Research within this program is mainly focused on the study of nonlinear partial differential equations (PDE) and differential systems that describe physical processes, motivated by the current issue of the involvement of mathematics in other sciences through interdisciplinary research. Theoretical research is related to applications, especially in environmental sciences, biology, medicine and engineering. Among the different applications we mention: diffusion in porous media, problems with free boundary, phase transitions, reaction-diffusion processes, population dynamics, epidemics, cell membranes, fluid dynamics, problems in continuum mechanics.
From a mathematical point of view, the objectives focus on the study of the following aspects: existence, uniqueness, regularity, asymptotic behavior, attractors for nonlinear PDE systems (hyperbolic, parabolic, elliptical) in Banach spaces, mainly in the deterministic case, but stochastic effects are taken into account as well.
An important part of the research of this group refers to optimal control, controllability, stabilization, inverse problems and optimization of systems of equations that model real-world phenomena.
Some members of the team deal with the analysis and development of innovative numerical methods for numerical approximation of partial differential equations, algorithms and simulations in the deterministic and stochastic case, with applications in various fields of interest, such as solids mechanics, fluid mechanics, interaction fluid-structure, environment and biology.

Dependence can be studied by stochastic order, non-commutative probabilities or by various copulas and the considered limit theorems refer to Markov chains, Gauss Kuzmin processes or fragmentation processes.

Research in this program is focused on the following directions:
1) Statistical methods of supervised and unsupervised classification for the analysis of multidimensional databases with a high degree of complexity (functional data, graphs, hierarchy, etc.)
Motivation: New technologies allow the acquisition of data with increasingly complex structures. We meet these data in all fields: medicine (modeling and analysis of heart rate, image analysis, patients’ medical history, prescriptions), economics (indicators that evolve over time, product / company rankings, etc.), sociology (social networks), etc. Statistical analysis of these data requires their standardization, size reduction, and other transformations. Depending on the purpose, statistical models of supervised (regression, classification) or unsupervised (clustering) analysis require major developments in this context. Among the modern methodologies for the analysis of such data we can mention Bayesian analysis, parametric methods with penalty criteria (curse of dimensionalty), nonparametric methods (SVM-RKHS, Neural Networks etc).
Objectives: Research in this direction is mainly focused on statistical methods (regression, clustering) for:
a) longitudinal (functional) data observed on time intervals of different lengths
b) qualitative longitudinal data (categorically) with error of observing the moments of state change.
c) Statistical methods for graphs with evolution over time.
2) Stochastic models in epidemiology
Research in this direction is mainly focused on the study and deepening of stochastic models that describe and analyze the spread of infectious diseases from person to person (omitting those transmitted through water, food). The models we want to investigate are: Stochastic models for homogeneous communities; Suspected – Infected – Healed Models, CRS, in structured populations; Stochastic models in inhomogeneous communities.
Basic bibliography: (Lecture Notes in Mathematics 2255) Tom Britton, Etienne Pardoux – Stochastic Epidemic Models with Inference-Springer International Publishing (2019)
3) Research on portfolio theory models: equivalent models, models with absolute optimums, models containing non-smooth functions.
Motivation: Portfolio theory is a chapter in financial mathematics that has seen an explosive development in recent years. Applications of this theory are found in economics, management, agriculture, forestry, fish farming, energy, information technology, etc. Many of the problems that arise in portfolio theory (for example, models that contain transaction fees) are described using optimization models that contain non-smooth functions. It can be shown that, in some cases, these models are equivalent to mathematical programming models with complementarity constraints, which in turn are equivalent to mixed mathematical programming models. Most models in portfolio theory take into account investor preferences.
Objectives: We aim to study models of portfolio theory that take into account the preferences of a category of investors that is characterized by families of utility functions that meet various conditions. The optimal portfolios for this type of models will be called absolute portfolios. Research in this direction is mainly focused on the study of financial markets for which such portfolios exist. Characterizations of the random vectors defining these financial markets will be obtained.
4) Optimization algorithms in continuous n-dimensional space
Motivation: The mathematical theory of this category of algorithms is far behind the practical applications (very varied, especially in the field of multi-objective optimization), mainly due to the limited traceability of the multi-dimensional integral calculus involved in analyzing the performance of algorithms on different types. objective functions.
The main research objectives are focused on the following issues: estimating the probability of success, average progress and convergence time for different types of probabilistic (evolutionary) algorithms with one or more individuals in the population, using different multi-dimensional distributions for mutation. Research in this direction is mainly focused on:
a) Analysis of evolutionary algorithms with several individuals in the population, with mutation and crossover operators, on quadratic functions
b) Analysis of evolutionary algorithms on asymmetric functions of RIDGE type
5) Study of the convexity of the functions defined on subsets of the set of square matrices
Another important direction of research is the study of the convexity of the functions defined on subsets of the set of square matrices. Such a study is based on the theorem of the mathematician Chandle Davis by enlarging the set of matrices to which this theorem can be applied. The results expected to be obtained have applications in mathematical programming on cone matrices.
6) Study of Markov chains using matrix theory
Research in this direction aims at the study of Markov chains with the help of matrix theory, materialized in the study of limit theorems for finite Markov chains.
The results expected to be obtained have applications in the calculation of limit matrices.
7) Special models of mathematical programming and nonlinear optimization with applications.