Rencontres Franco-Colombiennes de Mathématiques

Lundi 11 Décembre :

(1) 9h30-10h : WELCOME OF PARTICIPANTS

(2) 10h00-11h : Amador Martin Pizarro, Quotients and equations

Quotients are ubiquitous in Mathematics, and a general question is
whether a certain category of sets allows quotients. For the category of
definable sets in a given structure, the model theoretic approach is called
elimination of imaginaries. For algebraically closed fields, Chevalley’s theorem
and the existence of a field of definition of a variety imply that a
quotient of a Zariski constructible set by a Zariski constructible equivalence
relation is again a constructible set. Similar results hold for other
classes of fields, such as differentially closed fields.
In this talk, we will focus on separably closed fields of positive characteristic
p, and particularly those of infinite imperfection degree. Such a
field K has infinite linear dimension over Kp. It is unknown whether the
theory SCFp;1 of separably closed fields of positive characteristic p and
infinite imperfection degree has elimination of imaginaries. In joint work
with Martin Ziegler, we will provide a natural expansion of the language to
achieve this, by showing that the theory SCFp;1 is equational. Equationality,
introduced by Srour, and later considered by Srour and Pillay, is a
generalisation of local noetherianity. We will present the main ideas of the
proof, without assuming a deep knowledge of model theory.

(3) 11h15-12h15 : Dario Garcia, Pseudofinite structures, simplicity and unimodularity

The fundamental theorem of ultraproducts (Los’ Theorem) provides a
transference principle between the finite structures and their limits. It
states that a formula is true in the ultraproduct M of an infinite class of
structures if and only if it is true for "almost every" structure in the class,
which presents an interesting duality between finite structures and their
infinite ultraproducts.
This kind of finite/infinite connection can sometimes be used to prove
qualitative properties of large finite structures using the powerful known
methods and results coming from infinite model theory, and in the other
direction, quantitative properties in the finite structures often induce desirable
model-theoretic properties in their ultraproducts. These ideas were
used by Hrushovski to apply ideas from geometric model theory to additive
combinatorics, locally compact groups and linear approximate subgroups.
More examples of this fruitful interaction were given by Goldbring and
Towsner who provided proofs of the Szemerédi’s regularity lemma and Szemerédi’s
theorem via ultraproducts of finite structures.
In this talk I will review the main concepts of pseudofinite structures,
and present joint work with D. Macpherson and C. Steinhorn where we explored
conditions on the (fine) pseudofinite dimension that guarantee good
model-theoretic properties (simplicity or supersimplicity) of the underlying
theory of an ultraproduct of finite structures, as well as a characterization
of forking in terms of decrease of the pseudofinite dimension. I will also
present the concept of *unimodularity* (for definable sets) - which is satisfied
by both pseudofinite structures and omega-categorical structures -
and, if time permits, a result (joint with F. Wagner) about the equivalence
between difference notions of unimodularity.

(4) 13h30-14h30 : Johannes Huisman, Characteristic numbers and existence
of real solutions to systems of polynomial equations

For a given system of polynomial equations satisfying some mild conditions,
we introduce a finite sequence of characteristic numbers. We show
that if one of them is odd, then the system has a real solution. This is
a vast generalization of the elementary fact that a real polynomial in one
variable of odd degree has a real zero.

(5) 15h00-16h00 : Sylvie Paycha, Evaluating meromorphic functions at poles
while preserving locality

Evaluating meromorphic functions at poles is unavoidable when computing
Feynman integrals and the main difficulty lies in preserving locality.
There are sophisticated methods such as the forest formula or algebraic
Birkhoff factorisation à la Connes and Kreimer to make sure locality is
preserved. They typically use dimensional regularisation, thus leading to
meromorphic functions in one variable.
Instead we use a multiparameter regularisation à la Speer, which leads to
multivariate meromorphic functions, our object of study in this talk. We introduce
a locality structure on multivariate meromorphic germs with linear
poles at zero, which "separates" germs depending on orthogonal sets of variables
and call such germs independent of each other. In order to evaluate
these germs at the zero pole, we define generalised evaluators at zero which
factorise over independent germs and characterise them by means of (decorated)
cones using multivariate Laurent expansions. A typical generalised
evaluator is obtained by first projecting the germ onto its holomorphic part
and then evaluating this projected germ at zero. We shall discuss the underlying
multivariate projections and their central role in renormalisation.
This talk is based on joint work (partially in progress) with P. Clavier,
L. Guo and B. Zhang.

Mardi 12 décembre :

(1) 9h45-10h45 : Erwan Brugallé, Plane tropical cubic curves of arbitrary
genus, and generalisation

It has been known for a long time that a tropical curve in R2 of degree
d has genus at most (d 􀀀 1)(d 􀀀 2)=2. In this talk I will explain how to
construct a plane tropical cubic curve of arbitrary genus. In particular,
I will resolve the apparent contradiction of the last two sentences. More
generally, I will talk about (upper and lower) bounds on Betti numbers of
tropical varieties of Rn (and if time permits on tropical Hodge numbers).
Generalizing what is written above for cubics, I will show that there is no
finite upper bound on the total Betti numbers of projective tropical varieties
of degree d and dimension m. This is a joint work with B. Bertrand and L.
Lopez de Medrano.

