Jornada de Aritmética y Geometría en Valparaíso
Rational points on curves over finite fields
An application of smoothed zeta functions
In 1979, the French mathematician Pierrette Cassou-Noguès introduced, based on ideas of Shintani, an important technique that allows to construct p-adic L functions over totally real number fields by interpolating special values of Hecke L functions. This technique, currently known as Cassou-Noguès trick, has been extensively used in the p-adic world since then, and it can be thought as a smoothing process of zeta functions in the sense that it improves their analytic properties. In this talk we will introduce this trick, and we will see how it can be used in working with partial zeta functions of number fields as well. More precisely, we will discuss an application of it to the computation of derivatives of partial zeta functions at s = 0.
Algebraic spline geometry
a spline is a piecewise polynomial function defined on a polyhedral complex embedded in a real space. Besides being one of the most powerful tools for approximating the solution of partial differential equations, splines are also fundamental in geometric modeling, and in novel fields such as Isogemetric Analysis. They have the capability of providing the adequate flexibility for the representation of smooth surfaces required in computer aided geometric design. By keeping a low polynomial degree, the spline functions are a suitable tool in terms of storage and manipulation. Despite an extensive literature, there are many open questions concerning the construction of spline spaces meeting all the appropriate approximation properties.
Much of what is currently known about splines was developed by numerical analysts using classical methods, in particular the so-called Bernstein-Bézier techniques. However, by their very definition, the study of splines involves an interplay between their algebraic structure, and the underlying combinatorics and geometry of the polyhedral complex. In fact, problems related to splines, such as subdivision strategies, dimension formulas and construction of basis, can be studied by using homological algebra. This approach has led to novel results in the past few years, and yielded unexpected connections between splines and classical problems in mathematics. For instance, the dimension of a spline space is closely related to the dimension of ideals generated by powers of linear forms, whose determination has only partially been solved and dates back to Früberg’s Conjecture (1985) and the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture (2001).
In the talk, we will present this latter technique for the study of splines. We will explore the connection between the theory of splines and ideals of fat points, and apply related results from algebraic geometry into the computation of the dimension of spline spaces, and the construction of bases, which play a crucial role in the application areas where splines are employed.