M.Sc. in Mathematics

Mezuniyet Koşulları ve Ders İçerikleri

Graduation Requirements and Courses

Graduation Requirement for M.Sc. (Thesis) in Mathematics

Publication Requirements for M.Sc. (Thesis) in Mathematics

In addition to mentioned graduation requirements, a Masters candidate is requested to satisfy the following publication requirement before the thesis defense:

• Conference Acceptance with departmental approval OR,
• Journal (SCI-Expanded level) submission with the result of accept, minor revision or major revision OR,
• Journal (SCI-Expanded level) submission and internal Departmental Review Process, followed with departmental approval

Required Courses

• MATH 501 Analysis 1
  This course covers selected topics in real analysis concerning measure theory and integration. Topics include Lebesgue measure and integration on R^n, convergence theorems, Fubini’s theorem, L^p spaces, abstract measure and integration, decomposition of measures, Lebesgue-Radon-Nikodym theorem.
• MATH 502 Analysis II
  This course introduces the student to the core notions and techniques of functional analysis. Topics include Hilbert and Banach spaces, duality and Hahn-Banach theorem, open mapping and closed graph theorems, principle of uniform boundedness, weak and weak-* topologies.
• MATH 503 Numerical Linear Algebra
  This course covers selected topics in numerical linear algebra. Topics include matrix analysis, direct methods for linear systems, iterative methods for linear systems, methods for eigenvalue problems, iterative methods for nonlinear systems.
• MATH 504 Partial Differential Equations
  The course starts with explicit solutions of three classical equations: Laplace’s equation, heat equation and wave equation and continues with an introduction to Sobolev spaces and analysis of solutions of boundary value problems for second-order elliptic linear equations. The topics include existence of weak solutions by energy methods, regularity of solutions and maximum principles.
• MATH 506 Probability Theory
  This course introduces the fundamental concepts of probability theory based on measure and integration theory. The course is split into three parts. The first part is on the basics of measure and integration, along with the notion of independence. The second part includes more advanced topics of measure and integration that are central to probability theory such as moments, L^p-spaces, types of convergence, Radon-Nikodym theorem, product spaces and Fubini’s theorem. The last part includes the notions of weak convergence, characteristic functions, and conditional expectation, and covers basic ‘successes’ of probability theory such as laws of large numbers, central limit theorem, and martingale convergence theorem.
• MATH 693 M.Sc. Thesis Study I in Mathematics
• MATH 694 M.Sc. Thesis Study I in Mathematics
• MATH 695 M.Sc. Thesis Study I in Mathematics
• GSE 680 Graduate Study and Seminars for Research, Innovation and Ethics

Elective Courses

Students are required to take at least 2 courses from the below mentioned pool. If students want to take a course outside this pool, they apply to the Institute Executive Board and can take the course approved by the Institute Executive Board and count it as an elective course.

MATH 511 Algebra
This course covers selected topics in Abstract Algebra. Topics include groups, subgroups, quotient groups and homomorphisms, group actions, rings, Euclidian domains, principal ideal domains and unique factorization domains, polynomial rings.

MATH 512 Statistical Learning Theory
This course covers selected topics in statistical learning theory. Topics include concentration inequalities, PAC learning, empirical risk minimization, VC-dimension and Rademacher complexity, convex learning problems, support vector machines and kernel methods, neural networks, online learning.

MATH 513 Stochastic Optimal Control
This course introduces the students to the theory of stochastic optimal control and the solution techniques of stochastic optimal control problems. Topics include Markov decision processes, dynamic programming principle, partially observed Markov decision processes, Linear quadratic Gaussian problem and Kalman filtering, discounted and average cost Markov decision processes, team decision theory and decentralized control.

MATH 514 Information Theory
This course introduces the fundamental concepts of information theory. Topics include measures of information for discrete-alphabet systems, fixed-length lossless source coding, variable-length lossless source coding, channel coding, measures of information for continuous-alphabet systems, rate-distortion theory.

MATH 515 Stochastic Analysis
This course covers some of the fundamental topics of stochastic analysis. Topics include construction and properties of Brownian motion, martingales in continuous time, stochastic integration, Itô’s formula, stochastic differential equations, changes of measure and Girsanov’s theorem.

EE 501 Linear Systems
The students will learn the principles of continuous or discrete-time linear systems, simple estimation with and control of linear systems and analysis of dynamic linear systems with inputs and outputs. Topics include QR factorization, least squares, least norm, their applications, analysis of autonomous linear systems and systems with inputs and outputs.

EE 503 Stochastic Processes
The students will learn the principles of probability, random variables, and stochastic processes. Topics include principles of probability, random variables, expectation, maximum likelihood and maximum a-posteriori probability detection, minimum mean square error estimation, convergence and limit theorems, random processes, Markov chains, and queuing theory.

