Graduate Program Course Offerings

Approximately 35 courses that constitute the department's regular graduate curriculum are offered either annually or biennially. They are supplemented by an ongoing sequence of special-topic courses reflecting research interests of the faculty.

Courses in the 1000 series are advanced undergraduate courses that are frequently suitable for graduate credit. Those in the 3000 series are advanced graduate courses. Course content, prerequisites, frequency of offering, and requirements may change from year to year.

Selected 1000-Level Undergraduate Courses

Class Course Description Course prerequisites
MATH 1010
Putnam Seminar
The aim of this course is to develop the capacity to solve mathematical problems involving a substantial element of ingenuity and perseverance.  Training will involve the study of problems from previous Putnam competitions, for which this course can be regarded as a useful preparation. An attempt will be made to look for unifying mathematical ideas.  General strategies for solving problems will also be discussed. Instructor Consent
MATH 1020 Applied Elementary Number Theory This course will reveal the key role played by number theory in the development of mathematics.  Some applications of number theory will be covered in the course.


MATH 0430

MATH 1025
Introduction to Mathematical Cryptography

The course covers the theoretical underpinnings of cryptosystems and the analysis of their limitations and vulnerabilities. Special emphasis will be placed on public key cryptosystems, including elliptic curve based systems. Real world applications such as browser security and bitcoin will be discussed. MATH 0430
MATH 1050 Combinatorial Mathematics Topics covered include the binomial theorem, inclusion exclusion principle, recurrence relations, generating functions, and coloring problems. MATH 0413 or 0450 or 1185
MATH 1070 Classical Numerical Analysis This course, with MATH 1080 forms a two term introduction to numerical analysis at the advanced undergraduate level and includes interpolation, numerical differentiation and integration, solution of non-linear equations, numerical solution of systems or ordinary differential equations, and additional topics as time permits.  Emphasis is on understanding the algorithms rather than on detailed coding, although some programming will be required. MATH 0240 and some programming experience
MATH 1075
Topics in Numerical Analysis
This course provides honors recognition to students who wish to digress with the professor to study extra topics.  
MATH 1080 Numerical Linear Algebra This course, with MATH 1080 forms a two term introduction to numerical analysis at the advanced undergraduate level and includes interpolation, numerical differentiation and integration, solution of non-linear equations, numerical solution of systems or ordinary differential equations, and additional topics as time permits.  Emphasis is on understanding the algorithms rather than on detailed coding, although some programming will be required. MATH 0240, MATH 0280 OR MATH 1180 and some programming experience
MATH 1100
Linear Programming
Topics covered will include linear programming problems, the simplex method, quality, revised simplex method, and the transportation problem. MATH 0280 or MATH 1180
MATH 1101
An Introduction to Optimization
This course introduces students to the techniques of optimization. Applications will be emphasized, but some theory will be addressed and proofs will be discussed.  As well, students will be taught how to use available software to answer questions. Course topics will include linear programming, integer programming, nonlinear programming, convex and affine sets, convex and concave functions, unconstrained optimization, and combinatorial optimization (i.e. Network flow problems). MATH 0240 and one of MATH 0280, 1180
MATH 1110 Industrial Numerical Analysis This course is concerned with the approximate numerical solution of problems which arise in an industrial environment.  Topics covered include physical interpretation of a mathematical model, use of library software, preparation of software, analysis of results, and reporting on findings. One of MATH 0280, 1180, 1185
MATH 1119
Applied Probability for Actuarial Math
This course covers standard topics in probability and their applications to actuarial risk management. It prepares students for the probability exam offered by the Society of Actuaries.  
MATH 1120 
Actuarial Math 1
This course will cover the material listed in the syllabus for Exam 2 (Mathematics of Finance) of the Society of Actuaries. Specifically it will present the relevant topics in the theory of interest (interest and discount rates, cash flows, annuities, amoritization and sinking funds, bonds) and investment (stocks, capital asset pricing model, arbitrage pricing theory, portfolios, options). The material will be presented in the traditional academic format of lectures and help sessions along with optional sessions directed specifically at preparing students for the SOA exam.  MATH 0220 AND 0230
MATH 1121
Actuarial Math 2
This course will cover the material listed in the syllabus for exam m (3) (mathematics of life contingencies and financial economics) of the society of actuaries. Specifically it will present the relevant topics in life insurance and life annuities, including multiple decrement models as well as the black and Scholes pricing of derivative securities and risk analysis. The material will be presented in the traditional academic format of lectures and help sessions along with optional sessions directed specifically at preparing students for the SOA exam. MATH 0240, MATH 1120, STAT 1151
MATH 1122 
Actuarial Math 3
Life Contingencies MATH 1121
MATH 1123
Actuarial Math 4
Life Contingencies, part 2 Math 1122
MATH 1180 Linear Algebra 1

This course stresses the theoretical and rigorous development of linear algebra.  Major topics include the theory of vector spaces, linear transformations, matrices, characteristic polynomials, bases and canonical forms.  Other topics may be covered as time permits. Credit will not be given for both MATH 0280 and 1180.

MATH 0413
MATH 1240 Linear Algebra 2 This second course in linear algebra features an abstract development of the subject.  Abstract vector spaces, linear transformations, and matrix representations will be studied. Some applications and generalizations will also be investigated.  
MATH 1250 Abstract Algebra In this course the basic algebraic systems, groups and rings are studied in some detail.  Topics include:  subgroups, permutation groups, homomorphism's, subrings, ideals and quotient rings.  The emphasis is on theory with examples. MATH 0430
MATH 1270 Ordinary Differential Equations 1

This course covers methods of solving ordinary differential equations which are frequently encountered in applications. General methods will be taught for single n-th order equations, and systems of first order nonlinear equations.  This will include phase plane methods and stability analysis. Computer experimentation will be used to illustrate the behavior of solutions of various equations. Credit will not be given for both MATH 0290 and 1270.

