Applications of Algebraic Geometry

The 2006–2007 thematic program on Applications of Algebraic Geometry is well underway, with the fall quarter concentration on computational algebraic geometry drawing to a close. The fall quarter kicked-off with a superb tutorial on Algebraic geometric methods in engineering, and then continued with two very successful workshops on Algorithms in Algebraic Geometry, September 18-22, and Software for Algebraic Geometry,October 23-27. Both of the these programs were well attended and well received by the participants. Here is what the organizers for the Algorithms in Algebraic Geometry had to say about the workshop.

We had a very successful Workshop on Algorithms in Algebraic Geometry, the first of the Workshops in the Special Year 2006/2007 on Applications of Algebraic Geometry at IMA. There were 18 speakers, 13 interesting poster presentations, and more than 100 participants. A variety of theoretical and computational aspects were addressed: factorization of polynomials; counting solutions over finite fields and over the reals; numerical algebraic geometry; polytope methods in the solution of polynomial systems; computations on algebraic curves; moduli spaces of projective points; invariant rings; implicitization problems; applications to Computer Aided Design and reverse engineering of biologial networks; and complexity estimates. There will be an IMA Volume to reflect the activity at this Workshop, published by Springer.

—Alicia Dickenstein, Frank Schreyer and Andrew Sommese

Charles Wampler is a senior staff research scientist at General Motors who works in robotics and algebraic geometry. He spent the months of September and October resident at the IMA as a key participant in the program. In his talk on Mechanisms and Robot Kinematics: Numerical Algebraic Geometry, Charles described the basic constructs and algorithms of numerical algebraic geometry and some recent avenues of research regarding real solutions inside a complex curve which led to the image below, the poster image for the IMA thematic year program.

I think this workshop and the preceding tutorial were very important for me, as they would be for anyone else who is interested in computer algebra. They made me aware of the numerical methods in algebraic geometry that provide a very different way of looking at computations with polynomial ideals and algebraic varieties, which before I thought of only symbolically. I am glad to have met with the researchers who have actually developed these methods. I would like to understand them in detail and try to apply in differential algebra, where symbolic methods are known, but are very slow, and sometimes not applicable even theoretically, while the numerical methods would be completely new. It was a very nice surprise to see at this workshop more people interested in differential algebra than I usually meet at other conferences. Perhaps this is not an accident but yet another indication of the connection between the two areas.

—Oleg Golubitsky

One of the highlights of this workshop was the unveiling of Bertini, a software package that solves large systems of multivariate polynomial equations using continuation techniques. All four of the developers spent the fall in residence at the IMA, making the release possible.

This is what the organizers of this workshop had to say:

There were 89 registered participants at the workshop. At four talks a day, 20 regular 50 minutes talks were scheduled. On Monday evening, 10 posters were presented. On Wednesday and Thursday evening we had respectively 5 and 6 software demonstrations. The list of featured software packages includes Macaulay 2, SAGE, HomLab, Bertini, APAtools, PHClab, PHCmaple, PHCpack, KNOPPIX/Math, D-modules for Macaulay 2, Singular, Risa/Asir, CRACK, diffalg, RIFsimp, Gambit, FGb/RS, CoCoALib, 4ti2, PHoMpara, SYNAPS, DEMiCs, Magma, Kronecker, SOSTOOLS, Gfan, Maple 11. The IMA systems group had installed many of these programs on the computers at the IMA. At the poster session, the participants were given the opportunity to install the featured software systems on their laptop. A demonstration cluster computer of Rocketcalc was running during the poster session and accessible to all participants during the workshop. The evening before the workshop dinner on Tuesday started with a problem session. Prior to this session we made a list of problem descriptions available on the web site. The workshop ended on Friday evening with some additional problems, discussion on the posted problems, and a presentation of Jiawang Nie about the application to semidefinite programming to solve systems of algebraic equations which arise from differential equations. Instead of the "second chances" (usual for IMA workshops), the participants were given the opportunity to test the software systems on Wednesday and Thursday evening. The evening session started with a one hour plenary session, where each software system on demo in the evening was briefly explained. Following this plenary session, the participants moved to the 4th floor of Lind Hall, to experience the software systems on the computers in the open poster area, or in parallel, in the classroom 409. The IMA systems group worked hard in the weeks leading up to the workshop to install the software systems. Their effort benefited not only the workshop participants, but will also be beneficial to all subsequent participants to the thematic year, as they will find their workstations equipped with the latest software tools in algebraic geometry.

—Michael Stillman, Nobuki Takayama, Jan Verschelde

I thought this workshop was very well organized and interesting. Before this conference, I had not really tried any other software programs for algebraic geometry other than MAPLE and Macaulay 2. I found the talks very beneficial to see what the other software packages had to offer. I am very intersted in trying to use several of these programs: namely, SAGE, Singular, CoCoA, and the KNOPPIX DVD.

—Rachelle R. Bouchat

At the end of the week I could get a very precise idea of the existing software for doing computer algebra, both free and commercial, and of software dedicated to specific applications of computer algebra. Several recent software were new to me and I found their presentations very useful. This workshop was also a very good place to share ideas with developers, for better understanding the needs of the users in terms of interface and performances, and also for better understanding how all the existing packages can be linked together.

—Anonymous