Graduate Program Course List

Compulsory Departmental Courses

Elective Courses

Service Courses Taken From Other Departments

Compulsory Departmental Courses

MATH500 - Graduation Project
Introduction to the topic, Finding sources of information. Study new definitions and concepts. Learning theoretical grounding. Giving relevant examples and solving related problems. Presentation of the subject in an informative scientific style using modern text formats (TEX, Word, WordScientific, etc.) Submission of the report to the advisor. Presentation of the report.

MATH541 - Algebra
Groups: quotient groups, isomorphism theorems, direct products, finitely generated abelian groups, actions, Sylow theorems, nilpotent and solvable groups; rings: ring homomorphisms, ideals, factorization in commutative rings, rings of quotients, polynomial rings; modules: exact sequences, vector spaces, tensor products; fields: field extensions, the fundamental theorem of Galois theory, splitting fields, the Galois group of a polynomial.

MATH587 - Applied Mathematics
Calculus of variations: Euler-Lagrange equation, the first and second variations, necessary and suffcient conditions for extrema, Hamilton's principle, and applications to Sturm-Liouville problems and mechanics; integral equations: Fredholm and Volterra integral equations, the Green’s function, Hilbert-Schmidt theory, the Neumann series and Fredholm theory and applications.

MATH589 - Graduation Seminar
Presentation involving current research given by graduate students and invited speakers.

MATH597 - Master's Thesis

Elective Courses

MATH521 - Numerical Analysis I
Matrix and vector norms, error analysis, solution of linear systems: Gaussian elimination and LU decomposition, condition number, stability analysis and computational complexity; least square problems: singular value decomposition, QR algorithm, stability analysis; matrix eigenvalue problems; iterative methods for solving linear systems: Jacobi, Gauss-Seidel and Relaxation methods, conjugate gradient type methods, convergence analysis.

MATH522 - Numerical Analysis II
Iterative methods for nonlinear equations and nonlinear systems, interpolation and approximation: polynomial trigonometric, spline interpolation; least squares and minimax approximations; numerical differentiation and integration: Newton-Cotes, Gauss, Romberg methods, extrapolation, error analysis

MATH524 - Finite Difference Methods for Partial Differential Equations
Finite difference method, parabolic equations: explicit and implicit methods, Richardson, Dufort-Frankel and Crank-Nicolson schemes; hyperbolic equations: Lax-Wendroff, Crank-Nicolson, box and leap-frog schemes; elliptic equations: consistency, stability and convergence of finite different methods for numerical solutions of partial differential equations.

MATH542 - Algebraic Number Theory
Integers, norm, trace, discriminant, algebraic integers, quadratic integers, Dedekind domains, valuations, ramification in an extension of Dedekind domains, different, ramification in Galois extensions, ramification and arithmetic in quadratic fields, the quadratic reciprocity law, ramification and integers in cyclotomic fields, Kronecker-Weber theorem on abelian extensions, Dirichlet’s theorem on the finiteness of the class group, Dirichlet’s theorem on units, Hermite-Minkowski theorem, Fermat’s last theorem.

MATH543 - Group Theory I
Review of elementary group theory, groups of matrices, normal closure and core of a group, group actions on sets, the wreath product of permutation groups, decompositions of a group, series and composition series, chain conditions, some simple groups, Sylow's theorem, the simplicity of the projective special linear groups, solvable groups and nilpotent groups, locally finite groups, locally nilpotent groups.

MATH545 - Introduction to Authenticated Encryption
Fundamentals of cryptography, block ciphers, DES, AES competition, authentication, mode of operations, cryptographic hash functions, collision resistance, birthday paradox, Merkle Damgard construction, MD5, SHA-1, SHA-3 competition, Keccak, authenticated encryption, CAESAR competition, success probability of cryptanalytic attacks, LLR method, hypothesis testing, randomness testing.

MATH547 - Algebraic Geometry
Affine spaces, the Hilbert's basissatz, the Hilbert's nullstellensatz, the Zariski's topology, irreducible sets, algebraic varieties, curves, surfaces, sheafs, ringed spaces, preschemes, affine schemes, the equivalence between affine schemes and commutative rings, projective varieties, dimension, singular points, divisors, differentials.

MATH555 - Bernstein polynomials
Uniform continuity, uniform convergence, Bernstein polynomials, Weierstrass approximation theorem, positive linear operators, Popoviciu theorem, Voronovskaya theorem, simultaneous approximation, shape-preserving properties, De Casteljau algorithm, complex Bernstein polynomials, Kantorovich polynomials.

MATH556 - Applied Functional Analysis
Review on basics in functional analysis, spectral theory of self-adjoint operators in Hilbert spaces, semigroups of operators and their applications to evolution equations, optimal control in Banach spaces.

MATH562 - Theory of Differential Equations
IVP: existence and uniqueness, continuation and continuous dependence of solutions; linear systems: linear (non)homogeneous systems with constant and variable coefficients; structure of solutions of systems with periodic coefficients; higher order linear differential equations; Sturmian theory, stability: lyapunov (in)stability, Lyapunov functions; Lyapunov's second method; quasilinear systems; linearization; stability of an equilibrium and stable manifold theorem for nonautonomous differential equations. 