one of MATH 0280, 1180, or 1185

MATH 1280 Ordinary Differential Equations 2 This is a course in stability and qualitative methods for analyzing ordinary differential equations which arise in realistic models.  Phase plane techniques, perturbation methods, and bifurcation theory are studied. MATH 1270
MATH 1310 Graph Theory The concept of a graph and the study of its theoretical properties and applications form the core of this course. Topics include paths, circuits, trees, planar graphs, coloring problems, digraphs, matching theory, and network flows. MATH 0413
MATH 1330 Projective Geometry Topics for this course are the algebraic, axiomatic, and/or synthetic development of projective geometry. MATH 1180 or MATH 1185
MATH 1350 Introduction to Differential Geometry Possible topics are the basic ideas of topology, description of curves in space, definition and local study of smooth surfaces in Euclidean space (fundamental forms, geodesics, and curvature), global properties of surfaces, gauss-bonnet formula and applications. MATH 0240 plus one of MATH 1180, MATH 1185
MATH 1360 Modeling in Applied Mathematics 1 This course introduces some of the fundamental approaches of applied mathematics.  The emphasis is on the model-building process and on developing an understanding of some of the unifying themes of applied mathematics such as equilibria, stability, conservation laws, etc.  The material is presented in the form of case studies. Math 0290 or Math 1270 or Math 1275

MATH 1365
Topics in Mathematical Modeling

This course provides honors recognition to students who wish to digress with the professor to study extra topics not covered in mathematics 1360, mathematical modelling.  
MATH 1370 Modeling in Applied Mathematics 2 This course presents contemporary mathematical theories of neuroscience, including single neurons and neuronal networks.  Attention will be given to the dynamics and the function of neural activity. MATH 0230 and some programming experience
MATH 1380
Math Biology
This course will provide a broad introduction to mathematical methods typically applied to problems in biology.  Models using calculus, ordinary differential equations, partial differential equations, discrete dynamical systems, stochastic dynamics, or a cellular automata framework will be presented and principal methods for their analysis will be described.  Computational methods will also be covered, including computing platforms such as XPPAUT.  Throughout the course, students will have extensive opportunities to practice the development and analysis of mathematical biology models. MATH 0280 or MATH 1180 plus MATH 0290 or MATH 1270
MATH 1410 Introduction to the Foundations of Mathematics 1 This course introduces the logical foundations of mathematics; it covers the propositional and predicate calculi, formal number theory, set theory, and beginning model theory. Math 0413 or Math 0450
MATH 1420 Introduction to the Foundations of Mathematics 2 Course is devoted to model theory and mathematical applications.  Items to be covered are:  completeness and compact ness theorems, and the Lowenheim-Skolem-Tarski theorems, ultra-products and ultra-limits, independence results, standard and nonstandard models, Categoricity in power, universal algebra, and Galois connections.  
MATH 1470 Partial Differential Equations 1 This is the first term of a two-term sequence in elementary PDE's.  The objectives of the course are to provide students with the techniques necessary for the formulation and solution of problems involving PDE's and to prepare for further study in PDE's.  The three main types of second order linear PDE's - parabolic, elliptic, and hyperbolic are studied.  In addition the tools necessary for the solution of PDE's such as Fourier series and Laplace transforms are introduced. MATH 0240 and one of MATH 0290 or 1270
MATH 1480 Partial Differential Equations 2 This is the second term of a two term sequence in PDES. Topics include Fourier transform, maximum principles, and existence, uniqueness and regularity of solutions to PDES. MATH 1470
MATH 1510
Mathematical Theory of Probability
This course is an introduction to the mathematical theory of probability.  Major topics include random variables, expecation, characteristic functions, conditional probability, and an introduction to Martingales and Markov chains.

Math 0420 or Math 0450, and one of 280, 1180 or 1185

MATH 1530 Advanced Calculus 1 This course contains a rigorous development of the calculus of functions of a single variable, including compactness on the real line, continuity, differentiability, integration, and the uniform convergence of sequences and series of functions.  Other topics may be included, such as the notion of limits and continuity in metric spaces. MATH 0420 or MATH 0450
MATH 1540 Advanced Calculus 2 This course, a continuation of MATH 1530, covers the theory of limits, differentiation, and integration of functions of several variables.


MATH 1530
MATH 1550
Vector Analysis and Applications
Topics covered include:  vector algebra, vector differentiation and integration, divergence, gradient, curl, the theorems of green, gauss and stokes, and curvilinear coordinate systems.  There will be an emphasis upon problem solving and applications in electromagnetic theory and fluid flow. MATH 0240 plus MATH 0280 or MATH 1180
MATH 1560 Complex Variables and Applications This course covers the following topics:  elementary operations with complex numbers, derivatives, integrals, Cauchy's theorem and consequences such as the integral formula, power series, residue theorem, applications to real integrals and series. MATH 0240 (with B or better) or MATH 1550
MATH 1570 Introduction to Fourier Analysis The course is a rigorous introduction to Fourier series and integrals with applications to heat flow, wave motion, physics, and number theory.  It is intended for students with a basic knowledge of real analysis including uniform convergence of sequences and series of functions.  No knowledge of the Lebasque integral is assumed. One of Math 0420, Math 0450 AND one of Math 0280, Math 1180, Math 1185
MATH 1651
Computer Methods Laboratory
This course will introduce students to the use of the computers in our undergraduate computing laboratory and to the numerical methods, symbolic algebra, and computer graphics which are helpful in the study of our upper level subjects in applied mathematics.  
MATH 1700 Introduction to Topology The topology of r1, as well as that of general metric spaces, will be studied.  Basic notions will be applied to obtain the fundamental existence theorem for first order ordinary differential equations.  The course will be run on a theorem proving and problem solving basis. MATH 0420 or MATH 0450
MATH 1801 Advanced Topics in Mathematics This is a topics course at the advanced undergraduate level. The topic will change each time the course is offered, and will generally reflect current interests of the faculty or a recent trend in mathematics.  

Graduate Course Listing

Class Course Description Course prerequisites
MATH 2000 Research and Thesis for the Master's Degree (1-15 Credits)

This course involves directed research and writing leading towards the completion of a Master's thesis.

