Contribution talks are presentations scheduled for a period other than February 26-28 during our Summer School. The speakers will discuss relevant topics related to their research, with the goal of enhancing the knowledge of our students and professors affiliated with PMA.

Patricio Almirón Cuadros - University of Granada

Date: February 06.

Time: 4:00 pm.

Venue: Auditorium of the Department of Mathematics.

Refreshments and snacks at 3:45 pm.

Title: On the miniversal deformation of complete intersection monomial curves.

Abstract: A monomial curve singularity is essentially the geometric representation of the semigroup algebra associated to a numerical semigroup. Complete intersection monomial curves (CIMC) are one of the richest examples when referring to the interplay between geometry of curve singularities and the combinatorics of the semigroup of values defined by the set of the possible intersection multiplicities with the curve. In 1976, Delorme provided an extremely useful combinatorial characterization of numerical semigroups whose semigroup algebra is a complete intersection (and hence its monomial curve). This characterization has been extensively used in the literature of numerical semigroup theory and semigroup algebras but surprisingly not used in the geometric context. The main aim of this talk is to show how Delorme's characterization can be used to study the miniversal deformation of a CIMC. The main part of the talk will focus precisely in this connection. Concretely, we will show that we can provide a surprising general decomposition result of a basis of the miniversal deformation of any CIMC. As a consequence, we can explicitly calculate this basis for some notable families of CIMC. An important topic related to the study of the base space of the miniversal deformation of a monomial curve is its connection with the moduli space of projective curves with a given Weierstrass semigroup. In 1974, Pinkham showed that the dimension of the negatively graded part of the miniversal deformation is related to the dimension of such a moduli space. If time permits, we will show how our explicit computation of the basis of the miniversal deformation yields some estimates for the dimension of the moduli space of the family. The talk  is based in a joint work with J.J. Moyano Fernández.

Leo Dorst  - University of Amsterdam 

Date: March 19.

Time: 10:00 am.

Online talk: https://meet.google.com/ptk-syfs-ifn 

Title: Least Squares Fitting of Spatial Circles 

Abstract: When you need to fit a sphere or circle to  3D data points, the non-linearity of the problem seemingly precludes the use of linear algebra.  We show how the problem can be reformulated into a recognizable form by means of CGA. We derive the complete solution, using geometric differentiation. Our reward will be a 5D orthogonal basis for all spheres in 3D space, of which the first is the best fitting sphere, the intersection of the first two the best fitting circle, and the intersection of the first three the best fitting point pair.