**Programme du jeudi 12 mai**

- 9h30 : Accueil Thé-café
- 10h30-11h30 : Michele Stecconi :
*Semicontinuity of Betti numbers and singular sets* - 12h: Repas en commun à la Biocoop
- 14h15-15h15 : François Bernard :
*Seminormalization and regulous functions on complex affine algebraic varieties.* - 15h30-16h30 : Jean-Philippe Monnier :
*Central seminormalization with elementary central gluings* - 16h30
*:*Pause Thé-café - 17h-18h : Anh Thi Ngoc Nguyen :
*Complex and real enumerative geometry in del Pezzo varieties*

Le soir, un repas en commun est prévu au restaurant Le Goût des Autres (rdv à 20h).

**Programme du vendredi 13 mai**

- 9h15-10h15 Anna Katharina Bot :
*A smooth complex rational affine surface with uncountably many nonisomorphic real forms* - thé-café
- 10h40-11h40 Fabien Priziac :
*On some properties of real algebraic groups and their actions on real algebraic sets* - Repas en commun à la biocoop
- 14h30-15h30 Jean-Baptiste Campesato :
*Cᵐ solutions of semialgebraic equations and the Whitney extension problem* - thé-café

### Titre et résumés des exposés :

- Michele Stecconi :
*Semicontinuity of Betti numbers and singular sets*

The topic of this talk is the study of the topology of generic singularities of smooth maps.

I will discuss how to use polynomial approximations in a quantitative way, to obtain an analogue of

Thom-Milnor inequality valid for all smooth maps, bounding the Betti numbers of the singular loci (in particular, the zero set) of a function f in terms of an appropriate pseudo-degree of f.

Here, the standard way of controlling the topology in the approximation: maintaining a transversality condition, produces a non-sharp inequality. I will present a result about the behavior of the Betti numbers under C^{^0} approximations that improves such method.

Based on the paper 'What is the degree of a smooth hypersurface?' Journal of Singularities (vol 23) 2021, with Antonio Lerario.

- François Bernard :
*Seminormalization and regulous functions on complex affine algebraic varieties.*

Let X be an affine complex algebraic variety. The "seminormalization of X" is an algebraic variety X^+ obtained by gluing together the points in the fibers of the normalization morphism. Its construction was inspired by the notion of “weakly normal analytic spaces”. One of the interest of the seminormalization comes from the fact that it has nice singularities in codimension 1 while being linked to X by a finite and birational homeomorphism. The main result of this talk is that one can identify the polynomial functions on X^+(C) with the rational functions of X which are continuous for the euclidean topology on all X(C). Those functions can be seen as complex regulous functions, a type of functions recently introduced in real algebraic geometry.

- Jean-Philippe Monnier :
*Central seminormalization with elementary central gluings*

We introduce a theory of seminormalization in the real setting called central seminormalization. It is related to the theory of regulous functions on real algebraic varieties. We provide a construction of this central seminormalization by a decomposition theorem in elementary central gluings.

- Anh Thi Ngoc Nguyen :
*Complex and real enumerative geometry in del Pezzo varieties.*

The enumerative problems with respect to counting (resp. real) algebraic curves passing through a (resp. real) configuration of points in (resp. real) algebraic varieties are usually known as Gromov-Witten invariants (resp. Welschinger invariants). In my talk, I will present some interesting relations between genus-0 Gromov-Witten-Welschinger invariants of some three dimensional del Pezzo varieties and that of del Pezzo surfaces. These are generalizations of results by Brugallé and Georgieva in 2016.

- Anna Katharina Bot :
*A smooth complex rational affine surface with uncountably many nonisomorphic real forms.*

A real form of a complex algebraic variety X is a real algebraic variety whose complexification is isomorphic to X. Many families of complex varieties have a finite number of nonisomorphic real forms, but up until recently no example with infinitely many had been found. In 2019, Lesieutre constructed a projective variety of dimension six with infinitely many nonisomorphic real forms, and this year, Dinh, Oguiso and Yu described projective rational surfaces with infinitely many as well. In this talk, I’ll present the first example of a rational affine surface having uncountably many nonisomorphic real forms.

- Fabien Priziac :
*On some properties of real algebraic groups and their actions on real algebraic sets.*

Considering real algebraic groups as real algebraic sets equipped with a polynomial group structure, we will present some of their general properties as well as the properties of their polynomial actions on real algebraic sets. In particular, we will highlight similarities and differences with complex algebraic groups. In this study, we will also be naturally brought to consider semialgebraic groups, that is semialgebraic sets equipped with a polynomial group structure. We will then focus on the case of compact real algebraic groups whose properties come closer to the properties of complex algebraic groups.

- Jean-Baptiste Campesato :
*Cᵐ solutions of semialgebraic equations and the Whitney extension problem.*

We address the question of whether geometric conditions on the given data can be preserved by a solution in

(1) the Whitney extension problem, which consists in determining whether a function g:X→ℝ defined on a closed subset X⊂ℝⁿ admits a Cᵐ extension on ℝⁿ, and,

(2) the Brenner-Fefferman-Hochster-Kollár problem, about the existence of a Cᵐ solution to A(x)G(x)=F(x), where A is a matrix of functions on ℝⁿ, and the unknown is a vector-valued function G.

In a joint work with E. Bierstone and P.D. Milman, we prove that, for both problems, when the data are semialgebraic (or, more generally, definable in a suitable o-minimal structure), the existence of a solution implies the existence of a semialgebraic (or definable) solution. Our results involve a certain loss of differentiability.

More precisely, for (1), we prove that given a semialgebraic closed subset X⊂ℝⁿ, there exists r:ℕ→ℕ such that if a semialgebraic function g:X→ℝ is the restriction of a Cʳ⁽ᵐ⁾ function then it is the restriction of a semialgebraic Cᵐ function.

For (2), we prove that given A a matrix of semialgebraic functions, there exists r:ℕ→ℕ such that if F is semialgebraic and A(x)G(x)=F(x) admits a Cʳ⁽ᵐ⁾ solution, then there exists a Cᵐ solution which is semialgebraic.