ICMS 2016 Session: Algebraic Geometry in Applications

ICMS 2016: Home, Sessions


Organizers


Aim and Scope

The aim of the section is to show how algebraic geometry can be applied outside algebraic geometry in many different areas, using the basic tools such as factorization, Gröbner bases techniques, symbolic-numeric techniques and others. We want to present applications in 

The section is open for all kinds of applications. It includes also software from non-commutative algebraic geometry. 


Publications

Submission Guidelines

Talks/Abstracts

           Calibration of accelerometers and the geometry of quadrics

Calibration of accelerometers in laboratories are expansive when accuracy is needed. In contrast, on field methods are usually simple and affordable, at the price of precision.  Mathematical methods, ie methods using only mathematical algorithms and well documented universal constants, fill the gap between the two approaches : Since a large number of decimals are computable on a small personnal computer, mathematical methods are both very precise and usable on field. We study a method of calibration of accelerometers usable on field. We prove that the calibration of an accelerometer with two axis is possible with 5 random measurements and that the calibration of an accelerometer with three axis is possible with 9 random measurments exactly when the sphere is the unique quadric containing the nine directions of measurements. In particular, checking the consistency of the random measurements is immediate with this criterion, without measurement tools. Once the geometry is understood, the software realisation consists of the determinations of quadrics and computations of base changes between quadrics.



      Jonathan Hauenstein (University of Notre Dame, hauenstein@nd.edu)

              Decomposing solution sets of polynomial systems using derivatives

A core computation in numerical algebraic geometry is the decomposition of the solution set of a system of polynomial equations into irreducible components, called the numerical irreducible decomposition.  One approach to validate a decomposition is what has come to be known as the `trace test'.  This test for irreducibility of a set of witness points, described by A. Sommese, J. Verschelde, and C. Wampler in 2002, relies upon path tracking and hence could be called the `tracking trace test'.  We will present a new approach which replaces path tracking with a purely local computation involving derivatives, the `local trace test'.  We will conclude by demonstrating this new local approach with examples related to kinematics and tensor decomposition. Joint work with Daniel Brake and Alan Liddell.


      Thomas Kahle  (OvGU Magdeburg, thomas.kahle@ovgu.de)

              Semi-algebraic geometry of Poisson regression.


Designing optimal experiments for non-linear regression is tricky because the result depends on the unknown parameters.  Based on

inequalities discovered by Kiefer and Wolfowitz in the 1960's in matrix analysis, we approach this chicken or egg problem with tools from real algebraic geometry.  For the Rasch Poisson counts model (a statistical model used in psychometry) we determine semi-algebraic parameter regions where a fixed design is optimal. We pose some challenges for the computational treatment of polynomial inequality systems.  Based on joint work with Kai-Friederike Oelbermann and Rainer Schwabe.


 

    Viktor Levandovskyy (RWTH  Aachen, viktor.levandovskyy@math.rwth-aachen.de)


             A commutative approach to the Bernstein data of a hypersurface"

Bernstein-Sato polynomial of a hypersurface F is an important object in several fields, such as D-module theory and singularity theory. For the case when F has non-isolated singularitiesall known general methods for the computation of the Bernstein-Sato polynomial usenoncommutative Groebner bases. But is it possible to perform the computations ina purely commutative way? We report on our investigations in both global and local algebraic situations. In particular, we show that in this way one obtains all Bernstein data, i.e. in addition to the Bernstein-Sato polynomial, also a corresponding Bernstein operator (important in theory and being a certificate for the algorithm) and the annihilator of the special function F^s in the ring of differential operators. Our implementation is made in computer algebra system Singular. This is a joint work with Daniel Andres.




       Fatemeh Mohammadi (TU Berlin, mohammad@math.tu-berlin.de)


            Combinatorial and geometric view of the system reliability theory.

Associated to every coherent system there is a canonical ideal whose Hilbert series encodes the reliability of the system. We study various ideals arising in the theory of system reliability. Using ideas from the theory of orientations, and matroids on graphs we associate a polyhedral complex to our system so that the non-cancelling terms in the reliability formula can be read from the labeled faces of this complex. Algebraically, this polyhedron resolves the minimal free resolution of these ideals. In each case, we give an explicit combinatorial description of non-cancelling terms in terms of acyclic orientations of graph and the number of regions in the graphic hyperplane arrangement. This resolves open questions posed by Giglio-Wynn and develops new connections between the theory of oriented matroid, the theory of divisors on graphs, and the theory of system reliability. Moreover, we show that the multiple failure and signature analysis of the system are encoded as algebraic invariants of certain deformations of the aforementioned ideal. In particular, we introduce a subdivision of the graphical hyperplane arrangement whose bounded regions are corresponding to multiple failures in the system.



                     3D printing dimensional calibration shape: Clebsch Cubic

3D printing and other layer manufacturing processes are challenged by dimensional accuracy. Various techniques are used to validate and calibrate dimensional accuracy through the complete building envelope. The validation process involves the growing and measuring of a shape with known parameters. The measured result is compared with the intended digital model. Processes with the risk of deformation after time or post processing may find this technique beneficial. We propose to use objects from algebraic geometry as test shapes. A cubic surface is given as the zero set of a degree $3$polynomial in $3$ variables. A class of cubics in real $3$D space contains exactly $27$ real lines. These lines can be used for dimensional calibration. Due to the thin shape geometry the material required to produce an algebraic surface is minimal. We provide a library for the computer algebra system Singular which, from $6$ given points in the plane, constructs a cubic and the lines on it. 


        Tomas Pajdla (Czech Technical University in Prague, pajdla@cvut.cz)


                 Computational Algebraic Geometry in 3D Computer Vision


We will review some recent developments in using computational algebraic geometry in 3D computer vision. In particular, we will explain how to construct fast and reliable solvers of systems of algebraic equations for engineering problems in computer vision by combining computational algebraic geometry with simulations. We will present some classical problems, e.g. computation of camera perspective models from image measurements, as well as some new and more difficult problems for rolling shutter cameras. We will point to issues that seem to be difficult for us and could motivate further developments in computational algebraic methods for solving engineering problems.





Talks
===  12.7. ==================================================================

10:30

         Tomas Pajdla (Czech Technical University in Prague, pajdla@cvut.cz)

         Computational Algebraic Geometry in 3D Computer Vision

11:05 

       Laurent Evain (Université Angers, laurent.evain@univ-angers.fr)

        Calibration of accelerometers and the geometry of quadrics

11:40

Jonathan Hauenstein (University of Notre Dame, hauenstein@nd.edu)

        Decomposing solution sets of polynomial systems using derivatives


===  13.7.  ==================================================================


16:20

         Fatemeh Mohammadi (TU Berlin, mohammad@math.tu-berlin.de)

         Combinatorial and geometric view of the system reliability theory.

16:55

         Viktor Levandovskyy (RWTH  Aachen, viktor.levandovskyy@math.rwth-aachen.de)

         A commutative approach to the Bernstein data of a hypersurface

17:30

         Thomas Kahle  (OvGU Magdeburg, thomas.kahle@ovgu.de)

         Semi-algebraic geometry of Poisson regression.