Aneiros Vivas, Eva Hilbert's 17th problem on singular surfaces We consider Hilbert's 17th problem for real analytic surfaces, i.e. the posibility of representing analytic functions over the surface as sums of meromorphic functions, and the Artin-Lang property, i.e., if it is equivalent for a finite number of analytic functions being positive at some point of the surface and being positive in some ordering of the ring of meromorphic functions. We give some results for certain special cases like real algebraic surfaces whose part of maximal dimension is bounded.

 Basu, Saugata Efficient Algorithms for Computing the Betti Numbers of Semi-algebraic Sets. In this talk I will describe some recent progress in designing efficient algorithms for computing the Betti numbers of semi-algebraic sets in several different settings. I will describe a single exponential time algorithm for computing the first few Betti numbers in the general case and polynomial time algorithms in case the set is defined in terms of quadratic inequalities. One common theme underlying these algorithms is the use of certain spectral sequences -- namely, the Mayer-Vietoris spectral sequence and the cohomological descent'' spectral sequence first introduced by Deligne. Certain parts of this work is joint with R. Pollack, M-F. Roy and (separately) with T. Zell.

 Bernig, Andreas Support functions and the construction of the normal cycle Let V be a finite-dimensional Euclidean space. To a definable subset S⊆V, one can associate a function on V with values in the group ring Z[R], which is called support function. In the case of a convex set S, this is just the classical support function. It was shown by Ludwig Bröcker that S is uniquely determined by its support function. We show that S is bounded if and only if its support function is Lipschitz continuous. This yields a new, elementary construction of the normal cycle of a bounded definable set. Besides the axioms of o-minimal systems, this construction only uses C2-cell decompositions.

 Bertrand, Benoit Polynomial systems with few real zeroes pdf

 Briand, Emmanuel The configurations of two real projective quadrics Two couples of proper real projective quadrics are said to be rigid isotopic, or in the same configuration, if they are connected by a deformation of their equations that conserves, in some sense, the singularities of their intersection. We will consider the problem of counting and enumerating (exhibiting one representant per class) the (generic and singular) configurations of couples of proper real projective quadrics, in any dimension

 Broglia, Fabrizio Artin Lang Property for Global Semianalytic Sets In this talk we underline some upgrading of a work in progress whose goal is to try to adapte the tool "real spectrum" to the algebra of global semianalytic functions on a real coherent set. We consider the subset of orderings in the usual real spectrum of this algebra whose support has non-empty zero set. This choice is encouraged by the fact that for this spectrum we prove the Artin-Lang property in some reasonable cases as curves, normal surfaces and coherent sets whose connected components are compact: in other words in those cases the boolean algebra of global semianalytic sets can be identified with the boolean algebra of costructible sets of this spectrum. These results are obtained as a consequence of a Positivstellentz, weaker than the algebraic one, but enough to get a one to one correspondence between the two boolean algebras under reasonable hypothesis. As a conclusion we will stress the fact that in the global analytic case the Hilbert's 17-th problem and the real Nullstellensatz share the same main obstruction: namely to know wheter there are real prime ideals whose zero set has dimension smaller than expected.

 Brugallé, Erwan Construction of real plane algebraic curves with asymptotically maximal number of even ovals An oval of a real plane algebraic curve of degree 2k is said to be even if it is enclosed in an even number of ovals of the curve. The question of the asymptotic maximal number of even ovals of such a curve with respect to k can be traced back to the work of Virginia Ragsdale one century ago. In this talk, I will explain how to construct a family of curves of even degree which achieves this maximal number. This construction is based upon a mix of three methods of construction : small perturbations, patchworking, and "dessin d'enfants". The first one has been known at least for 150 years, the second one is due to Viro in the late 70's and the latter has been recently introduced in real algebraic geometry.

 Cimprič, Jaka G-invariant preorderings. I will present the main ideas from the G-invariant semialgebraic geometry: Construction of the invariant presentation, defining inequalities of the orbit space, finite-generatedness of the preordering of G-invariant sums of squares on the field of rational functions and a recent proof that for the ring of polynomials this preordering need not be finitely generated

 D'Acunto, Didier Some quantitative results in semi-algebraic geometry I will recall the notion of ridge and valley lines of semialgebraic functions. This notion is a key tool for obtaining quantitative results in semialgebraic geometry: Łojasiewicz exponent, geodesic diameter, quantitative Morse-Sard Theorem,.. I will try to explain some of them

