The first week of September the 2nd ARCADES Doctoral School and the workshop ESR Days – organised by the ESRs, will take place in Barcelona.
The meeting will take place at the Institut de Matematiques of the University of Barcelona.
Registration is now open! Please register here no later than June 15th.
The events will start on Monday early afternoon and will close on Friday early afternoon.
The following invited speakers have confirmed their participation:
David Brander – Technical University of Denmark
Alicia Dickenstein – University of Buenos Aires
Bert Jüttler – Johannes Kepler University Linz
Rosa Maria Miro Roig – University of Barcelona
Hal Schenck – Iowa State University
Martin Sombra – ICREA & University of Barcelona
Carla Manni – University of Rome Tor Vergata
Preliminary schedule
Monday, Sept 3 | Tuesday, Sept 4 | Wednesday, Sept 5 | Thursday, Sept 6 | Friday, Sept 7 | |
9:30 | – | Bert Jüttler | Hal Schenck | Rosa Maria Miro Roig | Fellow’s presentations and discussions |
11:00 | – | Coffee break | Coffee break | Coffee break | Coffee break |
11:30 | – | Bert Jüttler | Hal Schenck | Rosa Maria Miro Roig | Fellow’s presentations and discussions |
13:00 | Lunch | Lunch | Lunch | Lunch | |
14:30 | David Brander | Carla Manni | Alicia Dickenstein | Cultural activity | – |
16:00 | Coffee break | Coffee break | Coffee break | – | |
16:30 | Martin Sombra | Carla Manni | Alicia Dickenstein | Cultural activity | – |
18:00 | University building tour | Supervisory Board meeting | Educational commitee meeting | – | |
Welcome reception (18:30) | Project dinner (20:00) |
All participants are kindly requested to make their own accommodation arrangements.
There are some University Residences in Barcelona where you can get some reasonably priced rooms. You can check in www.resainn.com/en/accommodation/.
But we suggest you to use some hotel or apartment searcher, like www.booking.com, www.airbnb.com or similar, to find other offers and options.
Designing with elastic curves
David Brander, Technical University of Denmark
Planar elastic curves were used in the design of ships and aircraft in pre-CAD years, via physical splines, i.e., elastic rods held in position at a number of interpolation points by so-called ducks. In the digital setting however, it turns out to be difficult to simulate physical splines, mainly because of non-uniqueness of solutions for a given boundary value problem. Hence the standard in CAD is to use polynomial and rational splines. However, planar elastic curves arise naturally in some manufacturing settings: for example hot-blade cutting, a generalization to non-ruled surfaces of the well-known hot-wire cutting. To design or rationalize for this process require the use of elastic curves.
In this talk I discuss some approaches that lead to reliable methods for designing with elastic curves and elastic splines.
Macaulay style formulae for the sparse resultant
Martin Sombra, ICREA & University of Barcelona
The sparse resultant is a classical object from elimination theory, that been widely used in polynomial equation solving and that has strong connections with combinatorics, toric geometry, residue theory, and hypergeometric functions.
In this talk, I will review some of the matrix formulae for this object. The first ones go back to Cayley and Sylvester in the univariate case, and to Macaulay in the dense multivariate case. Formulae for the sparse case were obtained by Canny-Emiris and by D’Andrea, and simplified in a recent work in collaboration with D’Andrea and Jeronimo.
Towards Efficient Matrix Assembly in Isogeometric Analysis
Bert Jüttler, Johannes Kepler University Linz
The framework of Isogeometric Analysis was introduced by T.J.R. Hughes et al. in 2005 in order to enhance the interaction between geometric design and numerical simulation. In particular, it aims to reconcile the representations of geometric objects which are used in software for Computer-Aided Design (CAD) with the mathematical technology of the finite element method (FEM). While this approach opens numerous new possibilities, it also creates additional computational challenges. These include the need to finde new methods for performing matrix assembly. The talk will present several approaches for improving the efficiency of this process in isogeometric analysis. They combine results from approximation theory and numerical tensor calculus.
Total positivity in CAGD
Carla Manni, University of Rome Tor Vergata
Total positivity is a powerful concept permeating different areas of mathematics including approximation theory, probability and statistics to mention a few.
In the context of Computer Aided Geometric Design (CAGD), total positivity is a key property: dealing with a total positive system/basis ensures variation diminishing and shape preserving properties of the considered representation. Moreover, among the all possible normalized totally positive bases (if any) of a given space it is possible to identify those which are “optimal” from the geometric point of view. Bernstein polynomials and B-splines are optimal normalized totally positive bases for the polynomial and spline spaces respectively.
After presenting the basic definition and properties of totally positive matrices, we will introduce totally positive (normalized) bases and discuss their variation diminishing properties. Finally, we will present the concept of geometrically optimal totally positive bases. The general theory will be discussed in detail for polynomial and spline spaces.
Geometric modeling and syzygies
Hal Schenck, Iowa State University
Understanding the implicit equation and singular locus of a parametric object embedded in projective or affine space is a central problem in geometric modeling; such objects are used by companies as diverse as Pixar and Boeing. In particular, this is a real-world problem of importance in manufacturing and image manipulation. There are typically three methods used to find the implicit equation(s) of the image of a map: Gröbner bases, resultants, and syzygies. We will focus on the third method. I’ll start with an overview of syzygies and Rees algebras, then move on to other tools such as Fitting ideals, the determinant of a complex, approximation complexes, and the McRae invariant. We’ll conclude by applying these tools to examples.
Iterated discriminants and singular space curves
Alicia Dickenstein – University of Buenos Aires
In general, two quadric surfaces intersect in a nonsingular quartic space curve, but under special circumstances this intersection curve may degenerate to a finite number of different possible types of singular curves. These degenerate space curves are important since they occur frequently in practice and, unlike the generic case, they admit rational
parameterizations. In the nice paper [3], the authors formulate the condition for a degenerate intersection, which refines the study of the real case and with an algorithmic point of view the classical treatise [1]. Independently, the condition for a degenerate intersection of two surfaces of tensor type (or more generally, of two hypersurfaces described by multilinear equations) is studied in [4].
Folllowing joint work with S. di Rocco and R. Morrision in [4], I will present a general framework of iterated discriminants to characterize the singular intersection of hypersurfaces with a given monomial support, which generalizes both previous situations. I will explain the notion of mixed discriminant and the relation with these iterated discriminants.
[1] T. J. I’A. Bromwich: Quadratic forms and their classification by means of invariant-factors. Cambridge Univ. Press, Cambridge, Jbuch 37, 1906.
[2] A. Dickenstein, S. di Rocco, R. Morrison: Iterated multivariate discriminants and mixed discriminants, Manuscript, 2018.
[3] R.T. Farouki, C.A. Neff, M.A. O’Connor: Automatic parsing of degenerate quadricsurface intersections, ACM Transactions on Graphics 8 (3) (1989) 174-203.
[4] L. Schläfli: Gesammelte mathematische Abhandlungen. Band II, Verlag Birkhäuser, Basel, 1953.
Liaison Theory with a view towards Algebraic Geometry
Rosa Maria Miro Roig, University of Barcelona
Two options are under consideration: a guided city tour and/or Sagrada Familia tour. This will be decided according to the preference of participants.