Algebraic Algorithms

  • Polynomial system solving, sparse/toric elimination theory, structured matrices. Resultants reduce system solving to linear algebra. Newton polytopes provide the bridge between algebra and combinatorial geometry. Optimal sparse resultant matrices for multihomogeneous polynomial systems (article), and MAPLE software MHRES. Macaulay-type formulae for sparse resultants of generalized unmixed systems (article). More Algebraic software.

  • Real solving, Real algebraic numbers. Solvers based on Continued Fractions and Sturm sequences. Software tools in C++ (Mathemagix): module Realroot, interval-Newton method, solvers for polynomial systems (subdivision methods). MAPLE software SLV for real algebraic numbers, real solving of equations and bivariate systems. Benchmarking of black-box univariate real solvers.

Geometric Algorithms

  • Geometric Modeling and Nonlinear Computational geometry. Exact and approximate implicitization of parametric hypersurfaces by interpolation using matrix operations, and sparse elimination for exploiting structure (article). Extensions to space curves and objects defined by point clouds (Wiki page). Voronoi diagrams of circles and ellipses (IMA nugget), arrangements of curved objects (talk) on CGAL. Python projects and CGAL-Python bindings, including visibility tools: webpage.

  • Convex geometry in general dimension. Random walks for approximating polytope volume (software). Convex hull and symbolic perturbation, mixed volume (software). Regular fine mixed subdivisions of Minkowski sums, regular triangulations, secondary polytopes, resultant polytopes (software). Minkowski decomposition (webpage): optimal algorithms with a fixed-size summand, approximation of general (NP-hard) problem. Polytopal computations: normals, faces, ridges, extreme or interior points (module Polytopix in Mathemagix).

  • Approximate geometric algorithms in high dimensions. Clustering for big data: we improve k-means to cluster 100 Mil SIFT images in <1hr (ICCV 2015, image). kd-GeRaf: randomized kd-trees for approximate nearest-neighbors (ANN) in very high dimensions. Dimensionality reduction by a weak version of the JL lemma, and nearly-record complexity for ANN (SoCG 2015).

Applications

  • Molecular conformations in Structural bioinformatics. Enumeration of all possible conformations of (small) molecules/proteins under geometric constrains. Determination of active sites by alpha-shapes, Sampling of rotamers, and clustering to deduce structural determinants. Graph embedding in Euclidean spaces, rigidity theory, distance geometry to compute conformations from NMR data.

  • Robot kinematics. Robust calibration of parallel robots (by applying elimination techniques), forward kinematics of Stewart/Gough platforms, and design of robotic platforms for medical applications such as physiotherapy.

  • Structure of Transmembrane proteins. Geometric modelling of β-barrels and detection of the transmembrane region of a β-barrel transmembrane protein. TbB-Tool is the software.

  • Lakes is a program for the prediction of protein binding sites. It analyzes the solvent and its contacts with proteins and defines clusters of water molecules, which mark potentially exposed interaction and binding sites of the protein. Contact: Dr Thanassis Tartas. Figure: black is the protein, colored are the oxygen atoms of clusters, red being the largest cluster.