 Degtyarev, Alex On maximal real elliptic surfaces The objective of our work is the deformation classification of real elliptic surfaces. In the study of elliptic surfaces over C, one passes first from arbitrary surfaces to Jacobian ones (via the Tate-Shafarevich group), and then, to trigonal curves on ruled surfaces (via Weierstraß equation). We attempt to apply this program to the real case. First, a real version of the Tate-Shafarevich group is constructed, and it is shown that its discrete part is determined by the topology of the real part of a Jacobian surface. The passage to the trigonal curves is straightforward, and it remains to study the latter. To this end, we apply a version of (decorated) designs d'enfents and show that any real trigonal curve can be cut into certain elementary building blocks (which are more or less plain cubic curves). Although in general it seems hopeless to obtain any kind of uniqueness of the above decomposition, for M-curves (and, hence, M-surfaces) we do give a complete classification.

 Delzell, Charles N. Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials Let h: Rn --> R be a continuous, piecewise-polynomial function (here, R denotes the real numbers). The Pierce-Birkhoff conjecture (1956) is that any such h is representable in the form supi infj fij, for some finite collection of polynomials fij in R[x_1,...,x_n], where sup and inf denote pointwise-supremum and infimum of functions, respectively. (A simple example, for n=1, is h(x) = |x| = sup{x,-x}.) In 1984, L. Mahe and, independently, G. Efroymson, proved this for n < 3; it remains open for n > 2. In this paper we prove an analogous result for "generalized polynomials" (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not just natural numbers; in this version, we restrict to the positive orthant, where each xi> 0. As before, our methods work only for n < 3. Mahe used induction on the degree of polynomials; this is not possible here, where degrees are arbitrary real numbers. Instead, we use induction on the number of xn-monomials, according to a trick used by C. Sturm in 1829 to extend the Fourier-Budan theorem to generalized polynomials. At one step, we also use C. Miller's cell-decomposition result for subsets of Rn definable by means of power functions xr (r an arbitrary real number, and x > 0).

 Dickmann, Max Quadratic Form Theory over Formally Real von Neumann Regular Rings pdf

 Domitrz, Wojciech Local classification of quasi-homogeneous varieties and maps by volume-preserving diffeomorphisms" (joint work with Joachim Rieger) In the talk I explain the relation between the local classification of singular subvarietes of Rp by volume-preserving diffeomorphisms and quasi-homogeneity with respect of a smooth submanifold. A subvariety V of Rp is called quasi-homogeneous with respect to a smooth submanifold if there exist a coordinate system (x1,...,xk,y1,...,yp-k) on Rp such that S={x1=...=xk=0} and V is quasi-homogeneous in this coordinate system with weights (w1,...,wn), where w1,...,wk are positive and wk+1=...=wn-k=0. I also consider AΩ classification of map-germs. Map-germs f,g:Rn--> Rp are AΩ-equivallent if there exist a diffeomorphism-germ Φ: Rn--> Rn and a volume-preserving diffeomorphism-germ Ψ: Rp--> Rp such that f=Ψ ° g Φ. I explain the relation between AΩ classification of map-germs and weak quasi-homogeneity of map-germs. I present examples of A simple weakly quasi-homogeneous map-germs.

 El Khadiri, Abdelhafid Weierstrass Division Theorem in definable C∞-Germs in a polynomially bounded O-minimal structure We give some examples of polynomially bounded o-minimal expansion R of the ordered field of real numbers where the Weierstrass Division Theorem does not hold in the ring of germs, at the origin of Rn, of definable C∞ functions.

 Fichou, Goulwen The corank of a Nash function germ is a blow-Nash invariant Is the corank an invariant of the blow-analytic equivalence between real analytic function germs? We give a partial positive answer in the particular case of the blow-Nash equivalence. The proof is based on the computation of some virtual Poincare polynomials and zeta functions associated to a Nash function germ.

 Fischer, Andreas Extending Peano-differentiable functions in o-minimal structures In classical Analysis there are examples of 2-times Peano-differentiable functions defined on a closed subset A of R which cannot be extended to a Peano-differentiable function on R such that the Peano-derivatives of the extended and the original function coincide on A. We discuss this extending problem in the o-minimal context. There, each definable m-times Peano-differentiable function defined on a closed subset A of Rn is the restriction of a definable m-times Peano-differentiable function which is defined on the whole affine space.

