# Algebraic Geometry and Geometric Modeling (13w5062)

Arriving in Banff, Alberta Sunday, January 27 and departing Friday February 1, 2013

## Organizers

Ron Goldman (Rice University)

Jorg Peters (University of Florida)

Frank Sottile (Texas A & M University)

## Objectives

The interactions between geometric modeling and algebraic geometry are driven by natural developments in each field. In algebraic geometry it is the study of computational methods, particularly those pertaining to real varieties and applications, while in geometric modeling, it is the drive to master and use more tools in the study of their basic objects. Likewise there have been two dominant trends in the interactions, one algebraic and one geometric. While furthering these existing interactions, we also hope to begin a third focused on finding real solutions to polynomial equations and inequations.

The original interactions between the subjects were algebraic. In modeling, the basic objects are parametric curves and surfaces. Determining intersections of these objects requires the computation of an implicit representation of the parametric object, which is a very difficult algebraic problem. In the 1980's Sederberg realized that the classical Dixon resultant solves this problem for general parametric surfaces. For the less general surfaces that often arise, significantly more subtle methods are needed, and the method of moving lines, (and planes) was developed, which turns out to be an innovative way to compute syzygies in computational algebraic geometry. There is a very well-developed connection and flow of ideas between the subjects related to resultants and computation of syzygies, whose impact reaches beyond the original areas of interaction. Some of the important work here has been done by Sederberg, Cox, Goldman, and others.

A second area of interaction has been geometry. Krasauskas realized that the basic objects in geometric modeling, Bezier curves and surfaces, in fact come naturally from an important class of algebraic varieties called toric varieties. The impact of this for geometric modeling has been mild---it has led to a deeper understanding of these basic objects in modeling, and a study of the properties of Krasuaskas' toric patches. These patches come from the positive part of a toric variety, which is an object common to several additional fields such as approximation theory (via general barycentric coordinates and splines, both univariate and multivariate), algebraic statistics (discrete exponential families), and toric chemical dynamical systems, so toric parches have a potential imipact on these other subjects. There are also strong links to nonlinear computational geometry. Some who have worked on these topics include Garcia-Puente, Ranestad, and Sullivant.

There is a third underveloped area of interaction which we feel needs nurturing. Geometric modeling often needs to find real roots of equations, or to solve inequations. It also needs to determine the shape of patches from constraints or control points. One approach to these problems involve their formulation as real number solutions to systems of polynomial equations. Solving and studying real solutiosn to systems of algebraic equations is a significant topic in algebraic geometry, as is understanding algebraic systems of inequalities. One goal of our workshop will be to bring together people from both sides with interest in equations and inequations, which will be the first time there has been such an interaction.

All three threads converge in some traditional topics (which are often studied separately) such as singularities (detection and analysis), intersection problems, and approximate methods.

The traditional, algebraic, interactions of the subjects will be a major theme of the workshop. In addition to people who have worked directly on these topics, we hope to invite people such as Greg Smith and others involved with software for computational algebraic geometry, such as Macaulay 2 or Singular, as well as some applied-friendly pure mathematicians.

An important theme of the workshop, and that which will involve people from the most disparate areas are the geometric interactions via toric patches and their connections to other areas, global geometric properties of patches (e.g. convexity, smoothness), and internal properties such as precision and barycentric coordinates. For example, barycentric coordinates are fundamental constructs in geometric modeling, computer graphics, and finite element analysis. They have applications to surface parameterization in graphics and even surface animation for entertainment applications (Pixar now uses barycentric coordinates to animate their surfaces). We even build higher order surfaces with barycentric coordinates. Besides those discussed above, there is relevant work of Floater, Warren, and Schaefer.

Approximation theory is related to modeling via barycentric coordinates and splines, and leads to similar interactions with algebraic geometry. For example, questions such as accuracy, precision and rate of convergence are considered in geometric modeling as well as in approximation theory. The matter of (linear) precision in toric algebraic geometry turned out to be a related tool---more interaction is needed in this area with potential benefits to the approximation theory community. Some people who are interested in deeper interactions include de Boor, Sorokina, and Schekhtman.

This workshop, while international, would have a positive effect on research in North America. While there are research groups working in these topics in this Hemisphere, and some woprkshops at the MSRI and Towson University (the latter focusing on Approximation Theory and Algebraic Geometry), as well as an IMA conference in 2007 on nonlinear computational geometry, there is not the depth of interaction seen, for example in Europe, which has enjoyed regular conferences on these and related topics, with a history of large interactive projects such as the current SAGA (ShApes, Geometry, Algebra).

Besides these specific topical scientific objectives, the main objective is to continue to engage in the hard work of bringing these different communities together. At the original meeting in Vilnius, as well three short workshops in the US: one at MSRI in 2004, and two at Towson University 2008 and 2009 and a broader one on nonlinear computational geometry at the IMA in 2007, the participants felt these were valuable, but that the gap between the fields had not yet been bridged. A longer period of sustained interaction at BIRS would help to further this trend.