MATH 2010 Teaching Orientation (1 Credits)

This course is for Teaching Assistants in the Department of Mathematics. The course emphasizes techniques; procedures and discussions, which prepare the TA to successfully, manage recitations and teach classes in Mathematics.

MATH 2020 Progress in Mathematics

This course will deepen the students' understanding of analysis through intensive training in problem solving followed by comprehensive study and dissection of the problems attempted.  Students preparing for the analysis portion of the preliminary exam are strongly encouraged to enroll.

MATH 2030 Iterative Methods for Linear and Nonlinear Systems (3 Credits)

Topics include matrix theory, matrix and vector norms, error analysis, factorizations, direct and iterative methods for solving linear and nonlinear systems, least squares, and the algebraic eigenvalue problem.

MATH 2050 Graph Theory Basic concepts and definitions from the theory are studied. In particular, enumerative problems are selected by the instructor.  
MATH 2055 Codes and Designs A unified theory of linear codes, combinatorial design and statistical design will be presented.  Hamming, BCH, Golay, quadratic residue and other classes of codes will be constructed. Several decoding schemes, including decoding by way of locator polynomials, will be done.  Applications to statistical design and quality control will be described in detail.  
MATH 2060 Combinatorics (3 Credits)

Topics in this course vary with the instructor's research interests. Focus will be placed on algebraic coding theory, construction of new nonlinear codes, Mobius inversion on posets, and symmetric functions. Techniques used involve ideals in polynomial rings, generating functions, and algebraic number rings. Basic knowledge of groups, fields and rings is highly desirable.

MATH 2070 Numerical Methods in Scientific Computing 1 (4 Credits)

This course is an introduction to practical numerical methods for science and engineering. The course is complemented with a fully integrated computer laboratory, where you learn to use available software and to implement your own solution methods. Topics include: roundoff errors and stability analysis, root finding for nonlinear equations, interpolation, approximation of functions and numerical integration. The techniques presented are frequently used to deal with problems in physics, chemistry and engineering. The lecture introduces a numerical method and elaborates on its applicability and expected behavior. Frequently you will be assigned a related laboratory exercise. Registration for the lab is required.

MATH 2071 Numerical Methods in Scientific Computing 2 (4 Credits)

The sequence M2070-M2071 gives an in-depth introduction to the basic areas of numerical analysis. The courses will cover the development and mathematical analysis of practical algorithms for the basic areas of numerical analysis. Math 2071 includes treatment of the topics of numerical linear algebra and numerical methods for differential equations. The course M2071 does not assume a knowledge of M2070; and material from M2070 that is needed in M2071 will be reviewed as necessary. These courses also include a Computational Laboratory that complement the lectures.

MATH 2090 Numerical Solution of Ordinary Differential Equations (3 Credits)

This course is an introduction to modern methods for the numerical solution of initial and boundary value problems for systems of ordinary differential equations and differential algebraic equations.  Numerical methods and their theory, including convergence and stability considerations, order and step size selection and the effects of stiffness are discussed. 

MATH 2160 Set Theory This is an introductory course intended to prepare the students for the various applications of set theory.  The contents include basic axioms, Boole algebra of classes, algebra of relations, ordinals, transfinite induction, equivalents of the axiom of choice, and of the continuum hypothesis.  
MATH 2170 Logic and Foundations The contents of this course include the propositional calculus, the predicate calculus, model theory and their applications to mathematical systems.  
MATH 2180 Introduction to Fractal Geometry This course will give an introduction to fractal geometry. Topics include metric spaces, measures, fractal dimensions, discrete dynamical systems and multifractal analysis.  
MATH 2190 Functions of Several Variables This course covers topics of the calculus of several variables from a more general and theoretical point of view. Topics include convergence, continuity, compactness, inverse and implicit function theorems and differential forms.  
MATH 2200 Real Analysis 1 This is the basic real analysis course for graduate students.  In particular, the theory of Lebasque integral and some of its many ramifications are studied.  This leads to differentiation and integration theories for real valued functions.  
MATH 2201 Real Analysis 2 This course is a continuation of 2200.  Topics include elements of the theory of Banach and Hilbert spaces, Radon Nikodym theorem, duality for the LP spaces, product measures, Fubini's theorem and differentiation.  
MATH 2210 Complex Analysis 1 This is the basic first course in complex analysis.  Topics include analytic functions, Cauchy-Riemann equations, elementary conformal mappings, Poisson formula, Taylor and Laurent series, argument principle, residues and the classifications of singularities.  
MATH 2211 Complex Analysis 2 This course is a continuation of 2210.  The major topics are normal families, the Riemann mapping theorem and harmonic functions.  Selected topics, by the instructor, will also be covered.  
MATH 2219 Dynamical Systems (3 Credits)

This course will introduce the student to new concepts from dynamical systems.  Invariant manifolds, normal form; Bifur cations and chaos will be discussed from the geometric point of view.  Some global analysis will also be described.  Numerical tools and methods for analyzing local bifurcations will also be discussed.  Emphasis will be on practical applications of these techniques.

MATH 2240 Analytic Number Theory Some of the topics covered in this course include residue classes, unique prime factorization, character mod n, the Riemann zeta function and its analytic continuation, poles and functional equations.  Also the prime number theorem and Dirichlet theorem will be covered.  
MATH 2245 Algebraic Number Theory In this course the theory of numbers will be algebraically, that is, as the study of algebraic numbers.  Particular attention will be paid to the development of quadratic and cyclotomic number fields and to factorization theorems. Other topics include ideal theory and the work of Kummer and Fermat's last theorem.  
MATH 2260 Potential Theory This is an elementary introduction to the subject.  Topics include harmonic functions, Poisson integral, maximum principle, classical Dirichlet problem, Harnack's inequality, boundary limit theorem, super harmonic functions and their properties and green's potentials.  
MATH 2280 Hardy Spaces A brief introduction to HP space theory in the disk.  Topics include Poisson formula, boundary behavior and Fatou's theorem, factorization theorems and Blaschke products.  
MATH 2301 Analysis 1 (3 Credits)

This course is an introduction to Real Analysis/Measure Theory with some Functional Analysis. Topics include: Lebesgue Measure and Integral; some Hilbert and Banach space theory; Monotone Convergence Theorem; Lebesgue's Dominated Convergence Theorem; Fatou's Lemma; Jensen, Holder and Minkowski Inequalities; absolutely continuous measures; the Radon-Nikodym Theorem; the Riesz Representation Theorem for L^p; the Hahn Decomposition Theorem; the Fubini-Tonelli Theorem.

