Multivariate Splines and Algebraic Geometry

Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-Bézier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems Mathematics Subject Classification (2010): 41A15, 13D40, 52C99. Introduction by the Organisers The workshop Multivariate Splines and Algebraic Geometry, was attended by 26 researchers interested in multivariate splines, polynomial approximation and algebraic geometry. A key problem in both pure and applied mathematics is to construct finite dimensional spaces of functions that are capable of approximating complicated or unknown functions well. Such spaces are especially important for scientific computing, where they are used in computer-aided geometric design, data fitting, and the solution of partial differential equations by the finite-element method. Historically, polynomials have played the central role, but more recently 1140 Oberwolfach Report 21/2015 it has been recognized that spaces of piecewise polynomials are much more efficient and effective. A C-differentiable piecewise polynomial function on a ddimensional simplicial complex ∆ ⊆ R is called a spline. Let S k(∆) denote the vector space of C splines on a fixed ∆, where each individual polynomial has degree at most k. But before we can use spline spaces, we need to solve several basic problems such as finding their dimension, constructing local bases, and determining their approximation power. Despite an extensive literature on the subject, there remain open questions in all of these areas. Much of what is currently known was developed by approximation theorists, using methods of classical analysis, in particular the so-called Bernstein-Bézier techniques. However, due to their many interesting structural properties, splines have also become of keen interest to researchers in commutative and homological algebra, geometry, combinatorics, and topology. Unfortunately, these various communities had not collaborated much. The main purpose of the workshop was to intensify the interaction between the different groups. We believe that the workshop brought together the two communities and fostered fruitful collaborations between individual researchers. We expect that such collaborations and the combined use of tools from the various mathematical fields will lead to essential breakthroughs on several of the above problems. The workshop began with a pair of introductory lecture series: Algebraic geometry for approximators and Approximation theory for geometers. These lectures established a firm grounding in common language and tools. We also held two open problems sessions that took place in the evenings. They highlighted the key conjectures as well. Several exciting new areas were discussed, such as T-Splines and the study of splines on polyhedral (rather than simplicial) complexes. We believe that the workshop generated strong ties between the two communities, and also emphasized to the younger participants the need for interdisciplinary techniques. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Multivariate Splines and Algebraic Geometry 1141 Workshop: Multivariate Splines and Algebraic Geometry