PhD in Mathematics

This course is an introduction to the modern theory of integration. The course begins with an introduction to measure theory, which is then used to introduce Lebesgue integration. After introducing the Lebesgue integral, we discuss the major theorems relating to Lebesgue integration. The course concludes with an introduction to function spaces and functional analysis.

Course learning outcomes
1. Prove basic propositions about measures and measure spaces.
2. Prove basic propositions about integrals with respect to a measure.
3. Prove basic propositions about simple function spaces.

This course aims to provide the students with a set of principles and skills that will be of paramount importance for conducting their research and disseminating the corresponding results. The course provides an overview of the scientific method and its application, project planning, working in teams, interaction with sources, and assessment of research results. Students are also taught the fundamental principles necessary for writing technical reports, scientific papers, and research proposals, and for preparing and delivering presentations: the focus of this part is on the connection between the employed research methods and the way in which the results are presented, together the writing/presentation style in itself. Finally, students are exposed to a general introduction on ethics, with subsequent focus on ethics in professional practice and research activity. The latter topic will focus on data management and intellectual property rights, and on the description of phenomena such as plagiarism and data fabrication.

Course learning outcomes
1. Collect and analyze mathematically information from various sources and prepare coherent presentation of his or her findings and summaries in research, and doing so in an ethical manner.
2. Use appropriate presentation techniques to present the selected topic(s) in mathematical sciences.
3. Demonstrate improvement in communication and writing skills by writing a review paper and giving a presentation.
4. Demonstrate the ability to make contributions in a team work environment in a scientific and ethical manner.
5. Understand and follow ethical standards and guidelines pertinent to mathematical research.

This course is designed to monitor progress and develop understandings, skills, and outlooks to conduct original, independent research at the PhD level. The student will develop (with the advisor's guidance) a research plan at the beginning of the semester that will state a research problem/question/hypothesis, its background, outline a research strategy and experimental approach, methods of proofs, interpretation and validation, and method of communication of the project results to others. The research plan is used as the basis for assessment of the student's research progress.

Course learning outcomes
1. Discover and pursue a unique topic of research in order to construct new knowledge.
2. Design and conduct an original research project.
3. Develop skills in designing a discipline specific research methodology.
4. Develop a working knowledge of relevant literature in the thesis discipline.
5. Practice academic writing appropriate to the discipline and learn how to participate in the peer review process.
6. Discuss research and other topics with academics in the thesis discipline.

This course is designed to monitor progress and develop understandings, skills, and outlooks to conduct original, independent research at the PhD level. The student will develop (with the advisor's guidance) a research plan at the beginning of the semester that will state a research problem/question/hypothesis, its background, outline a research strategy and experimental approach, methods of proofs, interpretation and validation, and method of communication of the project results to others. The research plan is used as the basis for assessment of the student's research progress.

Course learning outcomes
1. Discover and pursue a unique topic of research in order to construct new knowledge.
2. Design and conduct an original research project.
3. Develop skills in designing a discipline specific research methodology.
4. Develop a working knowledge of relevant literature in the thesis discipline.
5. Practice academic writing appropriate to the discipline and learn how to participate in the peer review process.
6. Discuss research and other topics with academics in the thesis discipline.

This is an upper level course in abstract algebra. It covers advanced group theory. In particular, field extensions, separability and inseparability, Galois theory, solvable and nilpotent groups and Sylow groups.

Course learning outcomes
1. Demonstrate a comprehensive understanding of the fundamental concepts in group theory, in particular, in Galois theory and the theory of field extensions and Sylow groups.
2. Demonstrate a comprehensive understanding of the fundamental concepts in ring theory, in particular, the notions of Noetherian rings, local rings, principal ideal domains, modules.
3. Operate with the concepts of group and ring homomorphisms and know their applications and significance.
4. Connect algebraic concepts and tools to solving problems in other areas of mathematics.
5. Develop their sense of rigour, logic and deduction when tackling a problem, and develop their analytical reasoning skills in problem-solving.
6. Demonstrate improvement in communication, reporting and writing skills by writing clear and soundly-argued solutions to problems.

