08.01.2020

Classically there is a clear distinction between theoretical and applied mathematics in the classification of different fields in the mathematical sciences. Bernd Sturmfels at the Max Planck Institute for Mathematics in the Sciences, along with Paul Breiding and Sascha Timme at the Technical University of Berlin present a novel approach that illustrates how this line can be blurred. They developed a software, which is not only able to compute numerical solutions for systems of polynomial equations, but can also be employed to answer classical questions of theoretical mathematics. Their results made the cover story of the January issue of the "Notices of the American Mathematical Society".

The article "3264 Conics in a Second" exemplifies how mathematical software can serve as a bridge between theoretical problems and applied methods. Besides creating the extensive numerical software package "HomotopyContinuation.jl", the authors also established an easily accessible website, which vividly showcases the various applications of the software.

One of these applications, in turn, constitutes the solution to a classical geometric problem, which was made even more accessible with a webapp.

Steiner's conic problem

The published paper revolves around a classical geometric problem from the perspective of modern numerical algorithms. In 1848 the mathematician Jakob Steiner posed the question of finding the number of conics tangent to five given conics.

A conic is a planar curve given by the intersection of the surface of a cone with a plane, yet it also constitutes the zero set of a quadratic equation in two variables. Even though Steiner's question initially seems quite academically rigorous in nature it is pertinent to modern applications. The conic problem is regarded as the origin of modern intersection theory.

This theory then forms the foundation for modern algorithms to compute the roots of polynomial systems, which is a fundamental problem in many applied fields: robotics, material sciences, machine learning, biology or dynamical systems theory are just a few exemplary fields, where polynomial equations need to be solved.

Software package HomotopyContinuation.jl

Exactly these numerical computations form the basis of the authors specialized software package HomotopyContinuation.jl, which was developed for Julia, a dynamic programming language focused on high-performance numerical analysis.

The accompanying website not only offers an all-encompassing software manual, but also contextualizes the software through an assortment of applications ranging from computer vision, robotics, chemistry, mathematics of data and algebraic geometry.

A variety of guides introduce the user to the software features and explain how they can be applied to numerous problems. Such as Steiner's conic problem, which can be formulated as a problem of finding the roots of a polynomial system. The customized software package is able to compute these solutions within a mere second.

The scientists at the Max Planck Institute and the Technical University of Berlin created a web interface accompanying their publication which allows the reader to easily compute the 3264 solutions for their chosen conics (see juliahomotopycontinuation.org/diy).

This innovative form of science communication is a novelty for publications in the Notices. The authors were able to use their software to compute exact equations for arrangements with real solutions to Steiner's conic problem.

Thus, they strikingly demonstrated how theoretical results involve numerical procedures, as well as the possible application of numerical methods in proofs of specific theoretical results.

Dr. Paul Breiding

Technical University of Berlin

Institute of Mathematics

Mail: p.breiding@tu-berlin.de

http://www.math.tu-berlin.de/~breiding

"3264 Conics in a Second" in the Notices of the American Mathematical Society

http://www.ams.org/journals/notices/202001/rnoti-p30.pdf

DOI: https://doi.org/10.1090/noti2010

http://www.juliahomotopycontinuation.org Information regarding the software HomotopyContinuation.jl

Jana Gregor | Max-Planck-Institut für Mathematik in den Naturwissenschaften (MPIMIS)

**Further reports about:**
> Max Planck Institute
> Naturwissenschaften
> algebraic
> conic
> geometric
> machine learning
> material sciences
> planar curve
> programming language
> web interface

Beyond 5G lab: Communication technology of the future

13.01.2020 | Friedrich-Alexander-Universität Erlangen-Nürnberg

Machine learning shapes microwaves for a computer's eyes

10.01.2020 | Duke University

Researchers from the University Hospital Zurich, ETH Zurich, Wyss Zurich and the University of Zurich have developed a machine that repairs injured human livers and keep them alive outside the body for one week. This breakthrough may increase the number of available organs for transplantation saving many lives of patients with severe liver diseases or cancer.

Until now, livers could be stored safely outside the body for only a few hours. With the novel perfusion technology, livers - and even injured livers - can now...

A balloon-borne scientific instrument designed to study the origin of cosmic rays is taking its second turn high above the continent of Antarctica three and a half weeks after its launch.

SuperTIGER (Super Trans-Iron Galactic Element Recorder) is designed to measure the rare, heavy elements in cosmic rays that hold clues about their origins...

One last time on Earth it has been turned on in France in December 2019. The next time the MOMA laser developed by the Laser Zentrum Hannover e.V. (LZH) is going into operation will be on Mars. The ExoMars rover into which the laser is integrated has now successfully passed the thermal vacuum tests at Airbus in Toulouse, France.

For 18 days the ExoMars rover Rosalind Franklin was subjected to thermal vacuum tests at Airbus. There, it had to withstand strong changes in temperature and...

The Atacama Desert in Chile is the oldest and most arid desert on earth. Organisms living in this area have adapted to the extreme conditions over thousands of years. A research team led by Dr Patrick Jung has now discovered and investigated a previously unknown biocoenosis of lichens, fungi, cyanobacteria and algae. It colonises tiny stones, so-called grit and its need for water is satisfied by fog and dew. These organisms also decompose the rock on and in which they live. The scientists believe that this is how they have shaped the landscape of the Atacama Desert. Their study was published in the renowned scientific journal "Gebiology".

Many desert areas have large black spots in the sand. These spots are mineral deposits, so-called desert varnish. In the Atacama Desert, which can be compared...

For the first time, physicists from the University of Würzburg have successfully converted electrical signals into photons and radiated them in specific directions using a low-footprint optical antenna that is only 800 nanometres in size.

Directional antennas convert electrical signals to radio waves and emit them in a particular direction, allowing increased performance and reduced...

Anzeige

Anzeige

Event News

„Advanced Battery Power“- Conference, Contributions are welcome!

07.01.2020 | Event News

International Coral Reef Symposium 2020 Holds Photo Competition

19.12.2019 | Event News

03.12.2019 | Event News

Latest News

Scientists in Mainz develop a more sustainable photochemistry

14.01.2020 | Life Sciences

Laserphysics: At the pulse of a light wave

13.01.2020 | Physics and Astronomy

New function for potential tumor suppressor in brain development

13.01.2020 | Life Sciences

VideoLinks

Science & Research

Science & Research

NASA | A Year in the Life of Earth's CO2

NASA Computer Model Provides a New Portrait of Carbon Dioxide

Black Holes Come to the Big Screen

The new movie "Interstellar" explores a longstanding fascination, but UA astrophysicists are using cutting-edge technology to go one better.

NASA's Swift Mission Observes Mega Flares from a Mini Star

NASA's Swift satellite detected the strongest, hottest, and longest-lasting sequence of stellar flares ever seen from a nearby red dwarf star.

NASA | Global Hawks Soar into Storms

NASA's airborne Hurricane and Severe Storm Sentinel or HS3 mission, will revisit the Atlantic Ocean for the third year in a row.

Baffin Island - Disappearing ice caps

Giff Miller, geologist and paleoclima-tologist, is walking the margins of melting glaciers on Baffin Island, Nunavut, Canada.

The Infrasound Network and how it works

The CTBTO uses infrasound stations to monitor the Earth mainly for atmospheric explosions.