Abstract : In this talk, we investigate the existence of weak solutions for elliptic problems involving Choquard nonlinearity. These equations have attracted significant attention due to their ability to model long-range interactions in various real-world applications. A key concept in solving PDEs is that of weak solutions. These solutions satisfy the integral form of the PDE and are useful when classical solutions may not exist or are challenging to compute. This makes them exceedingly valuable in practical applications. We will use Variational methods to solve the PDEs. This technique essentially transforms the problem of solving a PDE into the problem of finding critical points of the associated functional.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1

Abstract :-On dispersion generalized Benjamin-Ono equations

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1

Abstract : The problem of modeling dissipative effects in quantum physics dates back to the 1970s. After reviewing the main challenges associated with developing such models, I will present a specific model introduced by Bruneau and De Bièvre in the early 2000s. This model describes the interactions between a classical particle and an abstract environment, where the environment acts on the classical particle as a linear friction force. One of the key strengths of this approach is that it can be naturally extended to the quantum setting.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1

Abstract : In this talk, we will explore the role of generalized symmetries—symmetry groups that classify families of partial differential equations—in identifying fundamental symmetries in physics. We will also examine how this framework is crucial for defining conservation laws, building on Noether’s theorems and the contribution of Sophus Lie to the understanding of continuous symmetries.

Venue: Sala John Von Neumann, CMM. Beauchef 851, 7th floor.

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1

female protagonists, in commemoration of International Women’s Day.

Abstract :

Since 1965 when Penrose defined the concept of a trapped surface (compact and without boundary) in a 4-dimensional space-time, these surfaces have been an important object of study for geometricians and theoretical physicists, standing out for their mathematical properties as well as for their applications in general relativity. Trapped surfaces can be defined in terms of the causal character of their mean curvature vector, which allows us to generalize them to subvarieties, not necessarily compact and of any dimension and/or codimension, in space-time.

In this talk we show rigidity and non-existence results for parabolic spatial subvarieties with causal mean vector curvature in spacetimes that admit an orthogonal decomposition. These spacetimes contemplate, in particular, the family of globally hyperbolic spacetimes. On the other hand, we also give a result on the geometry of a more general family of subvarieties in such spacetimes, assuming the non-existence of local minima or maxima of a given function. As an application of our results in the field of general relativity, we obtain some results on trapped surfaces (not necessarily closed) in a very large family of spacetimes.

The results presented are part of a joint work with Professors Jónatan Herrera and Rafael M. Rubio of the University of Córdoba.

Venue: Sala John Von Neumann, CMM. Beauchef 851, 7th floor.

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1

female protagonists, in commemoration of International Women’s Day.

Abstract : In this talk I will discuss the non-existence of certain solutions to the equation that determines translating solutions to Mean Curvature Flow. This result completes the classification to semigraphical translators.

Venue: Sala John Von Neumann, CMM. Beauchef 851, 7th floor.Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09

female protagonists, in commemoration of International Women’s Day.

Abstract : – abstract-2

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09

Abstract : We consider the subcritical nonlinear Schrödinger (NLS) in dimension one posed on the unbounded real line. Several previous works have considered the deep neural network approximation of NLS solutions from the numerical and theoretical point of view in the case of bounded domains. In this paper, we introduce a PINNs method to treat the case of unbounded domains and show rigorous bounds on the associated approximation error in terms of the energy and Strichartz norms, provided reasonable integration schemes are available. Applications to traveling waves, breathers and solitons, as well as numerical experiments confirming the validity of the approximation are also presented as well.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09

Abstract :-Poblete-Felipe-2

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09

Abstract : The design of optimal feedbacks for control problems is a challenging task. The classical method for tackling this problem is based on dynamic programming. This involves finding the value function of the control problem by solving the Hamilton-Jacobi-Bellman (HJB) equation. However, this equation suffers from the “curse of dimensionality”, i.e., the computational cost of solving it grows exponentially with the dimension of the underlying control problem. For this reason, several methods based on machine learning have been proposed to solve HJB. Although numerical experiments have shown promising results, it is still necessary to find theoretical guarantees on the performance of this type of methods. In this regard, one of the main difficulties is the low regularity of HJB solutions.

In this talk we will present results related to the approximation of HJB solutions. These results allow us to find bounds for the performance of feedback generated by machine learning methods. It is important to note that these bounds only require the value function to be Hölder continuous, while similar results in the literature require the value function to be at least C^1. To illustrate the importance of bounds, a family of control problems indexed by a penalty coefficient will be presented. This coefficient controls the regularity of the value function, so that, for values close to zero the value function is C^2, whereas, it becomes non-differentiable when it is sufficiently large.  Additionally, the application of these results to the method called Averaged Feedback Learning Scheme (AFLS), which consists of solving an averaged version of the control problem, will be presented. Finally, the ability of this method to solve problems with high dimensionality will be shown through numerical examples.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09