Geometry of Polynomial System Solving, Optimization and Topology

Liste des contributions

1. Moment Method in Optimization by D. Henrion. 10:00-12:00. Amphitheater Darboux, IHP

09/10/2023 10:00

3. Moment Method in Optimization by D. Henrion. 10:00-12:00. Amphitheater Darboux, IHP

10/10/2023 10:00

4. Polynomials Systems and Real Geometry by I. Emiris and D. Plaumann. 14:00-17:30. Amphitheater Darboux, IHP

10/10/2023 14:00

5. Polynomials Systems and Real Geometry by I. Emiris and D. Plaumann. 10:00-12:00. Amphitheater Darboux, IHP

11/10/2023 10:00

8. General audience presentation "The quadratic equation revisited" by B. Sturmfels. 16:00-18:00. Amphitheater Darboux, IHP

11/10/2023 16:00

Abstract High school students learn how to express the solution of a quadratic equation in one unknown in terms of its three coefficients. What does this this formula matter? We offer an answer in terms of discriminants and data. This lecture invites the audience to a journey towards non-linear algebra.

6. Polynomials Systems and Real Geometry by I. Emiris and D. Plaumann. 10:00-12:00. Amphitheater Darboux, IHP

12/10/2023 10:00

7. Polynomials Systems and Real Geometry by I. Emiris and D. Plaumann. 14:00-17:00. Amphitheater Darboux, IHP

12/10/2023 14:00

9. Welcome coffee

16/10/2023 08:45

10. Opening

16/10/2023 09:15

11. Dimension results for sparse systems homogenized via rational polytopes by Pierre-Jean Spaenlehauer

16/10/2023 09:30

Abstract. A classical method to compute with sparse polynomials is to homogenize them with respect to Newton polytopes, regarding them as homogeneous elements of Cartier degrees in the Cox ring of a projective toric variety. In this talk, we investigate subvarieties defined by generic polynomial systems in the Cox ring when the degrees are non-necessarily Cartier, with a view towards...

12. Coffee break

16/10/2023 10:30

13. Chordal Graphs in Triangular Decomposition in Top-Down Style by Chenqi Mou

16/10/2023 11:00

Abstract. In this talk, I will present the connections between chordal graphs from graph theory and triangular decomposition in top-down style from symbolic computation, including the underlying theories, algorithms, and applications in biology. Viewing triangular decomposition in top-down style as polynomial generalization of Gaussian elimination, we show that all the polynomial sets,...

14. Modular algorithms for computing triangular decompositions of polynomial systems by Marc Moreno Maza

16/10/2023 14:30

Abstract. https://rtca2023.github.io/pages_Paris/files_m5/abstract_moreno-maza.pdf

15. Coffee break

16/10/2023 15:30

21. Polynomial systems arising in the formal verification of programs by Fatemeh Mohammadi

17/10/2023 09:30

Abstract. Multiple classical problems in the formal verification of programs such as reachability, termination, and template-based synthesis can be reduced to solving polynomial systems of equations. In this talk, I will describe the primary objects and these connections. In particular, I will show how the algebraic and geometric techniques can be applied, enhancing the scalability and...

18. Coffee break

17/10/2023 10:30

22. Positive solutions to polynomial systems and applications to reaction networks by Elisenda Feliu

17/10/2023 11:00

Abstract. The main object of study in the (algebraic) theory of reaction networks is the solution se of a system of parametric polynomial equations in the positive orthant. This system consists of polynomials with fixed support, the coefficients are linear in the parameters, but there might be some (proportionality) dependencies among the coefficients. The questions of interest concern...

23. Polynomial Systems Arising in Paradoxical 6R Linkages by Zijia Li

17/10/2023 14:30

Abstract. In this talk, we first provide a comprehensive definition of closed n-linkages and explain their mobility, typically denoted as n-6. We then focus on the intriguing subset of closed n-linkages with a mobility higher than n-6, known as paradoxical linkages. Based on the powerful tools of Bond Theory and the freezing technique, we present a thorough classification of n-linkages with a...

24. Coffee break

17/10/2023 15:30

17. Some advances in numerical algebraic geometry for computing real solutions by Jon Hauenstein

18/10/2023 09:30

Abstract. Numerical algebraic geometry provides a collection of algorithms for computing and analyzing solution sets of polynomial systems. This talk will discuss new techniques that have been developed in numerical algebraic geometry for focusing on real solution sets of polynomial systems. Several applications of these techniques will be presented such as computing smooth points on algebraic...