The Mathematical Analysis and Linear Algebra material in the Math Preliminary Exams syllabus (and in particular, the material in the U. Pitt. Math courses 1530 Advanced Calculus 1 and 1540 Advanced Calculus 2, as well as the graduate linear algebra classes 2370 and 2371) is assumed. Note that many graduate math courses implicitly assume that students are familiar with a wide range of undergraduate math courses and ideas: such as basic Set Theory and basic Topology; - and 2301 Analysis 1 is a graduate math course of this type.

MATH 2302 Analysis 2 (3 Credits)

In this course we continue on from 2301 Analysis 1 in Fall 2016 (Lebesgue measure and integration, and some functional analysis), stirring Fourier analysis, complex analysis, functional analysis and more real analysis into the mix...  In Fourier and Functional Analysis, topics include: more on Hilbert Spaces and L^p-spaces, Fourier Series and the Fourier Transform.  In Complex and Real Analysis, topics include: Power Series, Infinite Products, Holomorphic Functions, the Cauchy Integral Formula, and the Maximum Modulus Theorem.
The main prerequisite is the course 2301 Analysis 1; or an equivalent course.  The Mathematical Analysis and Linear Algebra material in the Math Preliminary Exams syllabus (and in particular, the material in the U. Pitt. Math courses 1530, 1540, 2370 and 2371) is assumed.

MATH 2303 Analysis 3 (3 Credits)

This is a basic course in Functional Analysis. It is assumed that students are familiar with Analysis 1 (measure theory). Analysis 2 (Complex Analysis) will be used occasionally, but it will not play any important role in the course. 

Topics include (in that order):

1. Basic theory of Banach and Hilbert spaces.

2. Bounded operators in Banach and Hilbert spaces.

3. Orthonormal bases in Hilbert spaces, Fourier series and spherical harmonics.

4. Baire category theorem, Banach-Steinhaus theorem, open mapping theorem and closed graph theorem.

5. Hahn-Banach theorem, separation of convex sets.

6. Reflexive spaces.

7. Weak convergence, Mazur's lemma, Banach-Alaoglu theorem in the separable case, direct methods in the calculus of variations.

8. Weak topology, Tychonov's theorem and Banach-Alaoglu theorem in the general case.

9. Compact operators, Fredholm operators, spectrum of compact operators, Fredholm-Riesz-Schauder theory.

10. Spectral theorem for compact self-adjoint operators.

11. Sobolev spaces and the eigenfunctions of the Laplace operator.

12. Banach algebras.

In the course I will show many applications of Functional Analysis in different areas of mathematics

MATH 2304 Analysis 4 (3 Credits)

In this course we will cover fundamental concepts of Harmonic Analysis in the Euclidean spaces. The following topics will be covered: Fourier transform, tempered distributions, the Marcinkiewicz and the Riesz-Thorin interpolation theorems, maximal functions, singular integrals, Hilbert transform, Riesz transforms, Calderon-Zygmind theory, fractional Laplace operators.

MATH 2370 Matrices and Linear Operators 1 (3 Credits)

Linear transformations on finite dimensional vector spaces are studied in a semi-abstract setting.  The emphasis is on topics and techniques which can be applied to other areas, e.g., Bases and dimension, matrix representation, linear functional, duality, canonical forms, vector space decom position, inner products and spectral theory.

MATH 2371 Matrices and Linear Operators 2 (3 Credits)

The course is a continuation of Math 2370 Matrices and Linear Operators I.

Topics will include spectral theory of self-adjoint mappings, calculus of matrix valued functions, matrix inequalities, convexity, duality theorem and normed linear spaces.

MATH 2400 Functional Analysis 1 A first course in the area, the emphasis of the course will be on normed linear spaces and linear operations, their basic properties will be discussed.  Also, inner product spaces and their properties will be covered.  
MATH 2401 Functional Analysis 2 This course is a continuation of 2400.  Topics to be covered include the spectral theory, compact operators, and distribution theory and Sobolev spaces.  
MATH 2410 Harmonic Analysis 1 This is a basic first course in harmonic analysis.  The major topic is the Fourier transform and Fourier series. In particular, various kernels and pointwise sum ability of Fourier series will be discussed.  
MATH 2480 Computational Approximation Theory Topics include fundamental theorems, polynomial approximation, splines, surface approximation, domain transformations and applications to science and engineering.  
MATH 2500 Algebra 1 (3 Credits)

The course is the first term of a two-term graduate algebra sequence. It covers the theory of groups, the theory of fields as well as Galois theory. Highlights of the course will include: Sylow's theorems, the structure of finitely generated abelian groups, the fundamental theorem of Galois theory, and applications (such as the solvability of polynomial equations by radicals and geometric constructions with a ruler and a compass).

MATH 2501 Algebra 2 (3 Credits)

In this course the fundamental properties of rings, fields and modules are studied.

MATH 2503 Matrix Groups The course is an introduction to some of the concepts of lie groups and lie algebra--all done at concrete level of matrix groups.  It centers around the isomorphism questions on matrix groups of small dimensions, and it leads to maximal torus, Clifford algebra, and Weyl group.  
MATH 2505 Algebra 3 (3 Credits)

This is a third course in algebra.  It covers the classical results on the structure and representation theory of associative algebras culminating with modern developments such as the theory of quiver algebras and categorification.  Highlights of the course include: Wedderburn-Artin theory, the structure of central simple algebras, the structure of finite dimensional algebras, semisimple algebras, the character theory of finite groups, theorems of Maschke, Frobenius, Burnside, quiver algebras and quiver representations, reflection functors, Gabriel's theorem, tensor categories, fusion categories.

