Overview
Overview

The degree of Master of Science in Applied Mathematics (MSc in Applied Mathematics) is awarded for successfully completing the requirements of a program of study, which includes taught courses and a thesis. The thesis is an independent investigation of specialized areas within the general field of Applied Mathematics.

The MSc in Applied Mathematics gives candidates the opportunity to deepen their knowledge in the broad field of Applied Mathematics and contribute to the process of discovery and knowledge creation through the conduct of original research. Candidates for this degree are taught and supervised by experienced faculty and are expected to demonstrate initiative in their approach and innovation in their work. In addition to successfully completing the taught course, candidates prepare and present a thesis on their chosen research area. Research may be undertaken on several topics corresponding to the areas of focus identified by the University. 

 

Program Educational Objectives
Program Learning Outcomes

Students graduating with the MSc in Applied Mathematics will be able to:

  1. Identify, formulate, and solve mathematical problems through knowledge and understanding of advanced mathematical concepts and computing
  2. Critically evaluate emerging technologies and assess how they can be applied to different practical problems
  3. Demonstrate the ability to advance his/her own knowledge and understanding through independent learning
  4. Conduct and document research as well as defend such research results
  5. Function in teams and communicate effectively
  6. Conduct themselves in a professional and ethical manner
Structure
Course Descriptions

MATH 621

Measure Theory (3-0-3)

Prerequisite: Undergraduate courses in Linear algebra, real analysis, and topology

Measure theory provides a foundation for many branches of mathematics, such as harmonic analysis, ergodic theory, theory of partial differential equations and probability theory. It is a central, extremely useful part of modern analysis, and many further interesting generalizations of measure theory have been developed. This course is an introduction to abstract measure theory and the Lebesgue integral. The Lebesgue integral is introduced, and the main convergence theorems are proved. Emphasis is given to the construction of the Lebesgue measure in Rn. Other topics treated in the course are Lpspaces, the Radon-Nikodym theorem, the Lebesgue differentiation theorem, and the Fubini theorem.

MATH 622

Real Analysis (3-0-3)

Prerequisite: Undergraduate courses in fundamental of mathematical reasoning, calculus II, and real analysis

This course is at a higher level compared to MATH 324 and covers more advanced topics as well as the standard topics more deeply. In this course, students will be introduced to the difference between pointwise and uniform convergence. The integration and differentiation of a sequence of functions will be treated in depth. The course provides a foundation for employing the Riemann-Stieltjes integral and using the Weierstrass M-test for a series of functions. Determining whether a series of functions converges uniformly will be discussed as well as the facility with power series of functions and their use in solving differential equations. This course also allows the students to demonstrate familiarity with common pathological counterexamples regarding sequences and series of functions. It introduces the students to Riemann integration and its use in analysis, various convergence theorems and their applications, equicontinuity and the Arzela-Ascoli theorem, the Arzela-Ascoli theorem to ODEs, and how to use the Stone-Weierstrass theorem in practical situations.

MATH 603

Multivariate Data Analysis (3-0-3)

Prerequisite: Undergraduate courses in linear algebra, differential equations, and engineering statistics (or equivalent)

Introductory graduate-level course in Multivariate Data Analysis. Multivariate Data Analysis is the study of multiple variables in a set of data where the relations among these variables and their structures are important. This course focuses on some of the most important techniques of data reduction and analysis of qualitative data.

MATH 605

Analytical Foundations of Risk and Optimization (3-0-3)

Prerequisite: Graduate standing, introductory probability, linear algebra, or by consent of instructor

The course provides a solid and rigorous introduction to analytical and optimization methods of primary importance in modern economic and financial applications. The recent evolution of the concept and taxonomy of risk provides a comprehensive mathematical framework to evaluate and assess the exposure to sources of risk whose impact may severely jeopardize effective decision-making. The course additionally includes recently developed methods for optimal decision-making under uncertainty.

MATH 606

Differential Equations (3-0-3)

Prerequisite: Undergraduate courses in calculus II, calculus III, engineering mathematics, and introduction to differential equations

This course will address the behavior of solutions: existence and uniqueness theorems for nonlinear systems, continuous dependence on data; and dynamical systems properties: long time existence, stability theory, Floquet theory, invariant manifolds and bifurcation theory. Many applications shall be discussed.

MATH 610

Model Estimation (3-0-3)

Prerequisite: Undergraduate knowledge of multivariate calculus, linear algebra, and probability and statistics

This course provides a rigorous introduction to statistical modeling. The topics covered include classical regression, nonparametric regression, penalized estimation, covariance parameters estimation, multivariate linear model, discrimination and allocations, and principal component analysis.