25. Coffee break

18/10/2023 10:30

19. u-generation: solving systems of polynomials equation-by-equation by Anton Leykin

18/10/2023 11:00

Abstract. We develop a new method that improves the efficiency of equation-by-equation homotopy continuation methods for solving polynomial systems. Our method is based on a novel geometric construction and reduces the total number of homotopy paths that must be numerically continued. These improvements may be applied to the basic algorithms of numerical algebraic geometry in the settings of...

20. Cocktail

18/10/2023 18:30

26. Sums of squares approximations in polynomial optimization: performance analysis and degree bounds by Monique Laurent

19/10/2023 09:30

Abstract. Polynomial optimization deals with optimizing a polynomial function over a feasible region defined by polynomial inequalities, thus modeling a broad range of hard nonlinear nonconvex optimization problems. Hierarchies of tractable semidefinite relaxations have been introduced that are based on using sums of squares of polynomials as a ``proxy” for global nonnegativity. These...

27. Coffee break

19/10/2023 10:30

28. Solving by duality by Bernard Mourrain

19/10/2023 11:00

Abstract. Finding the common roots of a set of polynomial equations is a problem that appears in many contexts and applications. Standard approaches for solving this difficult question, such as Grobner bases, border basis, triangular sets, etc. are based on polynomial reductions but their instability against numerical approximations can be critical. In this talk, we will describe a dual...

29. Non-negativity and rational sums of squares over zero-dimensional varieties by Teresa Krick

19/10/2023 14:30

Abstract. In this work in progress with Lorenzo Baldi and Bernard Mourrain, we extend previous results on univariate rational sums of squares, obtained with Bernard and Agnes Szanto, to the case of a non-negative rational polynomial on a basic zero-dimensional semi-algebraic set defined by rational polynomials.

30. Coffee break

19/10/2023 15:30

16. Conormal Spaces and Whitney Stratifications by Martin Helmer

19/10/2023 16:00

Abstract. We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer. We...

31. The geometry of the Vandermonde map at infinity by Cordian Riener

20/10/2023 09:30

Abstract. The Vandermonde map is the polynomial map given by the power-sum polynomials. We study the geometry of the image of the nonnegative orthant under under this map and focus on the limit as the number of variables approaches infinity. We will show, the geometry of this limit is the key to new undecidability results in nonnegativity of symmetric polynomials and deciding validity of trace...

32. Coffee break

20/10/2023 10:30

33. Recent advances in Gröbner basis algorithms and geometric applications by Christian Eder

20/10/2023 11:00

Abstract. The tasks of designing innovative mathematical software and of solving complex research problems using computational methods are strongly mutually dependent. Developing a new generation of algorithms to considerably push the computational boundaries of nonlinear algebra, notably addressing polynomial system solving, is thus envitable. One important task of this process is to no...

34. Topical day: Mechanism Design and Computer Algebra. Galois Theory for Planar Mechanisms by Josef Schicho. 9:30-10h30. Amphitheater Darboux

24/10/2023 09:30

Let G be a graph with n vertices and e edges. The computation of the position of n points in the plane such that for any two vertices in the graph connected by an edge, the distance between the two corresponding points is given, is equivalent to the inverse kinematic problem for a (highly parallel) planar mechanism with revolute joints. If the graph is a Laman graph, then the solution set is...

35. Mechanisms from Motions – Rational and Algebraic by H.-P. Schröcker. 11:00-12:00

24/10/2023 11:00

The factorization of rational motions has been introduced about a decade ago. On a kinematics level it corresponds to the decomposition of a rational motion into elementary motions (rotations, translations, ...) The mathematics behind is the factorization of special polynomials over non-commutative rings into linear factors. This talk gives an overview about the past decade of motion...

36. Inverse kinematics with computer algebra and the Lasserre hierarchy by Didier Henrion. 14:00-15:00

24/10/2023 14:00

The Inverse Kinematics (IK) problem consists of finding robot control parameters to bring it into a desired position under kinematics and collision constraints. We describe a global solution to the optimal IK problem for a general serial manipulator with 7 degrees of freedom (7DOF) with revolute joints. Classical modeling yields a polynomial optimization problem with constraints of degree...

37. Cloth State Representation Using a Derivative of the Gauss Linking Integral by Maria Alberich-Carraminana. 15:00-16:00

24/10/2023 15:00

Robotic manipulation of cloth presents a complex challenge due to the infinite-dimensional shape-state space of textiles. This complexity makes accurate state estimation a daunting task. To address this issue, we introduce the concept of dGLI Cloth Coordinates—a finite, low-dimensional representation of cloth states. This novel approach enables us to effectively distinguish among a wide range...

2. Moment Method in Optimization by D. Henrion. 10:00-12:00. Amphitheatre Darboux, IHP

38. Polynomials Systems and Real Geometry by I. Emiris and D. Plaumann. 10:00-12:00. Amphitheater Darboux, IHP