Abstract

Abstract

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 sets, approximate synthesis of mechanisms, and path planning for output mode switching.

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 show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. This is joint work with Vidit Nanda (Oxford).

Abstract

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 hierarchies give bounds on the global minimum of the original problem with asymptotic convergence (under a minor compactness assumption). In this lecture we discuss recent results on the performance analysis of these hierarchies and related effective degree bounds for dedicated sums of squares representations of positive polynomials on some classes of compact semi-algebraic sets (including the hypercube, the sphere or the ball).

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Abstract

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 completeness for such problems.

Abstract

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, including all the computed triangular sets, appearing in several typical top-down algorithms for triangular decomposition have associated graphs which are subgraphs of the chordal graph of the input polynomial set. These theoretical results can be interpreted as “triangular decomposition in top-down style preserves chordality” and are further used to design sparse triangular decomposition for polynomial sets which are sparse with respect to their variables. Sparse triangular decomposition is shown to be more efficient than ordinary one experimentally, and its application on computing equilibria of biological dynamic systems will also be reported.
This talk is based on the joint work with Yang Bai, Jiahua Lai, and Wenwen Ju.

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 approach which focuses on linear functionals vanishing at the roots. We will review the properties of Truncated Normal Forms, the connexion with classical computer algebra approaches and resultants. We will also detail the dual approach in the context of optimisation problems and for analysing isolated singularities. Examples from geometric modeling, robotics and tensor decomposition will illustrate the numerical behavior of these dual methods.

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Abstract

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 identifying alternative toric homogenizations that are suitable for practical computations. Joint work with Matías Bender.