 Ghiloni, Riccardo Rigidity and cardinality of moduli space in Real Algebraic Geometry Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Z over R, the set of nonconstant rational maps from Z to X is finite. A bijective map φ:\tilde{X} --> X from an affine algebraic variety over R to X is called a weak change of the algebraic structure of X if it is regular and φ-1 is a Nash map preserving nonsingular points. We prove the following rigidity theorem: Every affine algebraic variety over R is UMR up to a weak change of its algebraic structure. Let us introduce another notion. Let Y be an affine algebraic variety over R. We say that X and Y are algebraically unfriendly if all the rational maps from X to Y and from Y to X are trivial, i.e., Zariski locally constant. From the preceding theorem, we infer that, if dim(X)≥1, then there exists a set {φt:\tilde{X}t--> X}t∈R of weak changes of the algebraic structure of X such that, for each t,s∈R with t≠s, \tilde{X}t and \tilde{X}s are algebraically unfriendly. This result implies the following expected fact: For each (nonsingular) affine algebraic variety X over R of positive dimension, the natural Nash structure of X does not determine the algebraic structure of X. In fact, the moduli space of birationally nonisomorphic (nonsingular) affine algebraic varieties over R that are Nash isomorphic to X is equipollent to R. This result was already known when R is the field of real numbers and X is compact and nonsingular.

 Itenberg, Ilia Tropical geometry and asymptotic enumeration of real rational curves Joint work with V. Kharlamov and E. Shustin. The goal of the talk is to present some applications of tropical geometry in real enumerative geometry. An important link between the complex algebraic world and the tropical one is given by Mikhalkin's correspondence theorem. This theorem together with the discovery of the Welschinger invariants (the latter invariants can be seen as a real analog of the Gromov-Witten invariants) give rise to several results concerning the enumeration of real rational curves on algebraic surfaces.

 Janeczko, Stanislaw Local symplectic invariants and generalized Darboux Theorem pdf

 Jelonek, Zbigniew Nash manifolds with unique embedding into RN pdf

 Kaiser, Tobias An o-minimal version of the Riemann mapping theorem Let Ω⊂C be a bounded semianalytic domain. Let φ:Ω→B(0,1) be a Riemann map (i.e. a biholomorphic isomorphism) from the domain into the unit ball. Assume that the angles ∠∂Ωx∈π(R\Q) for all singular boundary points x of Ω. Then φ is definable in a (new) o-minimal structure.

 Klep, Igor A Nirgendsnegativsemidefinitheitsstellensatz Consider symmetric polynomials f,g1,...,gm in noncommuting variables. If f satisfies an identity ∑i pi* f pi =1+ ∑i qi* qi + ∑ij qij* gi qij, then f is obviously nowhere negative semidefinite on the class of tuples of operators defined by the system of inequalities g1≥0,...,gm≥0. We prove the converse when the quadratic module generated by the gi is archimedean. The converse, if true in general, would be a noncommutative counterpart to the Positivstellensatz from real algebraic geometry.

 Korovina, Margarita Pfaffian Hybrid systems In this talk we will introduce and investigate behaviour of Pfaffian hybrid systems. Pfaffian hybrid systems are a sub-class of o-minimal hybrid systems which capture rich continuous dynamics and yet can be studied using finite bisimulations. In particular Pfaffian hybrid systems include hybrid systems with continuous dynamics definable by functions such as exp, sin, cos, and other trigonometric functions in appropriate domains. The existence of finite bisimulations is crucial for most decidability results for hybrid systems. The existence of finite bisimulations for o-minimal hybrid systems has been recently shown by several authors. The next natural question to investigate is how the sizes of such bisimulations can be bounded. A First step in this direction was done in a joint paper where a double exponential upper bound was shown for Pfaffian hybrid systems. Recently we improved this bound to a single exponential upper bound. Moreover we show that this bound is tight in general, by exhibiting a parameterized class of Pfaffian hybrid systems on which the exponential bound is attained. The bounds provide a basis for designing efficient algorithms for computing bisimulations, solving reachability and motion planning problems.

 Kostov, Vladimir Root arrangements for polynomial-like functions and their derivations A real polynomial P of degree n in one real variable is called hyperbolic if all its roots are real. An arrangement of the roots of P, P',..., P(n-1) is defined by writing all these roots in a chain two consecutive roots being connected by the sign < or =. If there are no equalities at all, then the arrangement is called non-degenerate. We consider only arrangements compatible with the Rolle theorem (between two consecutive roots of P(k) there is a root of P(k+1)). In degree ≥4 not all arrangements are realizable by hyperbolic polynomials. Therefore we suggest generalizations of hyperbolic polynomials by means of which we expect to realize all arrangements. These are the hyperbolic polynomial-like functions - smooth functions whose n-th derivatives vanish nowhere and which have n real roots counted with the multiplicities. It turns out that they realize all arrangements in degree 4, but starting from degree 5 there are counterexamples.