EE 525 Machine Learning
Topics include linear regression and classification concepts and techniques, classification using neural networks, Gaussian mixture models and Expectation Maximization algorithm, principal component and factor analysis, support vector machines and multi-classifier methods.

IE 501 Linear Programming and Extensions
This course aims to provide students with a sound theoretical background on linear optimization and its extensions in other optimization areas. It builds upon previously acquired introductory knowledge on Linear Programming (LP), which includes development of LP models and the workings of the Simplex Algorithm. Students develop a deeper understanding of the mathematical underpinnings of the Simplex Algorithm and linear optimization in general. Topics include optimality conditions, duality theory, and methods for large scale optimization. Students also practice with using CPLEX, a state-of-the-art optimization software, through a project of their choice.

IE 502 Integer Programming
The aim of this course is to introduce both the practice and the theory of integer programming. The course covers integer programming’s scope and applicability, MILP models, linear inequalities and polyhedral, split inequalities, intersection cuts, valid inequalities, lift-and-project procedure, Benders’ decomposition, and enumeration.

IE 522 Mathematics of Operations Research
The aim of this course is to introduce students the mathematical sophistication of operations research. Topics to be covered include, methods of proof, convex analysis, sets and functions, metric spaces, and fundamentals of linear algebra. Students will be familiarized with several basic mathematical concepts that are utilized in the fields of engineering, management science and finance.

IE 532 Stochastic Models
This course provides an introduction to stochastic modeling. It covers various topics including conditional probability and expectation, discrete and continuous time Markov chains, and queuing theory. This is a Ph.D. level course and some background on probability is required. The focus of the course is more on thinking probabilistically on operations research/management science related applied problems, rather than a measure theoretic introduction.

IE 543 Optimization Under Uncertainty
This course is based on modeling uncertain mathematical optimization problems and solving these problems using exact or approximation methods. The course mainly addresses robust and stochastic optimization modeling and solution methodologies. Solution methods (exact or approximation) are coded in the computer environment using, MATLAB, YALMIP, and CPLEX.

IE 562 Game Theory
Game theory is a mathematical tool developed to understand not only economical market participants’ interactions but also social phenomena observed as a result of these interactions. The aim of this course is to introduce the students the analytical approaches to investigate the strategic interactions of the participants. Topics to be covered include, utility concept, games in normal form, dominance, Nash equilibrium, pure and mixed strategies, extensive form games, sequential games, games under asymmetric information. Students will be familiarized with the applications of the basic concepts in the fields of engineering, management and economics via case studies.

IE 581 Data Mining
The purpose of this course is to supplement engineering students with the basic knowledge of data mining techniques. These techniques will include descriptive ones such as clustering, association analysis and sequence analysis and, predictive ones such as decision trees and logistic regression. The theoretical lectures will be coupled by applied studies where the necessary skills for using a data mining software package will also be given. By the end of the course, the students will be able to identify the real life problems where a data mining approach will be useful and apply some of the alternative techniques that can be used to solve those problems.

ME 512 Finite Element Analysis and Engineering Applications
The basic mathematical theory of the finite element method is covered with related computational algorithms and computer implementation details. It is intended primarily for graduate students interested either in developing skills in the numerical solution of engineering problems, or in developing their basic skills in the finite element methodology. The course will emphasize the solution of real life problems using the finite element method underscoring the importance of the choice of the proper mathematical model, discretization techniques and element selection criteria. Finally, students will learn how to judge the quality of the numerical solution and improve accuracy in an efficient manner by optimal selection of solution variables.

The course material described above is complemented by a balanced set of theoretical and computational assignments.
ME 518 Computational Fluid Dynamics

This course provide students:

• Examples of contemporary applications of CFD in engineering and scientific practice,
• Physical and mathematical fundamentals of CFD,
• Computational strategies for solving fluid dynamics problems,

Application of STAR-CCM+ CFD software package for the solution of engineering problems.

ME 522 Advanced Fluid Mechanics
After successful completion of the course, the learner is expected to be able to:

• Know the fundamental theoretical, experimental and numerical methods (approaches) used in the field of fluid mechanics
• Analyze fluid dynamics problem systematically by using appropriate methods
• Design a set-up for experimental or numerical flow investigations
• Have insight into the boundary layer flows, turbulent flows, multi-phase flows and micro-fluid mechanics

Analyze, present and communicate the results of a complex engineering problem.

ME 589 Advanced Engineering Mathematics
This is an advanced course on engineering mathematics. Topics include mathematical modeling of nonlinear systems and linearization, batch processes, data-driven mathematical modeling, numerical mathematics, Kalman filter-based data processing, optimality, and vector calculus. Students make use of these techniques for their projects via MATLAB and Simulink.

CS 566 Introduction to Deep Learning
This course discusses fundamental deep learning subjects such as convolutional neural networks, supervised classification, logistic regression, cross entropy. It further analyzes numerical stability of training, its importance on measuring performance. Course also applies fine tuning performance parameters and introduce various methods for fast convergence of training.