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

• Cesare Bracco (University of Florence, Italy)
• Michael DiPasquale (Colorado State University, USA)
• Martina Lanini (University of Rome Tor Vergata, Italy)
• Bernard Mourrain (Inria Sophia Antipolis Méditerranée, France)
• Henry Schenck (Auburn University, USA)
• Tatyana Sorokina (Towson University, USA)
• Deepesh Toshniwal (Delft University of Technology, The Netherlands)
• Nelly Villamizar (Swansea University, UK)

 

Venue

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

 

Program

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

• Carla Manni (University of Rome Tor Vergata, Italy)
• Hendrik Speleers (University of Rome Tor Vergata, Italy)

 

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