Geometry Seminar, 2012

Previous / coming seminars:

May 4, Kai Hormann (Lugano, Switzerland)

Time/place:
13.15-14.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: Polynomial Reproduction for Univariate Subdivision Schemes of any Arity

Abstract: This talk is about the ability of convergent subdivision
schemes to reproduce polynomials in the sense that for initial data,
which is sampled from some polynomial function, the scheme yields the same polynomial in the limit. This property is desirable because the reproduction of polynomials up to some degree d implies that a scheme has approximation order d+1. We first show that any convergent, linear, uniform, and stationary subdivision scheme reproduces linear functions with respect to an appropriately chosen parameterization. We then present a simple algebraic condition for polynomial reproduction of higher order. All results are given for subdivision schemes of any arity m>1 and we use them to derive a unified definition of general m-ary pseudo-splines. Our framework also covers non-symmetric schemes and we give an example where the smoothness of the limit functions can
be increased by giving up symmetry.

March 12, Bruno Simões (Grafitech, Trento, Italy)

Time/place:
13.15-14.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: Gröbner basis computation revised

Abstract: Algebraic geometry is the mathematical field that studies geometric objects by means of algebra. Its origins go back to Descartes together with his introduction to the coordinate geometry. In the twentieth century, algebraic geometry became much broader and in many ways much more abstract due to the emergence of commutative algebra and homological algebra as the foundational language of the subject.

As the abstract theory of algebraic geometry was being developed in the middle of the twentieth century, a parallel development was taking place concerning the algorithmic aspects of the subject. The Gröbner bases theory emerged from that branch with the goal of providing a way to manipulate systems of equations systematically. Recent engineering applications of this theory include computer graphics, computer vision, geometric modeling, geometric theorem proving, optimization, control theory, statistics, communications, biology, robotics, coding theory, and cryptography.

In this talk we will provide an elementary introduction to Gröbner bases, comparing as well some of the best-known algorithms in this area (e.g. F5 and G2V). We will also describe a combination of ideas that can be used to improve the performance of signature-based Gröbner basis algorithms (e.g. using the information about syzygy module and Hilbert series).

February 27, Georg Muntingh (CMA)

Time/place:
13.15-14.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: A Classification of the Generalized Principal Lattices in Space

Abstract: In multivariate polynomial interpolation theory, the properties of polynomial interpolants depend very much on the configuration of the interpolation points in space. An important class is made up by the generalized principal lattices, which form corner stone in the classification of the meshes with simple Lagrange formula
and can be viewed as a generalization of the triangular meshes.

While generalized principal lattices are defined by an abstract combinatorial definition, all generalized principal lattices in the projective plane arise from a real cubic curve in the dual projective plane. As all such curves are of arithmetic genus 1, one can ask the question: Which space curves of arithmetic genus 1 and degree 4 give rise to generalized principal lattices in dual projective space?

In this talk we show how complete intersections of quadric surfaces can be used to define generalized principal lattices in space.

January 30, Gianpaolo Orioli (Università di Roma)

Time/place:
13.15-14.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: A quick journey through algorithms and polytopes for the stable set problem on claw graphs

Abstract: In his seminal paper, Edmonds provided a polynomial-time algorithm for solving the maximum weighted matching problem and a complete linear description of the matching polytope. Later, Padberg
and Rao provided a fast combinatorial algorithm for the separation problem over the latter polytope.

In the talk we will survey a few recents results on the maximum weighted stable set problem for a claw-free graph, that is a non-trivial generalization of the weighted matching problem. Namely, we will illustrate a polynomial-time algorithm for the problem, that is based on some graph decomposition results, and a linear description
of the stable set polytope (in a slightly extended space) that comes together with an efficient separation routine.

January 23, Michael Floater (IFI/CMA)

Time/place:
13.15-14.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: A smoothness criterion for monotonicity-preserving subdivision

Abstract: In this paper we study subdivision schemes that both interpolate and preserve the monotonicity of the data, and we derive a simple `ratio' condition that guarantees the continuous differentiability of the limit function. We then show that the condition holds for three specific non-linear, four-point schemes of this type: Kuijt and van Damme's scheme, based on rational functions,
Sabin and Dodgson's scheme, based on square roots; and a new scheme based on fourth roots. This is joint work with C. Beccari, T. Cashman, and L. Romani.

9 Jan: Heidi Dahl (SINTEF/SAGA)

Time/place:
13.15-14.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Disclaimer: This seminar will contain a fairytale. It will necessarily involve some slides with boring math, but also blowups of exceptional power as well as intriguing space travels.

Title: Rational rolling ball blends between natural quadrics

Abstract: Natural quadrics are the simplest primitive shapes used in Computer Aided Design (CAD): spheres, right circular cones and right circular cylinders. By combining natural quadrics with rolling ball blends between them, we are able to model the majority of mechanical
parts exactly. However, though the natural quadrics are rational surfaces, a rolling ball blend between two natural quadrics needs not be.

In this talk I will give a complete classification of the configurations of natural quadrics that can be blended with rational fixed radius rolling ball blends. Their parametrizations are constructed by applying results from canal surfaces. For variable radius blends, I will present a general algorithm for constructing minimal bi-degree rational parametrizations of patches on canal surfaces. Finally, we consider how to blend corners, where the blends of the adjoining edges intersect.

Organizers:

Georg Muntingh, CMA