Graduate Program Course Offerings

(1 Credit)

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.

(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.

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.

(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.

(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. 

The contents of this course include the propositional calculus, the predicate calculus, model theory and their applications to mathematical systems.

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.

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.

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.

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.

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.

(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.

(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

(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.

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.

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.

(3 Credits)

The course is the first term of a two-term graduate algebra sequence. Topics include rings and modules, in particular finitely generated modules over a PID, Galois theory and some commutative algebra and category theory. 

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.

(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.

(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.

(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. 

(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.

(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.

(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.

(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.

(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.

(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.

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.

(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.

Topics include:  definitions and basic properties, the Sobolev imbedding theorem, applications of the theory.

The general problem discussed in flows on networks.  Topics will include some graph theory, flows and potential differences, transport problems and flow algorithms.

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.

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.

(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.

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.

(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.

(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.

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.

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.

(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.

(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.

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.

(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.

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.

(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.

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.

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.

(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.

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.

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.

(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.

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.

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.