Mathematics finds applications in some of the most diverse and interdisciplinary research in the physical, engineering, and biological sciences. The Applied Mathematics specialization at FLAME is designed to expose students to those fields of Mathematics that are fundamental to the Modeling and Simulation of phenomena straddling diverse disciplines of the Physical and Natural sciences, the social sciences, and in business. The specialization lays emphasis on taking a practical problem and turn it into a mathematical problem, also known as Mathematical Modeling, and to use analytical and computational techniques that help in the solution of such mathematical problems.
The Applied Mathematics major enables students to become adept in the use of mathematical techniques to solve problems in diverse fields where mathematics has a role to play. A sequence of courses will serve to provide the background for students’ understanding of the essential principles of mathematics. It will then introduce them to advanced methods and techniques that will provide the necessary skills for solving problems in diverse areas.
At the introductory level the specialization lays the foundation for study of subsequent higher level courses. At the intermediate level, a strong theoretical foundation is set by introducing subjects that form the backbone of Applied Mathematics subjects while introducing the students to some of the applied fields. At the advanced level students are introduced to advanced computational techniques.
The specialization prepares the student for graduate level courses with specialization in fields such as Applied Mathematics, Economics, Finance, Statistics, Optimization, and Environment sciences among others. Equipped with modeling and computational skills, a student can pursue a career in research not only in the conventional Applied Mathematics areas but also in any area that requires Mathematical modeling and computational proficiency such as Modeling in Biology, Ecology, Social Sciences, Actuarial Sciences and Finance.
SPECIALIZATION AIMS
- To provide the necessary mathematical skills required to solve problems of practical interest
- To provide an understanding of the process undertaken to arrive at a suitable mathematical model
- To teach the fundamental analytical techniques and computational methods used to develop solutions to mathematical models.
- To expose students to a range of problems from diverse areas along with their associated conceptual models, and the appropriate methods employed to solve them.
COURSES (CORE AND ELECTIVE)
24 MAJOR COURSES
|
1. Introduction to Programming |
10. Abstract Algebra |
19. Applied Multivariate Statistics * |
|
2. Introduction to Discrete Mathematics |
11. Introduction to Algorithm Design |
20. Introduction to Mathematical Finance * |
|
3. Linear Algebra |
12. Partial Differential Equations |
21. Computational Methods in Differential Equations * |
|
4. Calculus of One Variable |
13. Mathematical Modelling |
22. Machine Learning and Forecasting * |
|
5. Introduction to Probability and Statistics |
14. Complex Analysis |
23. Applied Functional Analysis * |
|
6. Introduction to Real Analysis |
15. Applied Probability |
24. Modelling for Social Sciences * |
|
7. Fundamental of Data Structure |
16. Numerical Methods |
|
|
8. Intermediate Multivariate Calculus |
17. Applied Multivariate Statistics |
|
|
9. Ordinary Differential Equations |
18. Mathematical Optimisation |
* 4th year undergraduate courses
1. Introduction to Programming
This is a first course in computer programming for those with little or no previous programming experience. It equips the student with basic tools to efficiently solve problems on the computer. Programming places considerable emphasis on algorithms and requires you to specify every step of the solution process. By doing so, it hones your analytical and logical skills and in the end makes you a better problem solver.
2. Introduction to Discrete Mathematics
This course aims to cover the basics of discrete mathematics. Discrete mathematics is the sturdy of discrete mathematical structures which do not rely on the notion of continuity. It introduces fundamental mathematical structures and various proof techniques and methods for solving different kind of problems. This course prepares the student to do advanced courses in applied mathematics and computer science.
3. Linear Algebra
This course emphasizes matrix and vector calculations and applications. It delves deeply into the theory of Matrices and other algebraic constructs such as Vector spaces, Determinants and Linear Transformations with particular emphasis on understanding the underlying theory and develops the analytical skills to prove theorems.
