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.

Access the latest schedule of classes and course descriptions.

Undergraduate Course Listing

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 1020 Applied Elementary Number Theory (3 Credits)

syllabus

MATH 0430
MATH 1050 Combinatorial Mathematics MATH 0430
MATH 1070 Classical Numerical Analysis
MATH 1080 Numerical Linear Algebra
MATH 1100 Numerical Optimization Method
MATH 1110 Industrial Numerical Analysis
MATH 1180 Linear Algebra 1

Credit will not be given for both MATH 0280 and 1180

MATH 1240 Linear Algebra 2
MATH 1250 Abstract Algebra
MATH 1270 Ordinary Differential Equations 1

Credit will not be given for both MATH 0290 and 1270

MATH 1280 Ordinary Differential Equations 2
MATH 1310 Graph Theory
MATH 1330 Projective Geometry
MATH 1350 Introduction to Differential Geometry
MATH 1360 Modeling in Applied Mathematics 1
MATH 1370 Modeling in Applied Mathematics 2
MATH 1410 Introduction to the Foundations of Mathematics 1
MATH 1420 Introduction to the Foundations of Mathematics 2
MATH 1470 Partial Differential Equations 1
MATH 1480 Partial Differential Equations 2
MATH 1530 Advanced Calculus 1 (3 Credits)

Basic topological concepts in metric spaces will be discussed, including compactness and connectedness. Continuity of functions of several real variables, and uniform convergence of sequences and series of functions will be treated rigorously.

MATH 1540 Advanced Calculus 2 (3 Credits)

In this course, which is a continuation of the Fall Math 1530, the theory of differentiation and integration of functions of several variables will be developed rigorously. Topics will include the inverse and implicit function theorems, Fubini's Theorem, change of variables, and Stokes' Theorem.
Prerequisite(s): PREQ: MATH 1530

MATH 1560 Complex Variables and Applications
MATH 1570 Transformation Methods in Applied Mathematics
MATH 1700 Introduction to Topology
MATH 1730 Honors Algebra 1
MATH 1740 Honors Algebra 2
MATH 1750 Honors Analysis 1
MATH 1760 Honors Analysis 2
MATH 1800 Topics in Mathematics
MATH 1801 Acturial 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
MATH 2055 Codes and Designs
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
MATH 2170 Logic and Foundations
MATH 2180 Introduction to Fractal Geometry
MATH 2190 Functions of Several Variables
MATH 2200 Real Analysis 1
MATH 2201 Real Analysis 2
MATH 2210 Complex Analysis 1
MATH 2211 Complex Analysis 2
MATH 2219 Dynamical Systems (3 Credits)

THIS COURSE WILL INTRODUCE THE STUDENT TO NEW CONCEPTS FROM DYNAMICAL SYSTEMS. INVARIANT MANIFOLDS, NORMAL FORM, BIFURCATIONS, 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
MATH 2245 Algebraic Number Theory
MATH 2260 Potential Theory
MATH 2280 Hardy Spaces
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)

Math 2370 is a graduate level course in linear algebra which concentrates on developing the classical theory for linear operators on general finite dimensional vector spaces and finite dimensional inner product spaces. 

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
MATH 2401 Functional Analysis 2
MATH 2410 Harmonic Analysis 1
MATH 2480 Computational Approximation Theory
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
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
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
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
MATH 2980 Projects in Financial Mathematics
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
MATH 3010 Sobolev Spaces
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
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
MATH 3060 Topics in Combinatorics
MATH 3070 Numerical Solution of Nonlinear Systems
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
MATH 3211 Riemann Surfaces
MATH 3215 Quasiconformal Maps 1
MATH 3216 Quasiconformal Maps 2
MATH 3220 Several Complex Variables 1
MATH 3221 Several Complex Variables 2
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 3250 Singular Integral Theory 1
MATH 3251 Singular Integral Theory 2
MATH 3260 Topics in Fractal Geometry 1
MATH 3261 Topics in Fractal Geometry 2
MATH 3262 Topics in Fractal Geometry 3
MATH 3270 Iteration of Rational Maps 1
MATH 3271 Iteration of Rational Maps 2
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
MATH 3411 Hilbert Spaces of Entire Functions 2
MATH 3436 Fixed Points Wavelets & Fractals
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
MATH 3480 Topics in Spline Approximation
MATH 3500 Topics in Algebra
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

Math 3600, during the Spring semester 2016, will cover the topic "Introduction to Symplectic Geometry". Some familiarity with differential forms and manifolds will be assumed, but there are no other particular prerequisites. The course will develop a variety of topics, such as:
Sympletic linear algebra, Symplectic manifolds and cotangent bundles, Darboux’s  theorem, Contact manifolds, Sympletic capacities, Weinstein’s conjecture, Viterbo’s theorem, Gromov-Eliashberg rigidity, Gromov’s Symplectic Width, Hofer Geometry.

MATH 3750 General Topology 2
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
MATH 3921 Pseudodifferential Operators
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
MATH 3935 Topics in Applied Mathematics
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
MATH 3950 Nonlinear Dynamics, Chaos, and Oscillation
MATH 3951 Physical Methods in Mathematics
MATH 3960 Mathematics of Phase Boundaries

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