Dimension of Multivariate Splines: An Algebraic Approach
Online event, May 31, 2021


Piecewise-polynomial functions called splines are fundamental pillars that support modern computer-aided geometric design, numerical analysis, etc. These functions are defined on polyhedral partitions of ℝn. Their restriction to any polyhedron's interior is a polynomial, and these polynomial pieces are constrained to join with some desired smoothness across hyperplanes supporting the intersections of neighboring polyhedra. Computing the dimension of spline spaces is a highly non-trivial task in general for splines in more than one variable and involves an intimate interplay of algebra, topology, and geometry. Initiated by Strang and Schumaker, this is by now a classical topic in approximation theory and has been studied in a wide range of planar settings, e.g., on T-meshes, triangulations and more polygonal meshes. Important contributions has been obtained by following and generalizing the so-called homological approach introduced in this context by Billera in 1988. The one-day online workshop focuses on the most recent developments on this topic and offers the opportunity of interaction between researchers belonging to different research areas.


Invited Speakers



The talks of the workshop will be broadcasted through the platform Microsoft Teams. Join the workshop here.



14:50 - 15:00 Opening
15:00 - 15:30 Martina Lanini, Splines arising in topology, geometry, and representation theory
15:30 - 16:00 Bernard Mourrain, Algebraic tools for geometrically continuous splines
16:00 - 16:30 Deepesh Toshniwal, Dimension of splines of mixed smoothness
16:30 - 17:00 Tatyana Sorokina, Supersmoothness and dimension of multivariate splines
17:00 - 17:30 Coffee break
17:30 - 18:00 Henry Schenck, New bounds for planar splines
18:00 - 18:30 Michael DiPasquale, Homogeneous trivariate splines on vertex stars
18:30 - 19:00 Nelly Villamizar, A lower bound for the dimension of tetrahedral splines in large degree
19:00 - 19:30 Cesare Bracco, Tchebycheffian splines over T-meshes: the homological approach


Organizing Committee


The workshop is part of a series of scientific activities of the MIUR Excellence Department Project MATH@TOV (CUP E83C18000100006).