Plenary
Terence Tao
Universality and possible blowup in fluid equations.
We discuss some possible (and still speculative) routes to establishing finite time blowup in fluid equations (and other PDE), focusing in particular on methods based on establishing universality properties for such equations.
Carlos Kenig
Soliton resolution and channels of energy for nonlinear wave equations
We will discuss the soliton resolution conjecture and recent progress on it in the case of nonlinear wave equations, through the use of channels of energy.
Tatiana Toro
Geometry of Measures
(Abstract): We will discuss how the infinitesimal properties of measures yield information about their support in Euclidean space with a Riemannian metric. We will discuss the classical results and highlight the question that led to this work.
Michael Ruzhansky
Subelliptic pseudo-differential operators
on compact Lie groups
In this talk we present some modern development related to the theory of pseudo-differential operators. On compact Lie groups and under the presence of Riemannian and sub-Riemannian structures we are going to present global Hörmander classes based on the matrix-valued notion of a symbol. Then applications of these calculus to geometric analysis and PDE are given.
David Dos Santos Ferreira
On global Carleman estimates in Lebesgue spaces
Carleman estimates have proved to be a powerful tool used in establishing unique continuation properties for solutions of partial differential equations, in control and stabilisation theory and in inverse problems. Whenever a boundary is involved, global Carleman estimates, that is, Carleman estimates with boundary terms, are most often a necessary refinement.
Sharp Carleman estimates in Lebesgue spaces were first established by Jerison and Kenig to study unique continuation for solutions of partial differential equations with coefficients of low regularity. This talk is concerned with establishing global Carleman estimates in Lebesgue spaces, and is based on a joint work with Rémi Buffe.
Simon Donaldson
Some differential-geometric boundary value and mapping problems in dimensions 5, 6 and 7.
The setting for this talk is the consideration of special differential-geometric structures on manifolds of dimension 6 (Calabi-Yau structures) and 7 (torsion free G_{2} structures). We will begin by reviewing this background and in particular Hitchin’s variational approach. In the main part of the talk we will discuss variants of the theory on manifolds with boundary. Under dimension reduction these lead to a variety of PDE problems, some classical and some new.
Harald Helfgott
Grafos Expansores: de las telecomunicaciones a la teoría de Números
¿Qué son? ¿De dónde vienen? ¿Aplicaciones?
Enrique Zuazua
Control and Machine Learning
In this lecture we shall present some recent results on the interplay between control and Machine Learning, and more precisely, Supervised Learning and Universal Approximation.
We adopt the perspective of the simultaneous or ensemble control of systems of Residual Neural Networks (ResNets).
Roughly, each item to be classified corresponds to a different initial datum for the Cauchy problem of the ResNets, leading to an ensemble of solutions to be driven to the corresponding targets, associated to the labels, by means of the same control. We present a genuinely nonlinear and constructive method, allowing to show that such an ambitious goal can be achieved, estimating the complexity of the control strategies.
This property is rarely fulfilled by the classical dynamical systems in Mechanics and the very nonlinear nature of the activation function governing the ResNet dynamics plays a determinant role. It allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill.
The turnpike property is also analyzed in this context, showing that a suitable choice of the cost functional used to train the ResNet leads to more stable and robust dynamics.
This lecture is inspired in joint work, among others, with Borjan Geshkovski (MIT), Carlos Esteve (Cambridge), Domènec Ruiz-Balet (IC, London) and Dario Pighin (Sherpa.ai).