MATH563 - Difference Equations
The difference calculus, first order linear difference equations, second order linear difference equations, the discrete Sturmian theory, Green’s functions, disconjugacy, the discrete Riccati equation, oscillation, the discrete Sturm-Liouville eigenvalue problem, linear difference equations of higher order, systems of difference equations.

MATH564 - Impulsive Differential Equations
General description of IDE, systems with impulses at fixed times, systems with impulses at variable times, discontinuous dynamical systems, general properties of solutions, stability of solutions, adjoint systems, Perron theorem, linear Hamiltonian systems of IDE, direct Lyapunov method, periodic and almost periodic systems of IDE, almost periodic functions and sequences, bounded solutions of nonhomogeneous linear systems, integral sets of quasilinear systems.

MATH565 - Dynamic Systems on Time Scales
Differentiation on time scales, integration on time scales, the first-order linear differential equations on time scales, initial value problem, the exponential function on time scales, the second-order linear differential equations on time scales, boundary value problem, Green’s function, the Sturm-Liouville eigenvalue problem..

MATH571 - Topology
Topological spaces, homeomorphisms and homotopy, product and quotient topologies, separation axioms, compactness, connectedness, metric spaces and metrizability, covering spaces, fundamental groups, the Euler characteristic, classification of surfaces, homology of surfaces, simple applications to geometry and analysis.

MATH572 - Differentiable Manifolds
TTopological manifolds, differentiable manifolds, tangent and cotangent bundles, differential of a map, vector fields, submanifolds, tensors, differential forms, orientations on manifolds, integration on manifolds, Stokes's theorem.

MATH573 - Algebraic Topology
The fundamental group, covering spaces, the singular homology, the cellular homology, the simplicial homology, cohomology, universal coefficient theorems for homology, cup product and cross productand cohomology, the Künneth formula.

MATH574 - Riemannian Geometry
Review of differentiable manifolds and tensor fields, Riemannian metrics, the Levi-Civita connections, geodesics and exponential map, curvature tensor, sectional curvature, ricci tensor, scalar curvature, Riemannian submanifolds, the Gauss and Codazzi equations.

MATH575 - Calculus on Manifolds
Euclidean spaces, manifolds, the tangent spaces, vector fields, differential forms, integration on manifolds, Stokes’ theorem.

MATH576 - Differential Topology
Manifolds and differentiable structures, tangent space, vector bundles, immersions, submersions, embeddings, transversality, the Sard’s theorem, the Whitney’s embedding theorem, the exponential map and tubular neighborhoods, manifolds with boundary, the Thom’s transversality theorem.

MATH577 - An Introduction to Low Dimensional Topology
Knots, links and their invariants, Seifert surfaces, braids, mapping class groups, Heegaard decompositions, lens spaces and surface homeomorphisms, surgery of 3-manifolds, branched coverings.

MATH580 - Special Studies I
Conducting literature survey on the research subject. Understanding the latest developments in literature on the chosen subject. Writing and presentation of a report on the subject.

MATH581 - Special Studies II
Conducting literature survey on the research subject. Understanding the latest developments in literature on the chosen subject. Writing and presentation of a report on the subject.

MATH582 - Approximation Theory
Uniform convergence, uniform approximation, Weierstrass approximation theorems, best approximation, Chebyshev polynomials, modulus of continuity, rate of approximation, Jackson’s theorems, positive linear operators, Korovkin’s theorem, Müntz theorems.

MATH584 - Nonlinear Problems in Applied Mathematics
Boundary value problems for nonlinear second order ordinary differential equations on finite intervals, reducing nonlinear boundary value problems to a fixed point problem, application of the Banach and Schauder fixed point theorems, boundary value problems for nonlinear difference equations, application of the Brouwer fixed point theorem, positive solutions of nonlinear boundary value problems.

MATH585 - Mathematical Modelling
Modeling with first order differential equations: radioactivity, rate of growth and decay; single-species population models, a heat flow model, modeling RL and RC electric circuits; modeling with second order DEs: the motion of a mass on an elastic spring, modeling RLC electric circuits, diffusion models; modeling with systems of DEs: multiple-species population models, model of an electrical network; modeling with partial DEs: vibrating string, vibrating membrane, modeling of heat flow.

MATH591 - Analytical Probability Theory
Definition and properties of probability, conditional probability and independence, random variables, probability distributions, their types, classical distributions, moments, random vectors, independent random variables, moment-generating and characteristic function, sums of independent random variables, limit theorems.

Service Courses Taken From Other Departments

MDES600 - Research Methodology and Communication Skills
The principles and practice of research design and data collection. Research methodology, experimental design, career options, professional ethics and academic integrity, and oral and written presentation techniques. The core of the course will be a structured, supervised "mini-research project" on a topic of the student's choice. Students will be required to perform a literature survey on their topic, construct a research proposal that includes an experimental design, and write a paper summary in the style of a formal scientific paper. Incremental deliverables will provide structure and feedback over the course of the semester. Each student will be required to find an outside reader for their literature survey and final paper. Additional assignments will include participating in class discussions, giving in-class presentations, preparing a research portfolio (including a CV), and creating a personal website.