English

Control theory uses a very wide range of mathematical tools : algebra, analysis, geometry. As a consequence it is a topic of interest for many mathematicians from various areas. The aim of the school is to introduce mathematicians to some recent and very active subjects.

The program on Nonlinear Analysis consists of two parts : Holomorphic partial differential equations, variational methods for stationary and evolution partial differential equations. The aim of the school is to present recent results in these two branches of Nonlinear Analysis in a form which is accessible to graduate and doctoral students in Analysis and Applied Analysis.

Over the past thirty years there has been a growing need and interest for nonsmooth optimization methods. Nonsmoothness of objective or constraint functions may arise, for example, from heterogeneous material properties, from process controls and safety mechanisms that dynamically switch on and off; from variational inequality representations of free boundaries, phase transitions, contacts with friction; and from numerical schemes such as upwind finite differences and flux limiters. In multilevel optimization or very large scale applications nonsmoothness often arises by decomposition through the combinatorial nature of inequality constraints yielding piecewise smooth solution operators entering into the global objective. The participants of the school will be introduced to the relevant parts of nonsmooth analysis, the resulting optimality conditions in finite and infinite dimensions and the design of algorithms and software for their iterative solutions. Of particular concern are the numerical treatment of problems resulting from the discretization of differential equations.

This school is an introduction to subjects of algebraic coding theory and quasi cyclic codes. The purpose of this school is to introduce young mathematicians and students to the foundations of the the study of error correcting codes by means of algebra over finite rings and finite fields. Powerful decoding algorithms and connections with geometric codes will be emphasized when relevant. Applications to convolutional codes will be presented . It will consist of 6 courses which will be delivered by 6 courses given by San Ling, Ferruh Özbudak, Buket Özkaya, Joachim Rosenthal, Roxana Smarandache and Olfa Yemen. Moreover there will be around 15 talks on topics related to School’s concentration, given by different speakers.

This school is an introduction to subjects of current interest in algebraic geometry, with emphasis on group actions on algebraic varieties. The goal is to train Latin American students and young researchers on the different techniques in transformation groups (algebraic, symplectic, topological, etc.) and to highlight the many interactions between these viewpoints. There will be 5 courses, all of them delivered by well recognized specialists from France, USA and Mexico. Three of these courses will provide basic introductions to algebraic group actions, spherical varieties (which are generalizations of toric varieties and homogeneous spaces), and Mukai’s vector bundle method (for constructing certain K3 surfaces and Fano varieties as linear sections of homogeneous spaces). They will be followed by two courses on interactions between group actions, complex dynamical systems, and arithmetic geometry. Additionally, there will be several research talks given by the members of the Scientific Committee, researchers from PUCP, and invited researchers from South America, USA and Europe.

During the Research School the leading mathematicians will present a lot of lectures in the most important areas of modern algebra such as Jordan (super)algebras, conformal algebras, Sabinin algebras, composition algebras and GPI-algebras. There will be conferences about the last investigations of leading researches in modern algebra. New scientific relations will be established to produce in future new joint researches.

One of the goals of this school is to provide latin-american students with the possibility to attend courses and lectures related to Harmonic Analysis and Geometric Measure Theory, and their applications. This school will focus on those aspects of Harmonic Analysis which recently have had a huge impact, in particular in image and signal processing. A characteristic feature is that several technological deadlocks have been solved through the resolution of deep theoretical problems in harmonic analysis and Geometric Measure Theory. This school will present the new interlaces between Geometric Measure Theory and Harmonic Analysis and how these new understandings can be applied to solve real life problems. A special emphasis will be put on new relationships between Ergodic Theory and Geometric Measure Theory, EMD (empirical mode decomposition), and multifractal analysis based on p-exponents.

Research in the solvability of partial differential equations (PDE’s) leads us to a much wider scope. In the realization that alternative methods can be used for proving the existence and uniqueness of solutions of linear and nonlinear PDE’s, a new research area in Applied Mathematics was introduced. The theory of Sobolev spaces was developed, which turned out to be a suitable setting in which to apply ideas of functional analysis to glean information concerning PDE’s. But this research area is not developed well in Mongolia and there are many Ph.D (also post-Doc) students (large number of them are women) in our country are still facing issues concerning functional analysis. This is the main reason why we are keen on organizing the school in this research area.

The scope of problems accessible for a numerical treatment has been constantly broadened over the last fifty years. In particular, there has been a lot of research activity in the recent decades aimed at the problems with multi-scale and multi-physics features. The dedicated numerical methods (model reduction, micro-macro models and model coupling, non standard FEM) stem from diverse techniques and ideas such as homogenization, asymptotic analysis, statistical physics, domain decomposition methods, etc. In the same time, much effort has been devoted to the development of a posteriori error estimators which can now not only guide the computational mesh adaptation, but also help to choose the correct model or to minimize the number of iterations in the complicated multi-physics solution process. The courses in this school are aimed to cover the state of the art numerical approaches mentioned above and to present both the underlying mathematical ideas and the real life applications.

The main Research School will be preceded by a Preschool with an introduction to Finite Element Method (FEM) including conforming and non conforming variants, mixed FEM; a posteriori error control including Goal-Oriented approaches and adaptive FEM; stochastic computational methods and micro-macro approaches; homogenization and optimal control. The main School will contain advanced courses on numerical homogenization techniques, optimal control, error estimation and mesh adaptivity for multi-physics problems, finite volume methods for dissipative problems, asymptotic models for thin structures, and computational statistical physics. The possible applications of these techniques are in porous media flows, fluid-structure interaction, multiphase and other complex flows, modelling of vesicles and red blood cells, financial mathematics etc.

Our principal target would be young mathematicians, preferably with a master’s degree in mathematics or applied mathematics. The courses will be taught in English.