulay geometrical fibers.
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Abstract
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Let E be an elliptic curve defined over a number field k. Canonical height on E in a certain sense measures arithmetic complexity of points of E(k). Given a real number B, it is often useful to have good bounds on the number of points of E(k) with height at most log(B), which we denote by N(B). While classical results give good bounds for a fixed elliptic curve, in general it is hard to get uniform results. This problem can be simplified if we assume the existence of a nontrivial point of order two in E(k). We will present a strategy for uniformly bounding N(B) in this family of curves, following methods developed by Bombieri and Zannier and later Naccarato in the rational case, as well as new results on how this can be generalized to arbitrary number fields.
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Tropical geometry has been proven to be a powerful computational tool in enumerative geometry over the complex and real numbers. In this talk we present an example of a quadratic refinement of this tool, namely a proof of the quadratically refined Bézout's theorem for tropical curves. We recall the necessary notions of enumerative geometry over arbitrary fields valued in the Grothendieck-Witt ring. We will mention the Viro's patchworking method that served as inspiration to our construction based on the duality of the tropical curves and the refined Newton polytope associated to its defining polynomial. We will prove that the quadratically refined multiplicity of an intersection point of two tropical curves can be computed combinatorially. We will use this new approach to prove an enriched version of the Bézout theorem and of the Bernstein–Kushnirenko theorem, both for enriched tropical curves. Based on a joint work with S. Pauli.
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Registration is closed.
Marc Abboud (Rennes)
Ahmed Abouelsaad (Basel)
Sami Al Asaad (Grenoble)
Jefferson Baudin (EFPL)
Vladimiro Benedetti (Dijon)
Marta Benozzo (LSGNT London)
François Bernard (Angers)
Fabio Bernasconi (EPFL)
Angelo Bianchi (Sao Paulo)
Jérémy Blanc (Basel)
Aurore Boitrel (Angers)
Antoine Boivin (Angers)
Anna Bot (Basel)
Alice Bouillet (Rennes)
Alberto Calabri (Ferrara)
Jean-Baptiste Campesato (Angers)
Jung Kyu Canci (Lucerne)
Xavier Caruso (Bordeaux)
Alessio Cangini (Basel)
Nguyen-Bac Dang (Orsay)
Julian Demeio (Basel)
Adrien Dubouloz (Dijon)
Marta Dujella (Basel)
Mani Esna-Ashari (Basel)
Andrea Fanelli (Bordeaux)
Stefano Filipazzi (EPFL)
Pascal Fong (Orsay)
Douglas Guimarães (Dijon)
Isac Hedén (Uppsala)
Liana Heuberger (Bath)
Victoria Hoskins (Nijmegen)
Andrés Jaramillo Puentes (Essen)
Matilde Maccan (Rennes)
Frederic Mangolte (Marseille)
Irène Meunier (Basel)
Lucy Moser-Jauslin (Dijon)
Lucas Moulin (Dijon)
Léo Navarro Chafloque (EPFL)
Gianni Petrella (Luxembourg)
Antoine Pinardin (Edinbourgh)
Linus Rösler (EPFL)
Gerold Schefer (Basel)
Julia Schneider (EPFL)
Ronan Terpereau (Dijon)
Christian Urech (EPFL)
Immanuel van Santen (Basel)
Egor Yasinsky (Paris)
Sokratis Zikas (Poitiers)
Susanna Zimmermann (Paris)
Andrea Fanelli (University of Bordeaux)
Philipp Habegger (University of Basel)
Julia Schneider (EPFL Lausanne)
Ronan Terpereau (University of Burgundy)
Susanna Zimmermann (University of Paris-Saclay)
Logistic support: Adrien Dubouloz (University of Burgundy)
Here are the previous ones:
1st, 2nd , 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, Swiss-French Workshop in Algebraic Geometry
We gratefully acknowledge support from:
ANR FIBALGA
Institut de Mathématiques de Bourgogne
Swiss Academy of Sciences
Swiss mathematical society
University of Basel