MATH 2506 Algebra IV (3 Credits)

This is a fourth course in algebra.  It covers introductory topics in algebraic geometry, number theory, and representation theory, selected by the instructor.

MATH 2601 Advanced Scientific Computing 1 (3 Credits)

This course studies the mathematical analysis and practical implementation of discontinuous Galerkin methods for approximating elliptic, parabolic, and hyperbolic partial differential equations.

MATH 2602 Advanced Scientific Computing 2 (3 Credits)

The course will cover several topics in discretizations of partial differential equations and the solution of the resulting algebraic systems. Topics in discretizations will include mixed finite element methods, finite volume methods, mimetic finite difference methods, and local discontinuous Galerkin methods. Topics in solvers will include domain decomposition methods and multigrid methods. Applications to flow and transport in porous media, as well as coupled fluid and porous media flows will be discussed.

Advanced Calculus, Linear Algebra, and Differential Equations
MATH 2603 Advanced Scientific Computing 3

The Advanced Scientific Computing sequence covers topics chosen at the leading edge of current computational science and engineering for which there is sufficient interest.  The course requirements consist readings, homework, a term project and its presentation. Please contact the instructor if you have questions about your  preparation.

The fall 2016 course will  consider advanced topics from the analysis and numerical analysis of fluid motion. One specific topic considered in this course will be large eddy simulation of  turbulence.

MATH 2604 Advanced Scientific Computing 4 (3 Credits)

The course focuses on the fundamental mathematical aspects of numerical methods for stochastic differential equations, motivated by applications in physics, engineering, biology, economics. It provides a systematic framework for an understanding of the basic concepts and of the basic tools needed for the development and implementation of numerical methods for SDEs, with focus on time discretization methods for initial value problems of SDEs with Ito diffusions as their solutions. The course material is self-contained.   The topics to be covered include background material on probability, stochastic processes and statistics, introduction to stochastic calculus, stochastic differential equations and stochastic Taylor expansions. The numerical methods for time discretization of ODEs are briefly reviewed, then methods for time discretization for SDEs are introduced and analyzed. 

MATH 2700 Topology 1 (3 Credits)

A first course in topology, some of the topics covered include separation axioms, bases and sub-bases, product and quotient topology, homomorphisms, compactness, the baire category theorem, the lindelof property, connectedness, topological spaces, and compactification.

MATH 2701 Topology 2 (3 Credits)

This course is a continuation of 2700. In this course, the basic concepts and results in algebraic topology will be covered, including both homotopy and homology theory. In particular, the calculation of the fundamental group and homology groups from chain complexes will be covered.

MATH 2750 General Topology The fundamental theorems of general topology will be studied in particular, those results concerning generalized metric spaces, coverings and mapping will be studied.  
MATH 2800 Differential Geometry 1 (3 Credits)

A first course in differential geometry.  Topics may include the geometry of curves and surfaces (eg. Gauss map, fundamental forms, curvature), differentiable manifolds, Lie groups, tangent and tensor bundles, vector fields, and Riemannian structures.

MATH 2801 Differential Geometry 2 (3 Credits)

This course is a continuation of Differential Geometry 1.  The initial focus will be on differential topology, covering topics such as such as Sard's theorem, transversality, degree of mappings, and differential forms and Stokes' theorem.  Further topics may include Lie groups, distributions and the Frobenius theorem, and bundles and connections.​

MATH 2810 Algebraic Geometry (3 Credits)

This course is an introduction to the basic ideas of Algebraic Geometry, the approach to the subject may be either of the following:  The linear series on a curve approach, the algebraic approach through fields of algebraic functions, or the Sheaf theoretic approach.  Applications may also be included.

MATH 2815 Discrete Geometry and Computers This course will discuss various classical problems in discrete geometry.  It will develop the theory of packings and coverings, the Kepler sphere packing theorem, the dodecahedral theorem, kissing number problems, lattice packing problems in higher dimensions, and the kelvin problem in foams.  
MATH 2900 Partial Differential Equations 1 (3 Credits)

The course covers some fundamental topics of partial differential equations, including  transport equation, Laplace's equation, heat equation, wave equation, characteristics, Hamilton-Jacobi equations, conservation laws and shock waves, some methods to represent solutions.

MATH 2901 Partial Differential Equations 2 (3 Credits)

This course will cover Sobolev spaces, second order elliptic equations, weak solutions, linear evolution equations, semigroup theory, and Hamilton-Jacobi theory, and other topics in nonlinear PDE.

PDE 1 (Math 2900) is not a pre-requisite but a good background in analysis is necessary.

MATH 2920 Ordinary Differential Equations 1 (3 Credits)

This is the first course in a two-term sequence designed to acquaint students with the fundamental ideas involved in the study of ordinary differential equations.  Basic existence and uniqueness of solutions as well as dependence on parameters will be presented.  The course will cover linear ODES and the matrix exponential, oscillations via an introduction to Poincare-Bendixson theory for planar systems and to Floquet theory, and  Sturm-Liouville problems.  Students will also be introduced to geometric concepts such as stability of fixed points and invariance.  This first term will provide an excellent introduction to ODE theory for students interested in applied mathematics.

MATH 2921 Ordinary Differential Equations 2 (3 Credits)

This course, which follows Math 2920, presents a dynamical systems approach to the study of ordinary differential equations.  Topics include geometric theory including proofs of invariant manifold theorems, flows on center manifolds and local bifurcation theory, the method of averaging, Melnikov's method, and an introduction to Smale horseshoes and chaos theory.

MATH 2930 Asymptotics and Special Functions (3 Credits)

This course is an introduction to the theory of asymptotics and special functions. Some of the functions studied are the gamma function, orthogonal polynomials, and bessel functions. This course also covers techniques for finding asymptotic expansions of integrals and of solutions to differential equations.

