Conferências

Image
Plenarista: Felipe Linares, IMPA
Titulo: On solutions to Interaction equations for short and long dispersive waves
Resumo: In this lecture we will be concerned with properties of solutions to two nonlinear dispersive models called the Schrödinger-Korteweg-de Vries and Schrödinger-Benjamin-Ono systems.
First we will describe the decay of long-time solutions of the initial value problem (IVP) associated with the Schrödinger-Korteweg-de Vries system. We use recent techniques in order to show that solutions of this system decay to zero in the energy space. Our result is independent of the integrability of the equations involved and it does not require any size assumptions.
In the second part of the talk we will discuss the local well-posedness of the IVP associated with the Schrödinger-Benjamin-Ono system.
Image
Plenarista: Giancarlo Urzúa, Pontificia Universidad Católica de Chile
Titulo: On singular algebraic surfaces and the Horikawa problem
Resumo: In a series of articles in the 70s, Horikawa classified all complex nonsingular projective surfaces on (and close to) the Noether line. He described the connected components and dimensions of their moduli spaces. He also identified their topological types (in particular, they all are simply-connected), and their diffeomorphism types up to one infinite family. In that case the corresponding moduli spaces have two connected components, and the problem is to decide whether surfaces in these two distinct components are diffeomorphic. It has been open since the appearance of the first article in 1976. 
One way to attack this problem is to construct common mildly degenerated surfaces to these components. This would happen in the Kollár--Shepherd-Barron--Alexeev compactification of the moduli space. Recently it was proved by Julie Rana and Sönke Rollenske (2022) that there are common degenerations for these components, but so far too singular. On the other hand, it is known that if the singularities are of Wahl type (particular quotient singularities), then the surfaces in these two components would be diffeomorphic. Surprisingly, in the literature there is only one degeneration with Wahl singularities and for one component, due to Yongnam Lee and Jongil Park (2011), which considers the smooth original construction of Fintushel--Stern (1997). 
In this line of attack, the first question would be: Is it possible to classify all singular surfaces for these potential common degenerations? In a joint work with Vicente Monreal and Jaime Negrete, both Masters students at UC Chile, we are able to classify every singular surface with Wahl singularities of Horikawa type. In particular, this gives all possible candidates for these potential common mild degenerations. (Still, one needs to show smoothings for them to be able to use it for the Horikawa problem.) In addition, we are able to classify all Wahl singular surfaces close to the Noether line, such as quintics (in projective space) and I-surfaces. In this talk, I will review this famous Horikawa problem, the strategy via degenerations, and I will show the ideas for the classification of Horikawa surfaces with Wahl singularities.
Image
Plenarista: Gisèle Mophou, Université des Antilles
Titulo: Quasi-reversibility methods of optimal control for ill-posed heat equations
Resumo: We consider optimal control problems associated to generally non-well posed Cauchy problems in a general framework. Firstly, we approximate the ill-posed problem with a family of well-posed one and show that solutions of the latter one converge to solutions of the former one. Secondly, we investigate the minimization problem associated with the approximated state equation. We prove the existence and uniqueness of minimizers that we characterize with the optimality systems. Finally, we show that minimizers of the approximated problems converge to the minimizers of the optimal control subjected to the ill-posed state equation that we characterize with a singular optimality system. This characterization is obtained as the limit of the optimality systems of the approximated minimization problem. We use the techniques of quasi-reversibility developed by Lattès and Lions in 1969. Our results can be extended to classical elliptic second order operators with Dirichlet and Robin conditions, as well as the fractional Laplace operator with the Dirichlet exterior conditioen.
Image
Plenarista: Hanne Van Der Bosh, Universidad de Chile
Titulo: Bounds on the ground-state energy for Dirac Operators
Resumo: He classical Keller-Lieb-Thirring inequality bounds the ground state energy of a Schrödinger operator by a Lebesgue norm of the potential. This problem can be rewritten as a
minimization problem for the Rayleigh quotient over both the eigenfunction and the potential. It is then straightforward to see that the best potential is a power of the eigenfunction, and the optimal eigenfunction satisfies a nonlinear Schrodinger equation. This talk concerns the analogous question for the smallest eigenvalue in the gap of a massive Dirac operator. This eigenvalue is not characterized by a minimization problem. By using a suitable Birman-Schwinger operator, we show that for sufficiently small potentials in Lebesgue spaces, an optimal potential and eigenfunction exists. Moreover, the corresponding eigenfunction solves a nonlinear Dirac equatioen.
Image
Plenarista: Julián Fernandez Bonder, Universidad de Buenos Aires
Titulo: Eigenvalue problems for the generalized fractional laplacian
Resumo: In this lecture I will discuss some recent results on eigenvalue problems for the fractional g-Laplacian operator.
Image
Plenarista: Laura Ortiz, Universidad Nacional Autónoma de México
Titulo: 
Local models and analytic invariants of singular foliations by curves
Resumo: 
The solutions of analytic differential equations define partitions of the space by pairwise non intersecting curves. Such  partitions are said to define foliations by curves (the leaves of the foliation). When a differential equation has singular points, it defines a foliation by curves with singularities. The analysis of the differential equations in neighborhoods of their isolated singular points and their analytic classification has been a recurring task since the time of Poincaré.

When the linear part of the equation at the singular point has a matrix that satisfies some generic conditions, its classification is relatively simple, however when these conditions are not met, the analytical classification has been shown to be very complicated. Namely, in the analytical classification of differential equations in neighborhoods of their degenerate singular points, a set of singular curves must be considered as well as a group that describes the behavior of the dynamics transverse to the solutions.
In the seventies of the last century, René Thom conjectured that the so-called separatrices of the equation at the singular point was the set of curves to consider. This conjecture was soon disproved and the problem (Thom’s problem) of finding the invariants of analytical classification remained widely open for a long time.
In this talk we will talk about Thom's invariants for degenerate singular points that satisfy certain genericity conditions, as well as the geometry underlying such classification.