Applied Mathematics and Fuzzy Modelling

Degree Programme: Applied Mathematics

Faculty: Faculty of Science

Degree: Doctoral

Duration: 4 years

Form of Study: Full-time and Part-distance

Description:

PhD students in Applied Mathematics and Fuzzy Modelling then choose one of the following specializations:

  1. Differential equations: geometric, variational, and optimization methods
  2. Analytical number theory
  3. Fuzzy modelling

Ad 1) The study in the specialization “Differential Equations: Geometric, Variational, and Optimization Methods” of the PhD study field Applied Mathematics and Fuzzy Modelling focuses on one of the following areas:

  • global variational analysis,
  • differential equations on manifolds,
  • differential invariants and natural Lagrange structures,
  • optimization in abstract and infinite-dimensional spaces,
  • stochastic-heuristic algorithms in global optimization.

The study in this specialization also pursues applications, namely:

  • applications in physics and engineering (variational principles, dynamical control and optimization, non-holonomic mechanics and field theory,
  • real-life, everyday operational problems in business, industry, entrepreneurship, sustainable development, etc. The solution involves a mathematical formulation of the problem, (finding a suitable method for its solution), acquiring real data, and solving the problem with a practical recommendation based on the computed solution.

The focus of the student is supported by electing at least two compulsory elective subjects of this specialization, where the student acquires deep theoretical knowledge of mathematical theories and methods (topology, differential geometry, global analysis, geometric mechanics, variational equations, stochastic algorithms) of the given areas, and the ability to use the modern methods and means of mathematics to solve real problems.

Ad 2) The study in the specialization “Analytical Methods in Number Theory” of the PhD study field Applied Mathematics and Fuzzy Modelling focuses on one of the following areas:

  • Irrationality of infinite series
  • Diophantine approximations
  • Densities and measures of sets of positive integers or real numbers
  • Characteristics of distribution of number sets and sequences

The study in this specialization is mostly theoretical, nevertheless it involves also applications, namely:

  • applications of uniformly distributed sequences,
  • construction of low discrepancy sequences and their applications,
  • Monte Carlo and Quasi Monte-Carlo methods in mathematics, physics, economics.

The focus of the student is supported by electing at least two compulsory elective subjects of this specialization:

  • Diophantine approximations
  • Uniformly distributed sequences
  • Prime number theory
  • Monte Carlo and Quasi-Monte Carlo Methods and their applications
  • Selected topics in Number Theory

The students will acquire deep theoretical knowledge of mathematical theories and methods of the given areas, and the ability to use the modern methods and means of mathematics to solve real problems. They will be prepared for their scientific research work as well as for further pedagogical work.

Ad 3) The study in the specialization “Fuzzy Modelling” of the PhD study field Applied Mathematics and Fuzzy Modelling focuses on one of the following areas:

  • Algebraic structures of truth-values (residuated lattices, MTL-, BL-, MV-algebras, etc.).
  • Fuzzy logics.
  • Fuzzy modelling.
  • Time series analysis.
  • Data analysis.
  • Nature-inspired methods (evolutionary algorithms, swarm intelligence, etc.),

A substantial part of the study is devoted to practical development of the methods mentioned above and their applications, especially in the following fields of activities:

  • Data processing (data mining, time series analysis and forecasting, image processing, etc.).
  • Control and decision making based on expert knowledge.
  • Artificial intelligence and common sense reasoning.

Language of Instruction: English

Minimum Credit Requirements: 120 ECTS