This course examines ordinary differential equations (ODEs) from a geometric point of view and involves significant use of phase-plane diagrams and associated concepts, including equilibrium points, orbits, limit cycles, and domains of attraction. Various methods are discussed to determine existence and stability of equilibrium points and closed orbits. Methods are discussed for analyzing nonlinear differential equations (e.g., linearization and bifurcation analysis). An introduction to chaos theory is also presented. The techniques learned will be applied to equations from physics, engineering, biology, and ecology (as time permits).

Course learning outcomes
1. Sketch and interpret phase diagrams.
2. Distinguish between stable and unstable equilibrium points.
3. Use linear approximation methods for analyzing dynamical systems.
4. Find limit cycles and describe their stability.
5. Understand and apply the concepts of Lyapunov stability.

This course is an introduction to the theory of linear and nonlinear partial differential equations. The course begins with a discussion of the theory of linear and nonlinear first-order equations. This is followed by a discussion of the theory of linear second-order equations. The course concludes with an introduction to function spaces which are used in the modern study of partial differential equations.

Course learning outcomes
1. Solve transport eq.
2. Compute solutions of Laplace and heat eq. using Green's function.
3. Calculate solutions of the wave eq. by spherical means, Kirchhoff's and Poisson's formula.
4. Analyze uniqueness of solutions of main PDEs by energy methods.
5. Examine regularity and properties of solutions of main PDEs.
6. Model easy physical phenomena using PDEs.
7. Understand distribution theory and characterize Sobolev spaces.

The course covers theoretical foundations and applications of machine learning models. Topics include supervised methods for regression and classification (linear models, trees, neural networks, ensemble methods, instance-based methods), Bayesian parametric learning, density estimation and clustering, sequential models. Programming projects covering a variety of real-world applications are assigned.

Course learning outcomes
1. Derive mathematical expressions for a few simple algorithms like ordinary least squares regression, ridge regression, k-means and PCA.
2. Implement clustering, regression and classification algorithms using modern programming languages.
3. Apply gradient descent to solve simple convex optimization problems, and stochastic gradient descent to solve nonconvex optimization problems.
4. Assess whether the solution to a real-world machine learning problem might involve one or more of clustering, regression or classification.
5. Prepare reports and presentations with reproducible code.

A selection of advanced topics of mathematics. Topics will vary depending on the instructor and the level of the audience. In-depth content on selected mathematical topics, including their relation and application to the classroom. Topics courses permit a group of students to benefit from special expertise offered by faculty members and the opportunity to address particular learning interests of groups of students that may be identified.
Course learning outcomes1. By the end of the course the student will be able to give the key notions, definitions and results that are particular to the topic in question.
2. During the course students will research, prepare, and present one or more aspects of the material in the form of a written and/or oral report.
3. Students will demonstrate problem solving ability using the tools and materials in the course.
4. By the end of the course students should be able to explain the relevance of the topic and applications.

This course is a graduate introduction to probability theory with emphasis on rigorous mathematical theory. This is a measure-theoretic probability course, so students are expected to have solid background on at least undergraduate real analysis as well as elementary probability theory. Topics include Probability Space, Random Variables and Distributions, Limit theorems, Convergences of random variables, Generating functions, Characteristic functions and Zero-one law. Other topics may be included if time permits.

Course learning outcomes
1. Understand all concepts mathematically rigorously, based on measure theory.
2. Can state all definitions covered in class and key theorems.
3. Can apply the proof techniques covered in class to other problems.
4. Can apply the probabilistic concepts to other areas of studies.

