As palestras do Colóquio de Geometria e Aritmética (COLGA) do Rio de Janeiro serão realizadas em 30 de agosto (sexta-feira), no Instituto de Matemática, UFRJ - CT sala C116 – Ilha do Fundão.
Programação:
09:30 às 10:30: Applications of curves over finite fields to polynomial problems, Daniele Bartoli (Università degli Studi di Perugia - Itália).
Resumo: Algebraic curves over finite fields are not only interesting objects from a theoretical point of view, but they also have deep connections with different areas of mathematics and combinatorics. In fact, they are important tools when dealing with, for instance, permutation polynomials, APN functions, planar functions, exceptional polynomials, scattered polynomials, Moore-like matrices. In this talk I will present some applications of algebraic curves to the above mentioned objects.
10:30 às 11:00: Pausa para o café.
11:00 às 12:00: Inflection of linear series on hyperelliptic curves over arbitrary fields (joint with I. Biswas, I. Darago, C. Han and C. Garay López), Ethan Cotterill (UFF).
Resumo: According to Plucker's formula, the total inflection of a linear series (L,V) on a complex algebraic curve C is fixed by numerical data, namely the degree of L and the dimension of V. The problem of describing the k-rational inflectionary locus of a Gal(k-bar/k)-linear series (L,V) when k is a non-algebraically closed field is significantly more subtle. For example, the topology of the real inflectionary locus of a real linear series depends in a nontrivial way on the topology of the real locus of C. I will describe joint work with Biswas and Garay López in which we study this dependency when C is hyperelliptic and (L,V) is a complete series. Our main tool is a nonarchimedean degeneration, which allows us to relate the (real) inflection of complete series to the (real) inflection of incomplete series on elliptic curves. I will also describe work in progress with Darago and Han in which we compute k-rational inflectionary loci valued in the Grothendieck--Witt group GW of an arbitrary field k. To do so, we apply the A^1 homotopy theory of Morel, Voevodsky, Levine, Kass and Wickelgren.
