# Joint Summer School of the

UC|UP, MAP-PDMA and PDMat-UA PhD Programs

## University of Minho, Braga

### September 9 - 13, 2019

The Summer School will take place at University of Minho, Campus of Gualtar, and is jointly organised by the UC|UP Joint PhD Program in Mathematics of the Universities of Coimbra and Porto, the PhD Program in Applied Mathematics of the Universities of Aveiro, Minho and Porto (MAP-PDMA) and the PhD Program in Mathematics of the University of Aveiro (PDMat-UA).

The Summer School will consist of three intensive courses (8h each) and sessions where the students of the three PhD programs that organize the School may present their work.

## Courses

An introduction to Algebraic Geometry

Francesco Polizzi (Università della Calabria, Italy)

Motivations and first examples.

Affine varieties.

Projective varieties.

Birational equivalence.

Tangent space, non-singularity, dimension.

Applications and further examples.

Optimal Control of Sweeping Processes

Boris Mordukhovich (Wayne State University, Detroit, Michigan, USA)

This course is devoted to a novel class of optimal control problems governed by the so-called sweeping (or Moreau) processes that are described by discontinuous dissipative differential inclusions. Although such dynamical processes, strongly motivated by applications, have appeared in 1970s, optimal control problems for them have been formulated quite recently and occurred to be rather complicated from the viewpoint of developed control theory. Their study and applications require advanced tools of variational analysis and generalized differentiation, which will be presented in the lectures. Combining this machinery with the method of discrete approximations leads us deriving new necessary optimality conditions and their applications to practical models in elastoplasticity, traffic equilibria, and robotics.

Morrey-Besov spaces, heat equations and Navier-Stokes equations

Jan Vybíral (Czech Technical University, Prague, Czech Republic)

We present the application of Besov spaces built on Morrey spaces in the theory of heat equations and Navier-Stokes equations. After a brief introduction into Morrey spaces and Besov spaces, we join the two subjects and define the Morrey-Besov spaces. We review their basic properties with respect to embeddings and multipliers. The main part of the series will then be devoted to the application of these spaces in the study of heat equations and Navier-Stokes equations. We show the existence of global-in-time solutions when the norms of the initial conditions are sufficiently small.

## Scientific Committee

Alexandre Almeida (Univ. Aveiro), Carlos Rito (Univ. Porto), Maria Irene Falcão (Univ. Minho), Susana Moura (Univ. Coimbra).

## Local Organization

Maria Irene Falcão (Univ. Minho), Fernando Miranda (Univ. Minho).

#### Contacts

map.pdma@math.uminho.pt

## Poster