Gibson 1st Floor SR

Tonelli gave the first rigorous treatment of one-dimensional variational problems, providing conditions for existence and regularity of minimizers over the space of absolutely continuous functions.  He also proved a partial regularity theorem, asserting that a minimizer is everywhere differentiable, possible with infinite derivative, and that this derivative is continuous as a map into the extended real line.  Some recent work has lowered the smoothness assumptions on the Lagrangian for this result to various Lispschitz and H\"older conditions.  In this talk we will discuss the partial regularity result, construct examples showing that mere continuity of the Lagrangian is an insufficient condition.

Free-discontinuity problems describe situations where the solution of

interest is defined by a function and a lower dimensional set consisting

of the discontinuities of the function. Hence, the derivative of the

solution is assumed to be a "small function" almost everywhere except on

sets where it concentrates as a singular measure.

This is the case, for instance, in certain digital image segmentation

problems and brittle fracture models.

In the first part of this talk we show new preliminary results on

the existence of minimizers for inverse free-discontinuity problems, by

restricting the solutions to a class of functions with piecewise Lipschitz

discontinuity set.

If we discretize such situations for numerical purposes, the inverse

free-discontinuity problem in the discrete setting can be re-formulated as

that of finding a derivative vector with small components at all but a few

entries that exceed a certain threshold. This problem is similar to those

encountered in the field of "sparse recovery", where vectors

with a small number of dominating components in absolute value are

recovered from a few given linear measurements via the minimization of

related energy functionals.

As a second result, we show that the computation of global minimizers in

the discrete setting is an NP-hard problem.

With the aim of formulating efficient computational approaches in such

a complicated situation, we address iterative thresholding algorithms that

intertwine gradient-type iterations with thresholding steps which were

designed to recover sparse solutions.

It is natural to wonder how such algorithms can be used towards solving

discrete free-discontinuity problems. This talk explores also this

connection, and, by establishing an iterative thresholding algorithm for

discrete inverse free-discontinuity problems, provides new insights on

properties of minimizing solutions thereof.

Singular monopoles are solutions to the Bogomolny equation with prescribed singularities of Dirac monopole type. Previously such monopoles could be constructed only by the Nahm transform, with some difficulty. We therefore formulate a new construction of all singular monopoles. This construction relies on two ideas: Kronheimer's correspondence between singular monopoles on R^3 and self-dual connections on the multi-Taub-NUT space, and Cherkis' recent construction of self-dual connections on curved spaces using bow diagrams. As an example of our method we use it to obtain the explicit solution for a charge one SU(2) singular monopole with an arbitrary number of singularities.

Hilbert Sixth Problem of Axiomatization of Physics is a problem of general nature and not of specific problem. We will concentrate on the kinetic theory; the relations between the Newtonian particle systems, the Boltzmann equation and the fluid dynamics. This is a rich area of applied mathematics and mathematical physics. We will illustrate the richness with some examples, survey recent progresses and raise open research directions.