# 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.