4. Calculus of One Variable
Calculus forms the foundation for a variety of subjects and finds applications in fields like Physics, Engineering, Economics, and Finance among others. In this course students will learn the concepts and techniques of single variable Differential and Integral Calculus
5. Introduction to Probability and Statistics
This course provides an elementary introduction to probability theory and its application to statistics with emphasis on the theorems and proofs of univariate statistics. Addressed to a beginning Mathematics Major, it provides a foundation for advanced courses in probability and statistics.
6. Introduction to Real Analysis
This is a first course in mathematical analysis. In this course a rigorous analysis of the real numbers, and provides training in the methods of mathematical proof. It is the first course in the analysis sequence and is followed by the introduction to Functional Analysis course.
7. Fundamental of Data Structure
This course will introduce students to the common data structures and their applications. It introduces the concepts and techniques of structuring and operating on Abstract Data Types in problem solving. It will equip them with a range of approaches and established algorithms for solving common classes of problems. Common sorting and searching algorithms will be discussed, and the complexity and comparisons among these various techniques will be studied. The course takes a practical approach, focusing on coded examples and applications.
8. Intermediate Multivariate Calculus
This course deals with functions and calculus of several variables. It follows the course on Single Variable calculus. Topics covered include geometry of 2 and 3 dimensions, Partial differentiation, scalar and vector fields and multiple integration. His course aims to provide students with working knowledge of functions in two or more variables, their partial derivatives, geometric interpretation of the derivatives, demonstrate the applications of these concepts in problems of finding extrema with or without constraints, and introduce the tools and techniques of evaluating Multiple Integrals.
9. Ordinary Differential Equations
This course aims to introduce students to the basic theory of ordinary differential equations and the modelling of diverse practical phenomena by ordinary differential equations by a variety of examples. Students will learn both quantitative and qualitative methods for solving these equations.
10. Abstract Algebra
Algebra is language of Mathematics. This course aims to introduce abstract objects such as Groups, Rings and Fields and its applications in other disciplines. This course covers some properties of groups, rings and fields. Permutation groups and polynomial rings are included also included.
11. Design and Analysis of Algorithms
This course explores efficient problem-solving methods that are useful for data science and Scientific computations. It will review common data structures and their applications and introduce a wide range of approaches and established algorithms for solving common classes of problems. It will cover common programming paradigms like Divide and Conquer, Greedy algorithms, Dynamic Programming to solve a wide variety of problems. Some common Graph algorithms will also be covered.
12. Partial Differential Equations
Partial Differential Equations are differential equations with more than one independent variable. They arise naturally in the modelling of physical and natural phenomena such as waves, diffusion of heat and fluid flow. This course is an introduction to partial differential equations.
13. Mathematical Modelling
This course equips the student with some of the methods and techniques of modelling continuous systems. It builds on knowledge gained through previous courses on Calculus and introduces new techniques for analysing and solving problems that arise in the application of mathematics in various disciplines.
14. Complex Analysis
This course provides an introduction to the theory of function of a complex variable. Residue Theorem and its applications to the integrals and sums and also conformal mappings and their applications will be discussed. This course aims to introduce students to the principal techniques and methods of analytic function theory. This is quite different from real analysis and has much more geometric emphasis. It tries to show how complex analysis can be used to evaluate real integrals, investigate the location of roots of polynomial equations and also introduce students to some applications of complex analysis, for example in fluid flow
15. Applied Probability
This course introduces probabilistic distributions and stochastic processes. It builds on knowledge acquired from elementary courses in probability and equips them to understand and apply advanced concepts to relatively more complex problems arising in diverse fields where uncertainty is a decisive factor.
16. Numerical Methods
This course gives an introduction to the basic techniques for solving problems in science and engineering using numerical methods. It provides students with an understanding of the concepts and knowledge of the theory and practical application of numerical methods.
