Durante o Programa de Verão, também teremos algumas palestras por visitantes, conforme listado abaixo.
Débora Lopes (UFS)
Geometria diferencial clássica: um ponto de vista singular
14 de fevereiro de 2023, às 10:00, sala 3-012
Em sua obra clássica “Mémoire sur la Théorie des Déblais et des remblais” (1784), Monge começa a estudar famílias a dois parâmetros de retas no espaço, congruência de retas, buscando resolver um problema de minimizar o custo para transportar uma quantidade de terra de um lugar para outro, preservando o volume.
Na tentativa de entender este problema, muitas técnicas de equações diferenciais, análise e geometria foram desenvolvidas. A obra de Monge pode ser considerada um dos textos fundadores da geometria diferencial. Nesta palestra, buscamos mostrar como a Teoria das Singularidades, Geometria Diferencial e Equações Diferenciais se entrelaçam, a partir do surgimento de um problema, ainda em aberto, proposto há mais de dois séculos.
Vanderley Alves Ferreira Junior (Instituto Tecnológico de Aeronáutica)
Sinal das soluções da equação parabólica de ordem superior
16 de fevereiro de 2023, às 14:00, sala 3-012
Descreveremos a variação de sinal da solução do problema de valor inicial para uma equação parabólica de ordem superior quando o dado inicial é não-negativo e tem suporte compacto.
Carles Bivià-Ausina (Universitat Politècnica de València)
Log-canonical threshold of ideals and the sequence of mixed multiplicities
24 de fevereiro de 2023, às 16:00, sala 3-012
We characterize the ideals \(I\) of finite colength of the ring \(\mathcal O_n\) of complex analytic functions germs \((\mathbb C^n, 0)\to \mathbb C\) whose integral closure is equal to the integral closure of an ideal generated by pure monomials. This characterization, which is motivated by an inequality proven by Demailly and Pham, is given in terms of the log canonical threshold of \(I\) and the sequence of mixed multiplicities of \(I\). Moreover, we relate this topic with the question of characterizing the ideals of \(\mathcal O_n\) whose multiplicity is equal to the product of the sequence of mixed Lojasiewicz exponents of \(I\).
Konstantinos Kourliouros (Imperial College London)
Bruce-Roberts numbers and quasi homogeneous functions on analytic varieties
13 de março de 2023, às 16:15, sala 3-012
Given a germ of an analytic variety \(X\) and a germ of a holomorphic function \(f\) with stratified isolated singularity with respect to the logarithmic stratification of \(X\), we show that under certain conditions on the singularity type of the pair \((f, X)\), the following analog of the well known K. Saito’s theorem holds true: equality of the relative Milnor and Tjurina numbers of \(f\) with respect to \(X\) (also known as Bruce-Roberts numbers) is equivalent to the relative quasihomogeneity of the pair \((f, X)\), i.e. to the existence of a coordinate system such that both \(f\) and \(X\) are quasihomogeneous with respect to the same positive rational weights.
Ethan Cotterill (UNICAMP)
Enriched inflection and secant planes for linear series on algebraic curves
22 de março de 2023, às 17:00, sala 3-012
The fundamental local invariant of a linear series on an algebraic curve is its inflection, as measured by the vanishing of local derivatives in a point. Secant planes, which emerge from the rank behavior of tuples of points on the curve, may be regarded as (the manifestations of) a global analogue of inflection. Here we report on joint work with Ignacio Darago and Naizhen Zhang in which we compute the classes of inflection points and of tuples of points spanning secant planes in the Grothendieck–Witt ring of an arbitrary perfect base field of characteristic not equal to 2.
Pietro Speziali (UNICAMP)
Transitive actions on Weierstrass points of Riemann Surfaces and Beyond
29 de março de 2023, às 17:00, sala 3-012
Let \(X\) be a Riemann Surface of genus \(g\) bigger than \(1\). A point \(P\) of \(X\) is Weierstrass if its gap-set \(G(P)\) differs from \(\{1,…,g\}\). A classical result states that \(X\) has a finite number of Weierstrass points, whose number (counted with their weight, which measures how far a Weierstrass point is from being an ordinary point) equals \(g^3-g\). Next, assume that \(X\) admits a non-trivial group of automorphisms \(G\). By the famous Hurwitz bound, in our context \(G\) is finite and of order at most \(84(g-1)\). Since \(G\) must act on the set \(W\) of Weierstrass points of \(X\), it is natural to ask whether this action can be transitive. As it turns out, this is a rare possibility, with very few known examples in the literature. In this talk, we give an overview of the literature on this subject and our contributions. Also, we discuss some possible developments to curves of positive characteristic, and the challenges they pose.