(2) 11h00-12h00 : Andres Jaramillo Puentes, Rigid isotopies of degree 5
nodal rational curves in RP2.

In order to study the rigid isotopy classes of nodal rational curves of
degree 5 in RP2, we associate to every real rational quintic curve with
a marked real nodal point a trigonal curve in the Hirzebruch surface 3
and the corresponding nodal real dessin on CP=(z 7! z). The dessins are
real versions, proposed by S. Orevkov, of Grothendieck’s dessins d’enfants.
The dessins are graphs embedded in a topological surface and endowed
with a certain additional structure. We study the combinatorial properties
and decompositions of dessins corresponding to real nodal trigonal curves
C   in real ruled surfaces . Uninodal dessins in any surface with
non-empty boundary and nodal dessins in the disk can be decomposed in
blocks corresponding to cubic dessins in the disk D2, which produces a
classification of these dessins.

(3) 13h15-14h15 : Ilia Itenberg, Lines on quartic surfaces

We study the possible values of the number of straight lines on a smooth
surface of degree 4 in the 3-dimensional projective space. We show that the
maximal number of real lines in a real non-singular spatial quartic surface is
56 (in the complex case, the maximal number is known to be 64). We also
give a complete projective classification of non-singular complex quartics
containing more than 52 lines: all such quartics are projectively rigid. Any
value not exceeding 52 can appear as the number of lines of an appropriate
complex quartic. (Joint work with A. Degtyarev and A. S. Sertöz.)

(4) 14h30-15h30 : Maria Carrizosa, Counting polarizations of bounded degree
on abelian varieties

We know that there are only finitely many polarizations of given degree
(modulo automorphisms) on an abelian variety. We will give a bound for
this number.

(5) 16h-17h : Marco Boggi, Endomorphisms of Jacobians of algebraic curves
with automorphisms

Let C be a very general complex smooth projective algebraic curve endowed
with a group of automorphisms G such that the quotient C=G has
genus at least 3. I will show that the algebra of Q-endomorphisms of the
Jacobian J(C) of C is naturally isomorphic to the group algebra QG. Time
permitting, I will then explain some applications of this result to the theory
of virtual linear representations of the mapping class group. This talk is
based on joint work with Eduard Looijenga.

Mercredi 13 décembre :

(1) 9h45-10h45 : Laurent Charles, From Berezin-Toeplitz operator to entanglement
entropy

The first part of my talk will be an introduction to Berezin-Toeplitz
quantization on Kähler manifolds. Then I will consider a particular class
of Berezin-Toeplitz operators whose symbols are characteristic functions. I
will discuss their spectral distribution. As an application, I will explain the
area law for the entanglement entropy in Quantum Hall effect.

(2) 11h00-12h00 : Paul Emile Paradan, Indices équivariants d’opérateurs de
Dirac et limites semi-classiques.

Considérons une variété munie d’une structure spin S équivariante par
rapport à l’action d’un groupe de Lie compact G. A chaque fibré en droites
équivariant L on associer :
 une famille de représentations V (k) de G qui correspond à la quantification
géométrique des données (S;Lk); k  1,
 une famille de distributions (k) sur le dual de l’algèbre de Lie de G
qui est un analogue géométrique de la famille V (k).
Dans cet exposé, nous verrons comment exprimer le comportement asymptotique
de (k) au moyen de mesures de Duistermaat-Heckman. Dans le
cas où la variété est non-compacte, nous expliquerons comment en déduire
des propriétés fonctorielles sur les représentations V (k).
Ce travail est une collaboration avec Michèle Vergne (voir arXiv:1708.08226).

(3) 13h15-14h15 : Daniel Massart, Measurable Finsler metrics

we show how Borel-measurable Finsler metrics provide weak solutions to
some optimization problems in Riemannian geometry, such as the systolic
problem.

(4) 14h30-15h30 : Alex Cardona

(5) 16h-17h : Clara Aldana

Jeudi 14 décembre :

(1) 9h45-10h45 : Alex Berenstein, Polish groups and automatic continuity.

In this talk I will give an introduction to Polish groups and automatic
continuity. We say that a topological group is Polish if it is separable and
completely metrizable. We say a Polish group has the automatic continuity
property if any algebraic morphism to any separable topological group is
continuous.
In joint work with I. Ben Yaacov (ICJ) and J. Melleray (ICJ) we studied
these concepts for some groups of isometries and gave a criteria for
automatic continuity. I will talk about these criteria and then I will discuss
ongoing work with Rafael Zamora (Ph.D. Paris 6) on the group of
isometries of some metric structures called randomizations.