MATH 2930 Asymptotics and Special Functions (3 Credits)

This course covers  Hypergeometric and Confluent Hypergeometric Differential Equations and Series (e.g.  Laurent series and geometric series)   which arise  in physics and engineering.  We also study Bessel functions, the Gamma function, the Beta function, the  Riemann Zeta function, and  Laplace's asymptotic expansion of integrals depending on a parameter. The prerequisite for this course is a one undergraduate semester course in complex variables with a grade of B or higher. The grade will be determined from assigned homework problems.

MATH 2940 Applied Stochastic Methods (3 Credits)

This course will provide an overview of stochastic methods that can be applied to problems in biology, finance and physics. Analytical and computational techniques will be presented which apply to both continuous and discrete stochastic models.

MATH 2950 Methods in Applied Mathematics (3 Credits)

This course covers methods that are useful for solving or approximating solutions to problems frequently arising in applied mathematics, including certain theory and techniques relating to the spectral theory of matrices, integral equations, differential operators and distributions, regular perturbation theory, and singular perturbation theory.

MATH 2960 Computational Fluid Mechanics Topics include a review of symmetric linear systems and matrix theory, equilibrium and the calculus of variations in discrete and continuous systems, orthogonal series, networks Fourier series and convolutions, Fourier integrals, complex variables and conformal mapping.  Also some numerical methods such as the fast Fourier transform will be covered.  
MATH 2980 Projects in Financial Mathematics Representatives from business or government will present real-world problems and issues. Students will select one of the problems and generate some solutions either alone or in teams. Oral and written reports of the results will then be presented to both the posers of the problems and to the faculty and students in the professional master's program.  
MATH 2990 Independent Study (1-15 Credits)

This course is for all graduate students not under the direct supervision of a specific faculty member. In addition to a student's formal course load, this study is for preparation for the preliminary, comprehensive and overview examinations.

MATH 3000 Research and Dissertation for the PhD Degree This course is taken by a student who is working on a Ph.D. dissertation under the direction of a student's thesis advisor.  
MATH 3010 Sobolev Spaces Topics include:  definitions and basic properties, the Sobolev imbedding theorem, applications of the theory.  
MATH 3020 Calculus of Variations (3 Credits)

This course will introduce students to the subject of calculus of variations and some of its modern applications. Topics to be covered include necessary and sufficient conditions for weak and strong extrema, Hamiltonian vs Lagrangian formulations, principle of least action, conservation laws and direct methods of calculus of variations. Extensions to the functionals involving higher-order derivatives, variable regions and multiple integrals will be considered. The course will emphasize applications of these ideas to numerical analysis, mechanics and control theory.

Prerequisite(s): single-variable and multivariable calculus, some exposure to ordinary and partial differential equations. All other concepts, such as function spaces and the necessary background for the applications, will be introduced in the course. Beginning graduate students and advanced undergraduates are welcome.

MATH 3031 Network Theory The general problem discussed in flows on networks.  Topics will include some graph theory, flows and potential differences, transport problems and flow algorithms.  
MATH 3040 Topics in Scientific Computing

The course objective is to introduce students to formulating, debugging and solving finite element simulations of practical applications, with a focus on the equations of fluid flow.  Two popular freely-available computer packages will be presented: FEniCS and FreeFem++.  FreeFem++ is an integrated program, with a special language for specifying the mathematical formulation as well as integrated mesh generation and graphical postprocessing facilities.  FEniCS is less tightly integrated, consisting of a collection of functions for specifying the mathematical formulation as well as functions for interfacing with other packages for mesh generation, post processing, and numerical solution.  These functions are tied together using either the Python or C++ programming languages.  This course will focus on using Python. Python is a widely-used language with applications far removed from finite element modelling and can be the subject of multiple-semester courses. Although previous experience with Python would be valuable, it is not necessary.  The basics of the language plus those features necessary for this course will be presented during the lectures.  Previous experience with finite element methods will be valuable, but is not required because the theory will be summarized during the lectures. Applications for which FreeFem++ or FEniCS will be used include steady and transient heat conduction as well as the Stokes and Navier-Stokes equations. Various boundary conditions and finite elements will be presented, as well as the effect of these choices on solution methods.

Prerequisites include a basic knowledge of one of the following programming languages: Python, C, C++, FORTRAN, JAVA, or MATLAB; Linear Algebra and Calculus; and, at least one introductory computational/numerical analysis class, such as 1070/1080 or 2070/2071 or the equivalent.

MATH 3055 Chromatic Polynomials and Graph Structure This course will focus on the relationship between chromatic polynomials and graph structures.  Some structure properties to be considered are connectivity, chromatic number, girth, graph components, and graph isomorphism and homeomorphisms.  
MATH 3060 Topics in Combinatorics This course covers a variety of topics in combinatorics.  
MATH 3070 Numerical Solution of Nonlinear Systems This course is an introduction to modern methods for solutions of such equations.  Topics include:  algorithms for one-dimensional equations, linearization methods for systems, method of Gauss-Seidel type, quasi-Newton methods, continuation methods and elements of unconstrained minimization methods.  
MATH 3071 Numerical Solution of Partial Differential Equations (3 Credits)

This course covers contemporary methods for solving initial and boundary value problems. Topics include properly posed problems, characteristics, finite difference and finite element methods, and error estimates.

MATH 3072 Finite Element Method (3 Credits)

This course is an introduction to the theoretical and computational aspects of the finite element method for the solution of boundary value problems for partial differential equations. Emphasis will be on linear elliptic, self-adjoint, second-order problems, and some material will cover time dependent problems as well as nonlinear problems. Topics include: Sobolev spaces, variational formulation of boundary value problems, natural and essential boundary conditions, Lax-Milgram lemma, approximation theory, error estimates, element construction, continuous, discontinuous, and mixed finite element methods, and solution methods for the resulting finite element systems.

Prerequisite(s): Good undergraduate background in linear algebra and advanced calculus. Familiarity with partial differential equations will be useful.