MATH 611

Quantitative Tools for Data Science (3-0-3)

Prerequisite: Undergraduate courses in introductory probability and statistics and a working basic knowledge of linear Algebra is recommended

This course introduces the basic concepts of data analysis and statistical computing that are frequently used in the social sciences and humanities. The course emphasizes practical applications of quantitative reasoning, visualization, and data analysis.

MATH 612

Computational Methods and Optimization in Finance (3-0-3)

Prerequisite: Undergraduate courses in linear algebra, programming, and introductory probability and statistics

This course introduces the main classes of optimization problems (linear, quadratic, convex, integer, stochastic, and robust) and the algorithms to efficiently compute the optimum in each case. The methods will be applied to financial problems such as asset/liability management, option pricing and hedging, risk management, and portfolio optimization. The students will learn to use software related to each technique.

MATH 613

Financial Risk and Portfolio Management (3-0-3)

Prerequisite: Undergraduate courses in linear algebra and introductory probability and statistics

This course explores how imperfect correlation between assets leads to diversified and optimal portfolios as well as its implications in asset pricing. Students will learn how to design an investor’s profile by building a perfect portfolio through the combination of strategic and tactical asset allocations. Additionally, students will thoroughly explore risk by examining its different facets as well as exploring tools and methods for measuring, managing, and hedging risk. Utilizing real data, students will implement these techniques using appropriate software.

MATH 614

Factor Models with Machine Learning (3-0-3)

Prerequisite: Familiarity with multivariate calculus and some basic linear algebra will be assumed as well as some prior knowledge of undergraduate probability and statistics. The knowledge of various standard results concerning probability and statistics is assumed. If you are not familiar with these standard results, please contact the instructor as soon as possible

Introduction to factor models and key machine learning concepts for financial applications. Includes regression and principal-component factor models, factor analysis, neural networks, the kernel trick, and classification. All methods will be implemented in R or Python.

MATH 615

Abstract Algebra (3-0-3)

Prerequisite: Undergraduate courses in fundamentals of mathematical reasoning

Group Theory

Basic arithmetic properties of Z. Equivalence relations; definitions: group, subgroups, order; homomorphisms, isomorphism; Cosets; Lagrange’s Theorem and applications; cyclic groups; group actions; orbits; stabilizers; orbit partition; normalizers; centralizers; conjugacy; symmetric groups; Cayley’s Theorem; Orbit-Stabiliser Theorem; Sylow’s Theorems; quotient groups; the canonical homomorphism; direct products; finite abelian groups; finite abelian p-groups; the Hall polynomial; Structure Theorem of finitely generated abelian groups.

Ring Theory

Basic examples and definitions; subrings; homomorphisms kernels and Ideals. Quotient Rings. Factorization; Reducibility; roots; special classes of rings; prime and maximal ideals; field extensions; minimal polynomials of finite algebraic extensions; field automorphisms and the Galois Group; the Galois correspondence; fundamental theorem of Galois Theory; a Burly example; application: solution by radicals; insolubility of the quintic.

MATH 616

Linear Algebra and Optimization (3-0-3)

Prerequisite: Undergraduate courses in linear algebra and programming

The course revises important notions from linear algebra, which leads to an introduction to the main classes of optimization models: linear, quadratic, second-order cone, robust and semidefinite optimization. The students will learn to use software related to each technique.

MATH 617

Numerical Solutions of Differential Equations (3-0-3)

Prerequisite: Undergraduate courses in differential equations, engineering mathematics, basic knowledge of functional analysis and partial differential equations, and some programming skills

This course focuses on numerical methods for partial differential equations (PDEs), with an emphasis on a rigorous mathematical basis. The fundamentals of numerical approximations and efficient numerical approaches are presented. Particular attention is given to a qualitative understanding of PDE models, the basic concepts of finite differences and finite elements, as well as important concepts such as stability, convergence, and error analysis.

MATH 618

Mathematical Biology (3-0-3)

Prerequisite: Undergraduate courses in calculus II, calculus III, engineering mathematics, and introduction to differential equations

This course provides an introduction to the use of continuous and discrete differential equations in the biological sciences. Biological topics include single species and interacting population dynamics, modeling infectious diseases, regulation of cell function, molecular interactions, as well as neural and biological oscillators. The course also exposes students to mathematical tools necessary to analyze and interpret biological models. Such tools include phase portraits, bifurcation diagrams, perturbation theory, and parameter estimation techniques.

