1. Phạm Tiến Sơn

Title: An introduction to Semialgebraic Geometry.

Abstract: A semi-algebraic set is a subset of Rn defined by a finite sequence of polynomial equations and inequalities, or any finite union of such sets; a semi-algebraic map is a map with a semi-algebraic graph. Semi-algebraic sets and maps have distinctive and recognizable structural properties which make them an attractive domain for various applications. These relate both to the power of results that can be obtained and the power of available analytic techniques. In this lecture we present some basic properties of semi-algebraic sets and maps.

[1] R. Benedetti and J.-J. Risler. Real algebraic and semi-algebraic sets. Actualités Mathématiques. Hermann, Paris, 1990.

[2] J. Bochnak, M. Coste, and M.-F. Roy. Real algebraic geometry, volume 36. Springer, Berlin, 1998.

[3] M. Coste. An Introduction to Semialgebraic Geometry. Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica. Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000.

[4] L. van den Dries and C. Miller. Geometric categories and o-minimal structures. Duke Math. J., 84:497-540, 1996.

[5] L. van den Dries. Tame topology and o-minimal structures, volume 248 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1998.

**************

2. Jean Bernard Lasserre

Title: The moment - SOS hierarchy and its applications.

Abstract: We briefly review the Moment-SOS hierarchy and some of its applications. Initially designed to solve optimization problems whose objective function and constraints are described by polynomials, it also applies to solve many important applications in various domains of Science & Engineering, viewed as instances of the generalized moment problem (whose list of applications is almost endless). We will describe some of such applications.

Title: The Christoffel function: links with optimization & new properties.

Abstract: We will first introduce the Christoffel-Darboux kernel and the Christoffel function (CF). Those mathematical objects are well-known in the theory of approximation and orthogonal, but we claim that they also provide a useful, powerful and easy-to-use tool for solving some problems in data analysis (e.g. outlier detection, support inferences, density approximation), and also for interpolation.  Moreover, we will describe links (in the author's opinion some surprising) of the CF with convex duality, certificates of positivity in real algebraic geometry, equilibrium measure of compact sets.

**************

3. Hồ Minh Toàn

Title:  Moment problems and polynomial optimization.

Abstract: In this lecture, we will give a short introduction to the Moment problem and related problem (sums of squares). The topics cover

• Classical Moment problem (Haviland’s Theorem);
• Moment problems on semi-algebraic sets (Schmudgen’s Theorems);
• Relation between Moment problem and Sum of Squares.

References

[1] H. V. Hà and T. S. Phạm, Genericity in polynomial optimization, vol. 3 of Series on Optimization and Its Applications, World Scientific, 2017.

[2] J. B. Lasserre, Moments, Positive Polynomials and their Applications, Imperial College Press, London, 2009.

[3] M. Marshall, Positive polynomials and sum of squares, Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008.

**************

4. Nguyễn Văn Tuyên

Title: Vector optimization with polynomial data.

Abstract: In this talk, we present  connections between the Palais–Smale conditions, M-tameness, and properness for polynomial maps. Based on these results, we provide some sufficient conditions for the existence of Pareto solutions of polynomial vector optimization problems. We also introduce a generic class of polynomial vector optimization problems having at least one Pareto solution.

**************

5. Đinh Sĩ Tiệp

Title: Limits of real bivariate rational functions.

Abstract: We give some necessary and sufficient conditions for the existence of the limit of a real bivariate rational function at a given point. We also show that, if the denominator has an isolated zero at the given point, then the set of possible limits is a closed interval and can be explicitly determined.  As an application, we propose an effective algorithm to verify the existence of the limit and compute the limit (if it exists). Our approach is geometric and is based on Puiseux expansions.

**************

6. Nguyễn Hồng Đức

Title: Computation of the Łojasiewicz exponents of real bivariate analytic functions.

Abstract: The main goal of this talk is to present some explicit formulas for computing the Łojasiewicz exponent in the Łojasiewicz inequality on comparing the rate of growth of two real bivariate analytic function germs.