MATH 3075 Parallel Finite Element Method This course will cover major new developments in fully parallel finite element methods.  The reasons why this is a great advantage over traditional methods will be discussed and explained.  
MATH 3215 Quasiconformal Maps 1 This course will cover the quasi-conformal and quasi-regular mappings.  They relate to several areas of mathematics and some of these applications will be covered. The methods used in the course will included analysis, topology and geometry.  
MATH 3216 Quasiconformal Maps 2 This course is a continuation of math 3215.  It will be devoted to quasi-conformal analysis in Riemannian manifolds, the so called Donaldson - Sullivan point of view.  
MATH 3225 Mathematics of Finance 1 (3 Credits)

This course provides an introduction to the mathematical subjects required for the mathematical finance program, and assumes that the student has an undergraduate degree with some technical component (e.g. Engineering, Computer Science, Math, Statistics, Physics, etc.) Students are expected to have knowledge of Multivariable Calculus and Linear Algebra, and any sections on these topics will be presented as review. Topics to be covered include: Partial Differential Equations, Stochastic Analysis, Optimization and Numerical Methods. No financial background is required, but many of the examples and llustrations of the mathematics will be drawn from economics and finance.

MATH 3226 Mathematics of Finance 2 (3 Credits)

The course with its pre-sequel MATH3225 present fundamental principles and standard approaches used in mathematical finance. We will study   continuous-time stochastic models with applications in various fields of mathematical finance including prcing and hedging financial instruments, risk management and financial decision making etc. We will cover basic portfolio theory, pricing options and other derivatives, change of numeraire, term-structure models and etc  from Volume 2 of Shreve's book "Stochastic Calculus for Finance".  This course will investigate the mathematical modeling, theory and computational methods in modern finance. The main topics will be (i) basic portfolio theory and optimization, (ii) the concept of risk versus return and the degree of efficiency of markets, (iii) discrete models in options.

MATH 3227 Mathematics of Finance 3 (3 Credits)

This course covers special topics in mathematical finance. Topics will include stochastic control theory and stochastic differential games with applications to finance.

MATH 3228 Mathematics of Finance 4 (3 Credits)

This course covers advanced topics in modern mathematical finance. Topics will include advanced credit risk and interest rate models, stochastic control and stochastic optimization models for portfolio selection and option pricing, and numerical methods.

MATH 3260 Topics in Fractal Geometry 1 Topics to be covered include Hausdorff measure, various definitions for dimensions of a set, techniques for calculating these dimensions, also the local structure projections, products and intersections of fractals will be discussed.  Further topics include self-similar sets, self-similar measures, dynamical systems, Julia sets and random fractals.  
MATH 3261 Topics in Fractal Geometry 2 This course is a continuation of MATH 3260.  Some topics which will be covered include self-similar sets, self-similar measures, dynamical systems, Julia sets and random fractals.  
MATH 3270 Iteration of Rational Maps 1 This is an introductory course on complex dynamics.  In particular, the Fatou and Julia sets for a large number of examples will be worked out.  The general properties of these two sets will be given and relations to the Mandelbrot set and fractals will be discussed.  
MATH 3271 Iteration of Rational Maps 2 This is a continuation of math 3270.  Topics will include application of quasi-conformal maps to polynomial maps, Hermann rings, quasi-conformal surgery, the local geometry of the Fatou and Julia sets and the study of the Mandelbrot set.  
MATH 3370 Mathematical Neuroscience (3 Credits)

Course covers computational and mathematical neuroscience. It will include modeling and analysis of complex dynamics of single neurons and large-scale networks using a variety of methods from applied math.  No biology is required; some familiarity with differential equations will be helpful.

MATH 3375 Computational Neuroscience (3 Credits)

This course offers an introduction to modeling methods in neuroscience. Topics range from modeling the firing patterns of single neurons to using computational methods to understand neural coding. Some systems level modeling is also done.

MATH 3380 Mathematical Biology (3 Credits)

This course describes a number of topics related to mathematical biology. This year we will cover several areas of interest including pattern formation in reaction-diffusion and advection models with applications to immunology, chemotaxis, etc; evolutionary dynamics such as the evolution of cooperation, some game theory, and replicator dynamics; and some cell physiology modeling such as the cell cycle and simple circadian models.   The prerequisites are some simple differential equations, a bit of Fourier transforms, and some knowledge of software to numerically solve the various equations.

MATH 3410 Hilbert Spaces of Entire Functions 1 This course will cover Debranges theory.  The functional Hilbert spaces will be introduced and the salient properties of reproducing kernels will be studied.  Applications of this theory including a formal proof of the Riemann hypothesis will be covered.  
MATH 3411 Hilbert Spaces of Entire Functions 2 This course is a continuation of 3410.  The material will consist of a more careful study of the work of l. Debranges in this area.  An application will be the Riemann zeta function and the Riemann hypothesis.  
MATH 3436 Fixed Points Wavelets & Fractals The course will cover iterative image reconstruction using the wavelet transform, initiated by Mallat and Zhang.  
Math 3440 Fixed Point Theory in Bananch Spaces

(3 Credits)

We will begin with an overview of basic fixed point theory in banach spaces, from the banach contraction mapping theorem and schauder's theorem through to kirk's theorem. The course will continue with topics in metric fixed point theory and its  connections to banach space geometry and topology. This will include recent work of pei-kee lin, who showed that there exists a non-reflexive banach with the fixed point property for nonexpansive mappings; and tomas dominguez benavides, who proved that every reflexive banach space can be equivalently renormed to have the fixed point property for nonexpansive mappings. We will also discuss extensions of lin's work to the function space l^1 by maria japon pineda and carlos hernandes linares. The course will further include some of my (joint) research in this area, and related research of other authors.