MATH 619

Mathematical Methods for High-Dimensional and Discrete Data with Machine Learning Applications (3-0-3)

Prerequisite: Graduate standing, familiarity with multivariate calculus, linear algebra, and a solid course in undergraduate probability and statistics. The knowledge of various standard results concerning probability and statistics is assumed. If you are not familiar with these standard results, please contact the instructor as soon as possible.

Technological and scientific advances in our ability to collect, observe, and store data throughout science, engineering, and commerce call for a change in the basic understanding of how we are to learn and handle data.  This course rigorously surveys the modern literature concerning the mathematical foundations of several statistical learning and inference problems. A particular emphasis is on non-asymptotic results. Topics covered include sparse recovery, high dimensional PCA, and nonparametric least squares. The aim is to develop algorithms that are effective both in theory and in applications.

MATH 620

Advanced Statistical Inference (3-0-3)

Prerequisite: Undergraduate knowledge of multivariate calculus, linear algebra, and probability and statistics

This course delivers a rigorous introduction to classical statistical inference. Probabilistic concepts and tools are used to present inferential statistics methods. The following techniques will be covered: sampling distributions, parametric point estimators and their properties, interval estimation, hypothesis testing, and regression models.  Furthermore, an introduction to Bayesian statistics will be presented.

MATH 623

Health Data Science (3-0-3)

Prerequisite: Familiarity with multivariate calculus and some basic linear algebra will be assumed as well as some prior knowledge of undergraduate probability and statistics. The knowledge of various standard results concerning probability and statistics is assumed. If you are not familiar with these standard results, please contact the instructor as soon as possible.

This course provides an introduction to Health Data Science, with special emphasis on developing the knowledge and competencies necessary to understand the measurement and use of variables in health, their scales of measurement, and their use in biostatistics and Spatial Epidemiology.

MATH 624

Space-Time Data Science (3-0-3)

Prerequisite: Undergraduate course in Statistical Inference (or equivalent)

Space-time data are becoming available in overwhelming volumes and diverse forms as a result of growing remote-sensing capabilities, ground-based sensor networks, crowdsourcing, citizen science data, climate models, and novel medical sensing technologies.  Dealing with massive data sets having complex structures implies a collection of conceptual, methodological, and technical challenges, which are exacerbated by data diversity. Space-time statistical methods were not designed to deal with global, high-volume, hyper-dimensional, heterogeneous, and uncertain space-time data. In fact, the computational requirements of most available methods scale poorly with data size.  Space-Time Data Science (STDS throughout) is based on the integration of Statistics, Computer Science, and Machine Learning as fundamental vertices in a graph structure to be then synchronized with applied sciences, such as geography, physics, soil science, neuroscience, and epidemiology. Hence, the key of the success of STDS is to be able to tailor interdisciplinary approaches to the analysis of diverse and big space-time data. This course will introduce the statistical and computational aspects of STDS.

MATH 626

Financial Derivatives and Risk Management (3-0-3)

Prerequisite: Linear algebra, programming, and introductory probability and statistics

The concept of Financial Derivatives, which includes options, futures, and forwards, is crucial for risk management, speculation, and for arbitrage activities. This course covers the foundational theory in derivatives valuation and risk management from the mathematical modeling point of view. It demonstrates the strengths and weaknesses of different models. It also illustrates and exemplifies how valuation models and risk measures are applied in the financial industry.

MATH 630

Research Methods in Science (3-0-3)

Prerequisite: Graduate standing

This course provides sound knowledge and understanding of research methodology and project management skills and their application to science research and project development. Topics covered include aspects of MSc research, critical literature review, citations and references, technical writing, presentation skills, software and experimental methods, modeling and simulation methods, reliability and validity of results, analysis and interpretation of results, project management, and professional issues in research.

MATH 699

Master’s Thesis (12 credits)

Co-requisite: Approval of the Department Chair and the Associate Dean for Graduate Studies

In the Master’s Thesis, the student is required to independently conduct original research-oriented work related to important Applied Mathematics problems under the direct supervision of a main adviser, who must be a full-time faculty in the Department of Mathematics, and at least one other full-time faculty who acts as co-advisor. The outcome of the research should demonstrate the synthesis of information into knowledge in a form that may be used by others and lead to publications in suitable reputable journals/conferences. The student’s research findings must be documented in a formal thesis and defended through a viva voce examination. The student must register for a minimum of 12 credit hours of Master’s Thesis.

 

Study Plan