Contribution talks and minicourses are presentations or classes 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.

Marcio Antonio Jorge da Silva - State University of Londrina

Date: June 04

Time:10:00 am

Venue: Auditorium of the Department of Mathematics.

Refreshments and snacks at 9:45 am.

Title: Stability Analysis of Partially Damped Viscoelastic Systems

Abstract: Partially damped systems can be understood as those that exhibit both damping and undamped partial differential equations, which makes their stabilization a complex issue to be analyzed in several situations. The characterization of stability for such partially damped systems requires finding what conditions to take into account to determine whether the system is (uniformly) stable or not along the time. This can be done by firstly understanding where the damping feedback is coupled and looking for stability parameters through the structural coefficients of the system. Then, by making use of mathematical tools such as multipliers, Lyapunov functionals, frequency domain analysis, among others, we can provide the desired characterization of stability in many examples, which includes several (partially) damped systems with important applications in the field of control theory and engineering. In particular, a partially damped Timoshenko beam system with memory will be addressed in the present talk.

Keti Teneblat - University of Brasilia

Date: October 03

Time: 9:00 am

Venue: Auditorium of the Department of Mathematics.

Title: TBA

Marcos Tadeu de Oliveira Pimenta - São Paulo State University

Date: August 08-09

Time: 1:30-3:30 pm

Venue: Auditorium of the Department of Mathematics.

Title: Introdução à Teoria das Distribuições e aos Espaços de Sobolev (minicourse)

Abstract: Nesse minicurso, estudaremos a definição e propriedades básicas das distribuições. Essas últimas, são objetosmatemáticos que generalizam funções e medidas e que nos permitem efetuar diversas operações. Tal teoria é amplamente usada na área de Análise Matemática, sobretudo no estudo das equações diferenciais. Na segunda parte do minicurso, estudaremos a definição e propriedades elementares dos chamados espaços de Sobolev. Esses constituem importantes subespaços dos espaços de Lebesgue, formados por funções que possuem alguma propriedade de regularidade (ainda que em um sentido fraco). 

Elizabeth Terezinha Gasparim - Del North Catholic University

Date: November 27

Time: 10:30 am

Title: Fibrações Elípticas, hiperelípticas e a Teoria-F (talk in portuguese language).

Abstract: Curvas elípticas são grupos de Lie nos quais a operação de grupo se descreve de maneira muito geométrica. Exemplos básicos são os toros complexos (superfícies de Riemann de gênero 1). Curvas elípticas são populares em física por serem variedades Calabi–Yau. Já em dimensão complexa 2, as variedades Calabi–Yau favoritas para a física são as superfícies K3. Muitas delas têm estrutura de fibração elíptica, e com tal estrutura aparecem de forma essencial na Teoria-F (uma generalização da teoria de cordas do tipo IIB). As fibras singulares de tais fibrações determinam onde aparecem as famosas D-branas da teoria de cordas, e sua existência gera grupos de monodromia não triviais. Apresentarei algumas propriedades básicas das curvas elípticas e, em seguida, passarei a descrever curvas hiperelípticas, as quais têm gênero pelo menos 2. Mostrarei superfícies complexas fibradas por curvas hiperelípticas, contendo singularidades escolhidas de modo conveniente para aplicações específicas na física, que foram obtidas no artigo em colaboração com Ballico e Suzuk. Passarei então ao caso de variedades Calabi–Yau de dimensão complexa 3, apresentando muitos exemplos. As Calabi–Yau threefolds são famosas por causa do que se chama de Calabi–Yau Landscape. Estamos particularmente interessados no caso em que X=Tot(ωY)X = Tot(ω_Y)X=Tot(ωY​) é o espaço total do fibrado canônico de uma superfície Y. Mostramos a existência de tais X para qualquer possível valor de classes de Chern c2(X)c_2(X)c2​(X). Finalmente, comentarei sobre as aplicações à Teoria-F, observando o quanto elas mudam de acordo com a maneira como posicionamos as curvas hiperelípticas.

Luiz Gustavo Farah Dias - Federal University of Minas Gerais

Date: February 24 (2025)

Time: 4:00 pm

Title: On the mass-critical inhomogeneous NLS equation

Abstract: We consider the inhomogeneous nonlinear Schrödinger (INLS) equation

iu_{t}+Δu+|x|^{-b}|u|^{((4-2b)/N)}u=0,  x∈Rⁿ

with N ≥ 1 and 0 < b < 1, which is a generalization of the classical nonlinear Schrödinger equation (NLS). Since the scaling invariant Sobolev index is zero, the equation is called mass-critical. In this talk we discuss some blow-up results in the non-radial setting, obtained in collaboration with Mykael Cardoso (UFPI-Brazil).

Victor Hugo Gonzalez Martinez - Federal University of Pernambuco

Date: February 27 (2025)

Time: 3:00 pm

Title: Global Stabilization for the BBM-KP Equations on R^2

Abstract: In this talk, we present new results on the asymptotic behavior of the energy for the Ben-jam ́ın-Bona-Mahony Kadomtsev-Petviashvili (BBM-KP) equations (I and II) posed on the twodimensional plane R2, with localized damping. The

BBM-KP equations, which model wave propagation in fluid dynamics, are studied in the context of energy dissipation. This model provides an alternative to the classical Kadomtsev-Petviashvili (KP) equations, similar to the way the regularized long-wave (RLW) equation offers an alternative to the Korteweg-de Vries (KdV) equation. We analyze the asymptotic behavior of the total energy associated with the Cauchy problem for the BBM-KP equations under the influence of localized damping. Our main result shows that, in the presence of localized dissipation applied to a subdomain of the spatial domain, the energy tends to a steady state as time progresses. Specifically, we investigate the rate at which the energy approaches this asymptotic state, providing insights into the long-term dynamics of the solutions. To complement the theoretical analysis, we present numerical simulations

based on a spectralfinite difference scheme, combining the advantages of spectral methods for high accuracy in the frequency domain and finite differences for spatial discretization. These simulations validate the theoretical predictions and illustrate the behavior of the energy as it evolves over time, confirming the transition to an asymptotic state. Work in collaboration with F. A. Gallego (Universidad Nacional de Colombia (UNAL), Colombia) and J.C Munoz (Universidad del Valle, Colombia).