(2) 11h00-12h00 : Pablo Cubides, Exponentiation is easy to avoid (sometimes)

A celebrated theorem of Chris Miller states that if R is an o-minimal
expansion of the field of real numbers then either R is polynomially bounded
or the exponential function is definable in R. After introducing an analogue
of o-minimality for expansions of algebraically closed valued fields (called C-
minimality), the aim of the talk is to show that every C-minimal expansion
of a valued field (K; v) having value group Q is polynomially bounded. In
particular, we obtain that any C-minimal expansion of valued fields like Cp,
Falg((tQ)) are polynomially bounded. This is a joint work with Françoise
Delon.

(3) 13h15-14h15 : Otmar Venjakob, Regulator maps for Lubin-Tate extensions

Regulator maps à la Perrin-Riou play an important role in the Iwasawa
theory of cyclotomic fields: they map for instance very special (norm
compatible systems of) units to p-adic L-functions. Recently the Iwasawa
theory for Lubin-Tate extensions has become quite popular and I will report
on results towards the construction of regulator maps in this setting
using ('; 􀀀)-modules (joint work with Peter Schneider).

(4) 14h30-15h30 : Aurélien Galateau, The distribution of torsion on subvarieties
of abelian varieties.

The Manin- Mumford conjecture describes the distribution of torsion
points in subvarieties of abelian varieties. It was proven by Raynaud thirty
years ago, and explicit versions were later given by Coleman, Buium or
Hrushovski. I will describe a natural way to tackle this problem, by combining
algebraic interpolation with classical theorems on homotheties in the
Galois representation associated to the torsion of abelian varieties.

(5) 16h-17h : Guillermo Mantilla,

Vendredi 15 décembre :

(1) 9h45-10h45 : Maria Paula Gomez, The Baum-Connes Conjecture and
an Oka’s principle in Noncommtative Geometry

The Baum-Connes conjecture was introduced by Paul Baum and Alain
Connes in the 80’s; it gives a way of computing the K-theory of the reduced
C*-algebra of a locally compact group. This C*-algebra encodes the
topology of the temperate dual of the group and its K-theory is a topological
invariant of this space. The conjecture, and some generalizations,
are still open for many groups having a strong version of property (T); no
real progress have been done for 15 years. Strong property (T) is a rigidity
property on groups representations (e.g higher rank Lie groups and there
lattices have strong property (T)); it was introduced by Vincent Lafforgue
in his work on Baum-Connes as it prevents the methods that have been
used to prove the Baum-Connes conjecture to work. Nonetheless, a direction
that is still open concerns applying the ideas of Bost, who defined
a version of Oka principle in Noncommutative Geometry. In this talk, I
will give a short survey on the conjecture and I will explain the statement
linking it to Oka’s principle.

(2) 11h00-12h00 : Paul Bressler, On quasi-classical limits in deformation
quantization

Star-products (one parameter formal deformations of the usual product
on functions) serve as local models for DQ-algebroids. A DQ-algebroid is a
formal one-parameter deformation its "classical limit" which in general is a
twisted form of the structure sheaf of the the manifold. As is well known, a
star-product on functions on a manifold gives rise to a Poisson on the sheaf
of functions. I will explain what sort of additional structure arises on the
classical limit of a DQ-algebroid generalizing and extending the Poisson
structure.

(3) 13h15-14h15 : Yves Benoist, Recurrence on Affine Grassmannians

Let W be a k-dimensional affine subspace in the d-dimensional affine
space V , and let S be a symmetric set in the group G of invertible affine
transformations of V generating a Zariski dense subgroup of G. We prove
with C. Bruere that, if one chooses at random n elements of S and computes
their product g, the law of the image of W by g converges when n is going
to infinity to a measure m on the affine grassmannian variety. This limit
measure m has mass 1 when 2k is at least d and is null otherwise.

(4) 14h30-15h30 : Alberto Medina, Transformations of flat affine manifold

A flat affine manifold is a manifold endowed with a flat and torsion free
linear connection. We will give a new characterization of flat affine manifolds
by means of affine representations of the group of the automorphisms
of the manifold. From the infinitesimal point of view the representation is
given by the connection form and the fundamental form of the bundle of
linear frames of the manifold. We will also show the existence of a finite
dimensional associative envelope of the Lie algebra of the Lie group group
of transformations of the flat affine manifold.

(5) 16h-17h : Omar Saldarriaga, Transformation of flat affine Lie groups

We will show the existence of Lie groups endowed with a flat affine biinvariant
connection whose Lie algebra contains the Lie algebra of complete
infinitesimal affine transformations of the given Lie group. This is a special
case of the characterization given in Medina’s talk. We exhibit some results
about flat affine manifolds whose group of diffeomorphisms admit a flat
affine bi-invariant structure. We finish the presentation exhibiting some
examples.