This advanced-level functional analysis course covers the general topic of the function and operator theory as well as their applications in mathematical physics and astronomy: Advanced concepts of functional spaces, Banach algebras, subspaces, embedding of spaces, separable spaces, reflexive spaces. Foundations of the operator theory, bounded operators, invertible operators, restrictions and extensions of operators, Fredholm operators, Volterra operators, unbounded operators. Spectral theory, resolvent and spectrum, Hilbert-Schmidt theory and its application, Schatten-von Neumann classes, spectral properties of non-self-adjoint operators, biorthogonal systems. Symmetric rearrangements and applications, Hardy-Littlewood inequality, Riesz inequality, Polya-Szego inequality, Brownian motion in a ball, Optimal membrane shape for the deepest bass note, Maximiser body of the gravitational field energy, Dynamical stability problem of gaseous stars. Stability of symmetric steady states in galactic dynamics. Additionally the course will focus on applications of functional analysis in integral and differential equations arising engineering and mathematical physics.

Course learning outcomes
1. Demonstrate a comprehensive understanding of the fundamental principles of functional spaces (Banach algebras/spaces, subspaces, embedding of spaces, separable spaces, reflexive spaces etc…).
2. Establish and solve the operator equations.
3. Learn qualitatively restrictions and extensions of operators.
4. Learn the basic properties of bounded operators (Fredholm operators, Volterra operators etc) and unbounded operators (e.g. ordinary differential operators).
5. Acquire a basic knowledge of the spectral theory.
6. Learn the various applications of spectral theory in science, engineering, industry.
7. Develop their sense of rigour, logic and deduction when tackling a problem, and develop their analytical reasoning skills in problem-solving.
8. Demonstrate improvement in communication, reporting and writing skills by writing clear and soundly-argued solutions to problems.

Introduction to the basics of programming and algorithms in statistics. Exploration data of storage, manipulation, plotting, and analysis. Modularization, conditional execution, looping, and function construction are covered along with program debugging techniques.

Course learning outcomes
1. Apply statistical principles to experiments in various scenarios and understand how to make sense out of the data that is generated.
2. Interpret and present data to a wide audience and turn insight into decisions.
3. Conduct analyses which incorporate data and machine learning techniques to generate more complex understandings of data.

This is a graduate level course on stochastic calculus and its use in mathematical finance. We will cover Ph.D. level mathematical concepts and random processes theories used in finance. The topics to be covered start from stochastic integration theory, martingales and continues with the landmark theory of financial mathematics, the Black-Scholes theory. At the end of the course, the students are expected to master in stochastic calculus and its various applications in finance.

Course learning outcomes
1. Learn stochastic calculus.
2. Lean notion of fair price and arbitrage.
3. Learn Ito Lemma and its use in pricing derivatives in Black-Scholes framework.

This is a graduate level course on operations research and optimisation theory and its use in various applied sciences such as finance and engineering. The course covers deterministic and stochastic optimisation, in particular the dynamic programming algorithm, deterministic discrete and continuous time optimisation and its stochastic analogues, pontryagin maximum principle, modelling of stochastic linear and nonlinear systems.

Course learning outcomes
1. Learn dynamic programming paradigm.
2. Learn pontryagin maximum principle.
3. Use optimisation theory in finance and engineering.

The course covers numerical methods for solving partial differential equations including the fundamentals of the finite element and finite difference methods. It also covers the practical implementations of the algorithms for several types of boundary value and initial-boundary value problems with applications in physics and engineering.

Course learning outcomes
1. Understand several major numerical methods (including finite element/difference methods) for solving initial/boundary value problems of second and fourth order problems.
2. Be able to perform several major discretization procedures to reduce initial/boundary value problems into systems of algebraic equations and to be able to solve such systems numerically.
3. Be able to discretize some nonlinear problems by the numerical methods and solve the corresponding nonlinear algebraic equations.
4. Be able to develop and implement major numerical procedures by using computer codes and software-based methods.
5. To be able to prove some theorems related to existence/unique and stability/convergence of the problems and the associated numerical algorithms.