Apolo

Atividade

140046 - General determination theory

Período da turma:

03/02/2026 a 06/02/2026

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Descrição:

Programa completo – ementa e referência bibliográfica:

The starting point of this course will be a recent result of convex analysis, namely that every smooth convex and bounded from below function f defined on a Hilbert space is determined, up to a constant, by the modulus of its derivative ∥∇f∥. In other words, mere knowledge of the norm of the gradient function ∥∇f∥ provides information also about its direction. We refer to this type of result, as a determination result. We shall first see that the above determination result extends to the nonsmooth case (via subdifferential theory), where the distance of the subdifferential to 0 plays the role of ∥Df∥. Then we shall gradually move to the framework of metric spaces, omitting the linear structure, and replacing ∥Df∥ by the De Giorgi metric slope. In this case, we shall see that a determination result can also be established for the class of bounded from below Lipschitz continuous functions, by adding information on the critical values. Finally, we shall consider more general situations (stochastic processes), by means of an abstract modulus of descent, encompassing both steepest and average descents) giving rise to a general determination theory. We shall also discuss analogies to the viscosity theory of PDEs (Eikonal equations) and to the w-KAM theory.

Bibliografia:

A. Daniilidis, D. Salas. Determination of functions by metric slopes. Proc. Amer. Math. Soc. 150 (2022), 4325-4333.

Carga Horária:

10 horas

Tipo:

Obrigatória

Vagas oferecidas:

100

Ministrantes:

Aris Daniilidis (Especialista)

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