Program Overview

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MSc in Applied Mathematics

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.

Overview

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

The educational objectives of the MSc in Applied Mathematics program are to produce graduates who will be able to:

• Advance professionally and be recognized as leaders in their chosen areas within the broad field of Applied Mathematics
• Master solid analytical, quantitative and computational skills where Mathematics and Statistics play a key role
• Apply their technical expertise to address the critical needs of society in a creative, ethical, and innovative manner
• Further develop their knowledge and skills through graduate education or professional schools

Program Learning Outcomes

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

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

Structure

The MSc in Applied Mathematics program consists of a minimum of 30 credit hours. The required program credits are distributed as follows: 12 credits of Program Core courses, 9 credits of Program Elective courses, and 9 credits of Applied Mathematics Master's Thesis work. Students will be expected to discuss their selection of the elective courses with the Program Coordinator/Supervisor to ensure that it is coherent and supports their thesis topic.

Program Requirements

Students seeking the degree of MSc in Applied Mathematics must successfully complete a minimum 30 credit hours as specified in the program requirements detailed below, with a minimum CGPA of 3.0. Course selection should be made in consultation with the student's main adviser. All courses have a credit rating of three credits each, except the Master's Thesis.

Program Core

The MSc in Applied Mathematics degree program core requires a minimum of 12 credits, consisting of the four courses, which are 3 credits each. The core courses are listed below.

• MATH 621 Measure Theory
• MATH 622 Real Analysis
• MATH 605 Analytical Foundations of Risk and Optimization
• MATH 606 Differential Equations

Program Electives

Students must complete a minimum of 9 credits of electives from the list of courses given below. Students can also select one elective course (3 credits) from other offered MSc programs at KU, subject to the approval of their research advisor and the Associate Dean for Graduate Studies.

• MATH 603 Multivariate Data Analysis
• MATH 610 Model Estimation
• MATH 611 Quantitative Tools for Data Science
• MATH 612 Computational Methods and Opt in Finance
• MATH 613 Financial Risk and Portfolio Management
• MATH 614 Factor Models with Machine Learning
• MATH 615 Abstract Algebra
• MATH 616 Linear Algebra and Optimization
• MATH 617 Numerical Solutions of Differential Equations
• MATH 618 Mathematical Biology
• MATH 619 Mathematical Methods for High-Dimensional and Discrete Data with Machine Learning Applications
• MATH 620 Advanced Statistical Inference
• MATH 623 Health Data Science
• MATH 624 Space-Time Data Science
• MATH 625 Nonlinear Oscillations and Nonlinear Waves
• MATH 626 Financial Derivatives and Risk Management
• MATH 630 Research Methods in Science

Master's Thesis

MATH 699 Master's Thesis (minimum 9 credits)

A student must complete a master's thesis that involves creative research-oriented work within the broad field of Applied Mathematics, under the direct supervision of the 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-adviser. The research findings must be documented in a formal thesis and defended successfully in a viva voce examination. Furthermore, the research should lead to publishable quality scholarly articles.

Course Descriptions

MATH 603 Multivariate Data Analysis

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

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

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

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

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

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

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

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

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

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

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

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

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

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 621 Measure Theory

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 Lp spaces, the Radon-Nikodym theorem, the Lebesgue differentiation theorem, and the Fubini theorem.

MATH 622 Real Analysis

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 623 Health Data Science

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

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 625 Nonlinear Oscillations and Nonlinear Waves

Prerequisite: Familiarity with ODEs and PDEs

This is an advanced course on nonlinear oscillations and nonlinear waves, emphasizing analytical modeling techniques for oscillatory dynamics and their applications in science and engineering. Certain applications from physics (e.g., mechanical vibrations), engineering (e.g., electric oscillations in RLC circuits), and biology (e.g., epidemics) are presented and discussed.

MATH 626 Financial Derivatives and Risk Management

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

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

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 9 credit hours of Master's Thesis.

Study Plan

Students must consult with their respective advisors on the courses that they will enroll in, the required pre-requisites, and the thesis topic selection. Full-time graduate students must register for 9 to 12 credits, including thesis credits, during a regular semester (Fall and Spring) and a maximum of 6 credits during a Summer term. In the case of part-time students, the credit load is normally 6 credits during a regular semester as well as the summer term.

Typical Study Plan for MSc in Applied Mathematics (Full-time)

| Fall Semester | | Spring Semester |
---|---|---|---|---
Year 1 | MATH Core Course 1 | 3 | MATH Core Course 4 | 3
MATH Core Course 2 | 3 | Elective Course 1 | 3
MATH Core Course 3 | 3 | Elective Course 2 | 9
Total credit hours for the first semester |9 |Total credit hours for the second semester |9
Summer | |
Year 2 | Elective Course 3 | 3 | Elective Course 5 | 3
Elective Course 4 | 3 | MATH 699 Master's Thesis | 6
MATH 699 Master's Thesis | 3 | |
Total credit hours for the third semester |9 |Total credit hours for the fourth semester |9

Typical Study Plan for MSc in Applied Mathematics (Part-time)

| Fall Semester | | Spring Semester |
---|---|---|---|---
Year 1 | MATH Core Course 1 | 3 | MATH Core Course 3 | 3
MATH Core Course 2 | 3 | MATH Core Course 4 | 3
Total credit hours for the first semester |6 |Total credit hours for the second semester |6
Summer | |
Year 2 | Elective Course 1 | 3 | Elective Course 3 | 3
Elective Course 2 | 3 | Elective Course 4 | 3
Total credit hours for the third semester |6 |Total credit hours for the fourth semester |6
Summer | |
Year 3 | Elective Course 5 | 3 | MATH 699 Master's Thesis | 6
MATH 699 Master's Thesis | 3 | |
Total credit hours for the fifth semester |6 |Total credit hours for the sixth semester |6