The feelings of many who have been contacted about this possible workshop are summed up by Scott Schaefer, a Computer Scientist at Texas A&M: "...Algebraic Geometry is a related field to my own, but I know little about it..... This type of meeting could be quite beneficial since I would likely never see any Algebraic Geometry talks at the conferences I attend or even meet people from that community. The workshop could certainly serve the purpose of broadening my own understanding of that field and helping me find new tools and connections to my own research." The same statement, mutatis mutandis, can be made by others coming at these interactions from a different field.

The original interactions between the subjects were algebraic. In modeling, the basic objects are parametric curves and surfaces. Determining intersections of these objects requires the computation of an implicit representation of the parametric object, which is a very difficult algebraic problem. In the 1980's Sederberg realized that the classical Dixon resultant solves this problem for general parametric surfaces. For the less general surfaces that often arise, significantly more subtle methods are needed, and the method of moving lines, (and planes) was developed, which turns out to be an innovative way to compute syzygies in computational algebraic geometry. There is a very well-developed connection and flow of ideas between the subjects related to resultants and computation of syzygies, whose impact reaches beyond the original areas of interaction. Some of the important work here has been done by Sederberg, Cox, Goldman, and others.

A second area of interaction has been geometry. Krasauskas realized that the basic objects in geometric modeling, Bezier curves and surfaces, in fact come naturally from an important class of algebraic varieties called toric varieties. The impact of this for geometric modeling has been mild---it has led to a deeper understanding of these basic objects in modeling, and a study of the properties of Krasuaskas' toric patches. These patches come from the positive part of a toric variety, which is an object common to several additional fields such as approximation theory (via general barycentric coordinates and splines, both univariate and multivariate), algebraic statistics (discrete exponential families), and toric chemical dynamical systems, so toric parches have a potential imipact on these other subjects. There are also strong links to nonlinear computational geometry. Some who have worked on these topics include Garcia-Puente, Ranestad, and Sullivant.

There is a third underveloped area of interaction which we feel needs nurturing. Geometric modeling often needs to find real roots of equations, or to solve inequations. It also needs to determine the shape of patches from constraints or control points. One approach to these problems involve their formulation as real number solutions to systems of polynomial equations. Solving and studying real solutiosn to systems of algebraic equations is a significant topic in algebraic geometry, as is understanding algebraic systems of inequalities. One goal of our workshop will be to bring together people from both sides with interest in equations and inequations, which will be the first time there has been such an interaction.

All three threads converge in some traditional topics (which are often studied separately) such as singularities (detection and analysis), intersection problems, and approximate methods.

The traditional, algebraic, interactions of the subjects will be a major theme of the workshop. In addition to people who have worked directly on these topics, we hope to invite people such as Greg Smith and others involved with software for computational algebraic geometry, such as Macaulay 2 or Singular, as well as some applied-friendly pure mathematicians.

An important theme of the workshop, and that which will involve people from the most disparate areas are the geometric interactions via toric patches and their connections to other areas, global geometric properties of patches (e.g. convexity, smoothness), and internal properties such as precision and barycentric coordinates. For example, barycentric coordinates are fundamental constructs in geometric modeling, computer graphics, and finite element analysis. They have applications to surface parameterization in graphics and even surface animation for entertainment applications (Pixar now uses barycentric coordinates to animate their surfaces). We even build higher order surfaces with barycentric coordinates. Besides those discussed above, there is relevant work of Floater, Warren, and Schaefer.

Approximation theory is related to modeling via barycentric coordinates and splines, and leads to similar interactions with algebraic geometry. For example, questions such as accuracy, precision and rate of convergence are considered in geometric modeling as well as in approximation theory. The matter of (linear) precision in toric algebraic geometry turned out to be a related tool---more interaction is needed in this area with potential benefits to the approximation theory community. Some people who are interested in deeper interactions include de Boor, Sorokina, and Schekhtman.

This workshop, while international, would have a positive effect on research in North America. While there are research groups working in these topics in this Hemisphere, and some woprkshops at the MSRI and Towson University (the latter focusing on Approximation Theory and Algebraic Geometry), as well as an IMA conference in 2007 on nonlinear computational geometry, there is not the depth of interaction seen, for example in Europe, which has enjoyed regular conferences on these and related topics, with a history of large interactive projects such as the current SAGA (ShApes, Geometry, Algebra).

Besides these specific topical scientific objectives, the main objective is to continue to engage in the hard work of bringing these different communities together. At the original meeting in Vilnius, as well three short workshops in the US: one at MSRI in 2004, and two at Towson University 2008 and 2009 and a broader one on nonlinear computational geometry at the IMA in 2007, the participants felt these were valuable, but that the gap between the fields had not yet been bridged. A longer period of sustained interaction at BIRS would help to further this trend.

The feelings of many who have been contacted about this possible workshop are summed up by Scott Schaefer, a Computer Scientist at Texas A&M: "...Algebraic Geometry is a related field to my own, but I know little about it..... This type of meeting could be quite beneficial since I would likely never see any Algebraic Geometry talks at the conferences I attend or even meet people from that community. The workshop could certainly serve the purpose of broadening my own understanding of that field and helping me find new tools and connections to my own research." The same statement, mutatis mutandis, can be made by others coming at these interactions from a different field.