Upon completion of the course, the student will be able to:

1. 1. Demonstrate highly specialised knowledge of fields, rings, groups, and algebras, including forefront developments, as a basis for original research and critical evaluation of current problems at field interfaces.
   2. Articulate advanced classical algorithms and basics of signal processing, with critical evaluation of knowledge gaps and integration from related fields.
   3. Explain properties of trigonometric and orthogonal series, and bases in general Banach spaces, demonstrating forefront knowledge for original and innovative thinking.
   4. Describe methods of model-based design and main problems in equations of mathematical physics, with critical awareness of interdisciplinary interfaces and emerging challenges.
   5. Present probabilistic models for random phenomena and industrial problems, integrating highly specialised knowledge for independent original research.
   6. Elucidate mathematical concepts in information technology, with forefront knowledge and critical insight into cross-field issues and novel applications.
2. 1. Construct advanced algebraic models and implement original algorithms in practice, developing innovative procedures through independent specialised problem-solving in complex contexts.
   2. Apply spectral decomposition of normal operators, factorization methods for differential and integral equations, and numerical methods, critically integrating knowledge from fields to innovate in unpredictable contexts.
   3. Develop wavelet bases for signal processing, investigate Fourier series convergence, implement fast Fourier algorithms, and perform complex domain approximations, generating new knowledge through independent research.
   4. Analyze and apply probabilistic models, use mathematical statistics in econometrics for predictions, and develop original procedures at field interfaces with critical evaluation.
   5. Build sophisticated mathematical models for unfamiliar problems in various fields, conduct independent research, and propose original solutions with strategic foresight.
   6. Design effective algorithms using programming tools for complex problems, and apply programming packages to innovate in theoretical and applied scenarios, leading transformative applications.
3. 1. process, analyze the obtained results, prepare reports presenting the results of research, conduct scientific debates;
   2. analyze the existing problems and propose ways to solve them, apply the knowledge in solving the problems related to the field;
   3. choose the necessary research methods and adapt the existing methods to the study of a specific problem, use scientific publications and data on the Internet, collect information in the field of specialization.