MATH 3450 Theory of Distributions Using the theory of topological vector scales the theory of distributions will be studied.  Some of the applications of this theory to Fourier transforms and the Paley Wiener theory will be discussed.  
MATH 3480 Topics in Spline Approximation Spline approximation is Piewise analytic approximation. Most cad-cam systems rely heavily on spline approximation. In this course topics to be covered include b-splines data fitting using splines and numerical integration and differentiation.  
MATH 3500 Topics in Algebra Symmetric spaces are manifolds admitting symmetries of a certain nature. The theory was developed by Cartan at the beginning of the last century and it constitutes the proper framework for many questions in modern mathematics. The course will cover compact, noncompact, hermitian symmetric spaces, the Cartan classification, invariant differential operators, the Darboux and Poisson equations, the  Plancherel inversion formula and the Paley-wiener theorem for the spherical transform.  
MATH 3550 Lie Groups and Lie Algebras (3 Credits)

The main goal of the course is to understand the structure and classification of complex semisimple Lie algebras as well as their basic representation theory and the relationship with Lie groups. Highlights will include, the theorems of Engel, Cartan and Weyl, root systems, the Harish-Chandra isomorphism and various formulas for characters and weight multiplicities.

MATH 3600 Topics in Pure Mathematics

This course covers special topics in pure mathematics.  The subject matter varies each semester.  Topics may include the theory of modular forms, automorphic representations, Galois representations, class field theory, trace formula, special topics in algebraic geometry, the Langland's program, discrete and algorithmic geometry, motivic integration, and the formalization of pure mathematics.

MATH 3750 General Topology 2 This is a second course in general topology in which the fundamental questions and open problems in the area are discussed.  In particular, generalized metri-spaces, covering theorems and mapping of spaces will be covered.  
MATH 3760 Topics in Topology 1 (3 Credits)

Cohomology is an important concept and tool in various areas of pure mathematics, such as topology, differential geometry, algebraic geometry, and representation theory. This course will start with rational and integral cohomology and then move to survey generalizations such as: topological and algebraic K-theory and elliptic cohomology. We will also describe equivariant and twisted versions. Along the way many techniques and tools will be explained, such as: spectral sequences, mapping spaces, homotopy computations, classification of bundles, topology of Lie groups and of their classifying spaces. As time permits, the associated higher geometric and categorical structures will also be discussed.

MATH 3761 Topics in Topology 2
The course will be concerned with topics of current research activity in analytic topology, especially in the areas of generalized metric spaces and topological algebra.
MATH 3900 Graduate Internship (1-9 Credits)

Internship and/or employment experience under the supervision and oversight of a faculty member. This experience is to be an integral part of the students individual course of study.

MATH 3902 Directed Study (1-9 Credits)

This course is for students normally beyond their first year of graduate study who wish to study in an area not available in a formal course. The work must be under the direct supervision of a faculty member who has approved the proposed work in advance of registration. A brief description of the work should be recorded in the student's file in the department.

MATH 3920 Nonlinear Methods in Differential Equations This course covers functional analytic methods in the theory of differential equations, including applications of the fixed point theorem and the implicit function theorem to existence results.  It also presents an introduction to bifurcation theory and discusses numerical problems, schemes.  
MATH 3921 Pseudodifferential Operators Course will give an introduction to pseudo differential operators and their applications to partial differential equations and topology.  Topics include hypo elliptic operators, elliptic complexes and the index theorem.  
MATH 3923 Topics in Partial Differential Equations (3 Credits)

This course will explore recent developments in the theory of partial differential equations centered around the notion of viscosity solution.  After a review of hamiton-jacobi equations, we will discuss how they can be used as a substi tute for convolution in non-linear problems.

MATH 3930 Fixed Point Theory Topics include:  Banach's fixed point theorem, Brouwer's fixed point theorem and various applications, topological degree theory in finite dimensions and constructive aspects by simplicial algorithms.  Also a brief introduction to the Leray-Schauder degree in infinite dimension will be discussed.  
MATH 3935 Topics in Applied Mathematics After briefly introducing very elementary theory of asymptotic expansions, the course will focus on its applications. In particular, boundary layer expansion, interior inter facial layer expansion, as well as multi-scale expansion techniques will be quite detailed by examples of a number of up-to-date research problems.  
MATH 3935 Topics in Applied Mathematics (3 Credits)

The theory of water waves embodies the equations of fluid mechanics, the concepts of wave propagation, and the critically important role of boundary dynamics. It has been a subject of intense research since Euler's derivation of the equations of hydrodynamics. The second part focuses more on the analysis perspective of the water wave equations.Zakharov's Hamiltonian formulation of the irrotational water wave problem will be dis-
cussed and applications of this formulation to the issue of wellposedness will be outlined.
The course also use some of the asymptotic (integrable) models as an example to describe the wave-breaking phenomenon. Finally, a particular ow pattern, namely the traveling (or steady) waves will be addressed. The method of calculus of variation, (global) bifurcation theory, topological degree theory, Schauder estimates, and Fredholm theory will be introduced to establish the existence of such waves.
Mathematically, the above mentioned topics draw on deep ideas from applied mathematics, analysis, and PDEs. The main ingredients and techniques involved include Fourier analysis, harmonic analysis, elliptic theory, and more. Prior exposure to any of these will be helpful but not necessary.

MATH 3940 Applied Analysis 1
Methods will be developed to analyze the behavior of bumps and waves in integral models arising in neuroscience, and also in reaction-diffusion biological type models.
MATH 3941 Applied Analysis 2 This course covers advanced methods in the theory of ordinary differential equations.  Topics include: shooting arguments in existence problems, energy arguments, reduction of invariant equations, a discussion of bifurcation theory and differential equations in the complex domain.  
MATH 3950 Nonlinear Dynamics, Chaos, and Oscillation This course gives a description of modern techniques for analyzing nonlinear differential equations.  It covers topics such as chaos, nonlinear oscillations, bifurcation theory, phase locking and invariant manifold methods.  Topics such as averaging, singular perturbation, and equations on tori are also discussed.  
MATH 3951 Physical Methods in Mathematics This course investigates the mathematical structure of various methods in physics.  Topics include renormalization scaling and interfaces.  Applications to differential and difference equations will be done.  
MATH 3960 Mathematics of Phase Boundaries This course will concentrate on developing the major models used in solidification theory.  Thus the study of systems of differential equations will be a major theme of the course.