 Kuhlmann, Salma Differential κ-bounded Exponential-Logarithmic power series fields We construct fields of generalized power series with bounded support which admit an exponential function. We give a natural definition of an exponential, which makes these fields into models of the theory of the reals with exponentiation and restricted analytic functions. We discuss how to introduce transexponential functions, derivation and composition operators on these models.

 Lasserre, Jean Bernard Sum of squares approximation of polynomials, nonnegative on a real variety. We first show that every nonnegative polynomial f is almost a sum of squares (s.o.s.), that is, one exhibits a sequence of polynomials {fj}, where each fj is obtained from f by adding marginal (monomial) squares with small coefficients. The resulting sequence of approximations {fj} converges to f in the l1 norm of coefficients. We also show that a polynomial f, nonnegative on an arbitrary real variety V, can also be approximated in the l1 norm, using the same type of perturbation, and each approximation fj has a simple certificate of positivity on V. Importantly, no compactness assumption on V is required. As a consequence for optimization, one thus obtains simplified converging SDP-relaxations for minimizing f on V.

 Mahé, Louis On the length of polynomials Let k be a formally real field. We describe several approaches to recognize that a nonnegative polynomial P∈ k[Y] is a sum of 3, 5 or 7 squares. We then show how to prove that some well chosen polynomials P of degree multiple of 4 cannot be sums of 3 squares of polynomials.

 Marshall, Murray Recent results on spaces of orderings and spaces of signs Spaces of orderings and spaces of signs (considered first in 1996 in the book by Andradas, Broecker and Ruiz) provide an abstract framework in which to study semialgebraic geometry and, more generally, constructible sets in the real spectrum of a ring. A first order axiomatization of a space of orderings (resp., space of signs) is given as a multifield (resp., multiring) satisfying some additional properties. This simplifies and clarifies earlier axiomatizations due to Dickmann and Miraglia and Marshall (resp., Dickmann and Petrovich) and unifies various proofs. The Isotropy Theorem for spaces of orderings, established first in the field case by Bröcker and Prestel (and later, in the general case, by Marshall), asserts that a quadratic form is isotropic if it is isotropic on each finite subspace. A large class of local-global principles of this sort are now known to hold. Namely, a similar result holds for any pp formula which is 1-related and product free (assuming the space of orderings in question has finite stability index, although it is not known if this latter assumption is really necessary). The simplest sorts of pp formula not covered by this result have the form ∃t1∃t2 t1∈a+b ∧ t2∈c+d ∧ t1t2∈e+f. Spaces of orderings of function fields of irreducible conics over Q without rational points provide examples of spaces of orderings where the above sort of local-global principle fails for pp formulas of this latter type. This is work in progress, joint with Gladki. One would guess that the field of rational functions in two variables over a real closed field would yield similar examples, but this has not yet been proven. It is known already, by results of Dickmann, Miraglia and Marshall, that such examples cannot exist in the stability index 1 case (e.g., the function field of an irreducible curve over a real closed field), or in the case of the field of rational functions in 1 variable over Q.

 Piękosz, Artur Homotopy theory and o-minimality; weakly definable spaces over o-minimal structures I am going to show how to get the Comparison Theorems about the homotopy sets for locally definable and weakly definable spaces over o-minimal expansions of real closed fields, and mention some facts and questions about such spaces.

 Powers, Vicki Progress on Pólya's Positivity Propositions Pólya's Theorem says that if f∈ R[x1,....,xn] is homogeneous and positive on the n-simplex Δn := {(x1,....,xn) | xi ≥ 0; ∑i xi = 1}, then for suffciently large N the coeffcients of (x1+....+xn)Nf(x1,....,xn) are positive. This elegant and beautiful result has many applications, both in pure and applied mathematics. In this talk we discuss the history of the theorem, different proofs, and applications. We will conclude with some recent work of the speaker and Bruce Reznick extending the theorem to the case where f has zeros on the boundary of Δn.

 Rolin, Jean-Philippe Non oscillating solutions of analytic differential equations We will discuss some recent results and open questions related to the non oscillating property of solutions of analytic differential equations. In particular, we will focus on desingularization, o-minimality and model completeness...