17. Applied Multivariate Statistics
This course will introduce the theory and applications of multivariate statistical methods. It while emphasis will be on conceptual knowledge of the statistical tools and techniques used to analyse multivariate data, it also focusses on applying these techniques to real world using statistical packages. A background in calculus, probability and statistics is desirable.
18. Mathematical Optimisation
Optimization is the process of maximizing or minimizing an objective function that models a quantity of interest (e.g cost, price, effort, distance capacity…) arising in various disciplines in the presence of complicated constraints. In this course students will learn various techniques of optimization for both constrained and unconstrained problems with applications to problems arising in various disciplines
19. Applied Multivariate Statistics
Multivariate statistics deals with data that arise when several interdependent variables are measured simultaneously. They are ubiquitous and are generated in all disciplines. The analysis of such multivariate data is challenging and requires advanced statistical techniques which are implemented using computers. This course aims to provide a good understanding of the conceptual ideas that underpin the analysis of multivariate data.
20. Introduction to Mathematical Finance
This course provides a practical introduction to the mathematics behind finance in both discrete and continuous time. Aimed at advanced undergraduate students of Mathematics and Economics, a strong foundation in some of the tools and techniques is introduced in this course. Some of the methods include stochastic processes, arbitrage theory and partial differential equations which are used to model financial processes and price financial products.
21. Computational Methods in Differential Equations
This course will provide an introduction to numerical methods for ordinary and partial differential equations. Topics and methods to be covered in Ordinary differential equations include multistep and Runge-Kutta methods; stability and convergence; systems and stiffness; boundary value problems and for Partial differential equations, finite difference methods for elliptic, hyperbolic and parabolic equations; stability and convergence. The course will focus on introducing widely used methods and their implementation and highlight applications.
22. Machine Learning and Forecasting
Machine Learning is an important computational tool to create knowledge and gain insights from large amounts of data. This course will provide a mathematical introduction to machine learning, datamining, and statistical pattern recognition using supervised and Unsupervised learning methods. Topics to be covered include Regression, K -Nearest Neighbors, Classification, Dimensionality Reduction, Decision Trees and Random Forests, Principal Component Analysis and Clustering Analysis, Time series. The approach will be to gain practical knowledge to quickly and effectively apply the concepts learned to new contexts. R and Python will be used extensively.
23. Applied Functional Analysis
Functional analysis plays an important role in applied sciences as well as mathematics. It builds the foundations for the study of higher-level Mathematics and Physics and fields related to these subjects. This course will be an introduction to the basic concepts of Functional Analysis together with their applications.
24. Modelling for Social Sciences
The world is often described to be complex in which novel phenomena emerge from the actions of elementary units, which in the context of social science are humans. Making sense of these phenomena requires a framework that captures the essential elements of the process whose interplay helps in better understanding. One way to do it is through models. Models abstract the relevant information while filtering out the noise and help us recognise what is important. They help in making better decisions for more effective outcomes. The models that will be discussed in this course will span the whole gamut of social science ranging from political science, economics, social science, policy or business and will be studied by leveraging advances in mathematics and computing.
24 MINOR COURSES
|
1. Introduction to Programming |
10. Abstract Algebra |
19. Applied Multivariate Statistics * |
|
2. Introduction to Discrete Mathematics |
11. Introduction to Algorithm Design |
20. Introduction to Mathematical Finance * |
|
3. Linear Algebra |
12. Partial Differential Equations |
21. Computational Methods in Differential Equations * |
|
4. Calculus of One Variable |
13. Mathematical Modelling |
22. Machine Learning and Forecasting * |
|
5. Introduction to Probability and Statistics |
14. Complex Analysis |
23. Applied Functional Analysis * |
|
6. Introduction to Real Analysis |
15. Applied Probability |
24. Modelling for Social Sciences * |
|
7. Fundamental of Data Structure |
16. Numerical Methods |
|
|
8. Intermediate Multivariate Calculus |
17. Applied Multivariate Statistics |
|
|
9. Ordinary Differential Equations |
18. Mathematical Optimisation |
* 4th year undergraduate courses
1. Introduction to Programming
This is a first course in computer programming for those with little or no previous programming experience. It equips the student with basic tools to efficiently solve problems on the computer. Programming places considerable emphasis on algorithms and requires you to specify every step of the solution process. By doing so, it hones your analytical and logical skills and in the end makes you a better problem solver.