 Rouillier, Fabrice Some answers in RealSolving In this talk we will give a progress report on some exact or certified methods for solving, in practice, academic or industrial problems known to be difficult. One of our objectives is also to show fundamental divergences (as well in the design as in the tools used - mathematical or algorithmic) between constructive theoretical method, even if they are of good complexity, and algorithms which are efficient in practice. For example, in most theoretical strategies, the singular locus of an algebraic variety is "computed" using the Jacobian criterion supposing that the studied variety is irreducible (or at least that the defining ideal is equidimensional and radical). Such assumptions must be avoid in practice as most as possible for efficiency reasons and thus many classical results need to be reformulated of simply become unusable for practical issues. We will take as example the resolution of polynomial systems of equalities and inequalities depending on parameters for non trivial application. The proposed method uses as black boxes methods for solving zero-dimensional systems as well as methods for computing one point in each connected component of a semi-algebraic set.

 Schweighofer, Markus Global optimization of polynomials using gradient tentacles To every real polynomial f in n variables, we associate a sequence of sets in Rn which we call the real gradient tentacles of f. Every gradient tentacle is defined by a single polynomial inequality involving the sum of squares of the partial derivatives of f. We propose a new method of global optimization of polynomials by relaxation to a semidefinite program based on the following three non-trivial facts. It follows from known results (Parusinski, Kurdyka et al.) that f has on each of its gradient tentacles only finitely many asymptotic values at infinity. This shows that a generalization of Schmüdgen's Positivstellensatz can be applied to give denominator free sums of squares certificates for the nonnegativity of f on its gradient tentacles. It follows from results of Kurdyka, Orro and Simon that if f is nonnegative on all its gradient tentacles, then it is globally nonnegative.

 Servi, Tamara On equivalent statements of the decidability problem for the real exponential field The open problem of the decidability of the real exponential field can be stated as follows: is there an algorithm which, given an elementary sentence involving the structure of the real ordered field and the real exponential function, stops, with output "yes" if the sentence is true, and with output "no" if the sentence is false? Macintyre and Wilkie have proved that the question has a positive answer, provided that some number theory conjecture (Schanuel´s Conjecture) is true. Using some ideas from their work, we provide other equivalent statements of the problem, and a simplified proof of their result.

 Shiota, Masahiro Semialgebraic singularity theory Very often singularity theory starts from polynomial or analytic functions, and semialgebraic, semianalytic and subanalytic sets appear. But after then the theory usually works in the Cr-category, r=0,1,..,∞. It is natural to construct a singularity theory in the semialgebraic category only. Then simple, clear and beautiful phenomena happen. One of them is that a semialgebraic controlled tube system of a semialgebraic stratification is unique. As an application we see a compact semialgebraic set X admits a semialgebraic Whitney triangulation. It is well-known X admits a semialgebraic triangulation and also a semialgebraic Whitney stratification one by one. It can do both at the same time.

 Sottile, Frank Bounds on real solutions to systems of polynomials Understanding the real solutions to systems of polynomial equations is a difficult question with many applications. In this talk, I will discuss recent work with Soprunova in which we prove the existence of lower bounds for many sparse polynomial systems, and give a method for computing these lower bounds in some instances. I will also discuss recent work with Bertrand and Bihan giving realistic upper bounds for the number of real solutions to polynomial systems supported on circuits.

 Ullrich, Peter Stratification of α-semialgebraic sets Let R be a real closed field, and α a (non-trivial) valuation on R which is compatible with the ordering. An α-semialgebraic set is a (finite) Boolean combination of sets of the form {f>0} and {α(g)>α(h)} with polynomials f,g,h∈ R[X1,...,Xn]. We provide a generalization of the well-known Stratification Theorem of semialgebraic sets to the case of α-semialgebraic sets. Furthermore, we formulate an analogue of Thom's Lemma in the context of α-semialgebraic sets. The results can also be generalized to finitely many order-preserving valuations on R.

 Vorobjov, Nicolai Topological complexity of definable sets The talk will explain the recent progress in proving upper bounds on Betti numbers of semialgebraic and sub-Pfaffian sets defined by first-order formulae with and without quantifiers.

 Wilkie, Alex J. An o-minimal version of Gromov's Algebraic reparameterization Lemma with a diophantine application Let M be an o-minimal expansion of a real closed field and S an M-definable subset of (0,1)n. Then for every natural number r, there exist finitely many r-times continuously differentiable, definable functions with domain (0,1)dim S, whose ranges exactly cover S, and whose derivatives (up to order r) are bounded by 1. This extends a theorem of Gromov (which itself built on work of Yomdin) from the semi-algebraic case. The proof here requires the development of the notion of "definably Banach" space. As a corollary, we show that a definable set contains rather few rational points of given height outside of the algebraic part of the set.This is joint work with Jonathan Pila.