2. Introduction to Discrete Mathematics
This course aims to cover the basics of discrete mathematics. Discrete mathematics is the sturdy of discrete mathematical structures which do not rely on the notion of continuity. It introduces fundamental mathematical structures and various proof techniques and methods for solving different kind of problems. This course prepares the student to do advanced courses in applied mathematics and computer science.
3. Linear Algebra
This course emphasizes matrix and vector calculations and applications. It delves deeply into the theory of Matrices and other algebraic constructs such as Vector spaces, Determinants and Linear Transformations with particular emphasis on understanding the underlying theory and develops the analytical skills to prove theorems.
4. Calculus of One Variable
Calculus forms the foundation for a variety of subjects and finds applications in fields like Physics, Engineering, Economics, and Finance among others. In this course students will learn the concepts and techniques of single variable Differential and Integral Calculus
5. Introduction to Probability and Statistics
This course provides an elementary introduction to probability theory and its application to statistics with emphasis on the theorems and proofs of univariate statistics. Addressed to a beginning Mathematics Major, it provides a foundation for advanced courses in probability and statistics.
6. Introduction to Real Analysis
This is a first course in mathematical analysis. In this course a rigorous analysis of the real numbers, and provides training in the methods of mathematical proof. It is the first course in the analysis sequence and is followed by the introduction to Functional Analysis course.
7. Fundamental of Data Structure
This course will introduce students to the common data structures and their applications. It introduces the concepts and techniques of structuring and operating on Abstract Data Types in problem solving. It will equip them with a range of approaches and established algorithms for solving common classes of problems. Common sorting and searching algorithms will be discussed, and the complexity and comparisons among these various techniques will be studied. The course takes a practical approach, focusing on coded examples and applications.
8. Intermediate Multivariate Calculus
This course deals with functions and calculus of several variables. It follows the course on Single Variable calculus. Topics covered include geometry of 2 and 3 dimensions, Partial differentiation, scalar and vector fields and multiple integration. His course aims to provide students with working knowledge of functions in two or more variables, their partial derivatives, geometric interpretation of the derivatives, demonstrate the applications of these concepts in problems of finding extrema with or without constraints, and introduce the tools and techniques of evaluating Multiple Integrals.
9. Ordinary Differential Equations
This course aims to introduce students to the basic theory of ordinary differential equations and the modelling of diverse practical phenomena by ordinary differential equations by a variety of examples. Students will learn both quantitative and qualitative methods for solving these equations.
10. Abstract Algebra
Algebra is language of Mathematics. This course aims to introduce abstract objects such as Groups, Rings and Fields and its applications in other disciplines. This course covers some properties of groups, rings and fields. Permutation groups and polynomial rings are included also included.
11. Design and Analysis of Algorithms
This course explores efficient problem-solving methods that are useful for data science and Scientific computations. It will review common data structures and their applications and introduce a wide range of approaches and established algorithms for solving common classes of problems. It will cover common programming paradigms like Divide and Conquer, Greedy algorithms, Dynamic Programming to solve a wide variety of problems. Some common Graph algorithms will also be covered.
12. Partial Differential Equations
Partial Differential Equations are differential equations with more than one independent variable. They arise naturally in the modelling of physical and natural phenomena such as waves, diffusion of heat and fluid flow. This course is an introduction to partial differential equations.
13. Mathematical Modelling
This course equips the student with some of the methods and techniques of modelling continuous systems. It builds on knowledge gained through previous courses on Calculus and introduces new techniques for analysing and solving problems that arise in the application of mathematics in various disciplines.
14. Complex Analysis
This course provides an introduction to the theory of function of a complex variable. Residue Theorem and its applications to the integrals and sums and also conformal mappings and their applications will be discussed. This course aims to introduce students to the principal techniques and methods of analytic function theory. This is quite different from real analysis and has much more geometric emphasis. It tries to show how complex analysis can be used to evaluate real integrals, investigate the location of roots of polynomial equations and also introduce students to some applications of complex analysis, for example in fluid flow
15. Applied Probability
This course introduces probabilistic distributions and stochastic processes. It builds on knowledge acquired from elementary courses in probability and equips them to understand and apply advanced concepts to relatively more complex problems arising in diverse fields where uncertainty is a decisive factor.
16. Numerical Methods
This course gives an introduction to the basic techniques for solving problems in science and engineering using numerical methods. It provides students with an understanding of the concepts and knowledge of the theory and practical application of numerical methods.
17. Applied Multivariate Statistics
This course will introduce the theory and applications of multivariate statistical methods. It while emphasis will be on conceptual knowledge of the statistical tools and techniques used to analyse multivariate data, it also focusses on applying these techniques to real world using statistical packages. A background in calculus, probability and statistics is desirable.
18. Mathematical Optimisation
Optimization is the process of maximizing or minimizing an objective function that models a quantity of interest (e.g cost, price, effort, distance capacity…) arising in various disciplines in the presence of complicated constraints. In this course students will learn various techniques of optimization for both constrained and unconstrained problems with applications to problems arising in various disciplines
19. Applied Multivariate Statistics
Multivariate statistics deals with data that arise when several interdependent variables are measured simultaneously. They are ubiquitous and are generated in all disciplines. The analysis of such multivariate data is challenging and requires advanced statistical techniques which are implemented using computers. This course aims to provide a good understanding of the conceptual ideas that underpin the analysis of multivariate data.
20. Introduction to Mathematical Finance
This course provides a practical introduction to the mathematics behind finance in both discrete and continuous time. Aimed at advanced undergraduate students of Mathematics and Economics, a strong foundation in some of the tools and techniques is introduced in this course. Some of the methods include stochastic processes, arbitrage theory and partial differential equations which are used to model financial processes and price financial products.
21. Computational Methods in Differential Equations
This course will provide an introduction to numerical methods for ordinary and partial differential equations. Topics and methods to be covered in Ordinary differential equations include multistep and Runge-Kutta methods; stability and convergence; systems and stiffness; boundary value problems and for Partial differential equations, finite difference methods for elliptic, hyperbolic and parabolic equations; stability and convergence. The course will focus on introducing widely used methods and their implementation and highlight applications.
22. Machine Learning and Forecasting
Machine Learning is an important computational tool to create knowledge and gain insights from large amounts of data. This course will provide a mathematical introduction to machine learning, datamining, and statistical pattern recognition using supervised and Unsupervised learning methods. Topics to be covered include Regression, K -Nearest Neighbors, Classification, Dimensionality Reduction, Decision Trees and Random Forests, Principal Component Analysis and Clustering Analysis, Time series. The approach will be to gain practical knowledge to quickly and effectively apply the concepts learned to new contexts. R and Python will be used extensively.
23. Applied Functional Analysis
Functional analysis plays an important role in applied sciences as well as mathematics. It builds the foundations for the study of higher-level Mathematics and Physics and fields related to these subjects. This course will be an introduction to the basic concepts of Functional Analysis together with their applications.
24. Modelling for Social Sciences
The world is often described to be complex in which novel phenomena emerge from the actions of elementary units, which in the context of social science are humans. Making sense of these phenomena requires a framework that captures the essential elements of the process whose interplay helps in better understanding. One way to do it is through models. Models abstract the relevant information while filtering out the noise and help us recognise what is important. They help in making better decisions for more effective outcomes. The models that will be discussed in this course will span the whole gamut of social science ranging from political science, economics, social science, policy or business and will be studied by leveraging advances in mathematics and computing.