Key - General Education Requirements, Dietrich School of Arts and Sciences

Key - Basic Skills Requirements, Dietrich School of Arts and Sciences

I   Workshop in Composition
A  Algebra

Other Keys: Term/Session Codes | Subjects | Special Indicators | Days | Classrooms

MATH Courses 2131

MATH 0010 covers the first half of the College Algebra course MATH 0031. MATH 0020 covers the second half of MATH 0031. This two term sequence has been designed for students who have had algebra in high school, but who need to learn (as opposed to review) the subject over an extended period of time. In this course, emphasis will be placed on both pencil and paper work and the visualization of algebra concepts with a graphing calculator. Topics include linear and quadratic functions and their applications. There will be two exams and a cumulative final exam. The instructor lectures twice a week, and undergraduate teaching assistants lead problem solving recitations once a week. A graphing calculator is required

Prerequisite(s): none

Check with the department on how often this course is offered.

MATH 0010 covers the first half of the College Algebra course MATH 0031. MATH 0020 covers the second half of MATH 0031. This two term sequence has been designed for students who have had algebra in high school, but who need to learn (as opposed to review) the subject over an extended period of time. In this course, emphasis will be placed on both pencil and paper work and the visualization of algebra concepts with a graphing calculator. Topics include linear and quadratic functions and their applications. There will be two exams and a cumulative final exam. The instructor lectures twice a week, and undergraduate teaching assistants lead problem solving recitations once a week. A graphing calculator is required

Prerequisite(s): none

Check with the department on how often this course is offered.

MATH 0010 covers the first half of the College Algebra course MATH 0031. MATH 0020 covers the second half of MATH 0031. This two term sequence has been designed for students who have had algebra in high school, but who need to learn (as opposed to review) the subject over an extended period of time. In this course, emphasis will be placed on both pencil and paper work and the visualization of algebra concepts with a graphing calculator. Topics include linear and quadratic functions and their applications. There will be two exams and a cumulative final exam. The instructor lectures twice a week, and undergraduate teaching assistants lead problem solving recitations once a week. A graphing calculator is required

Prerequisite(s): none

Check with the department on how often this course is offered.

Math 0020 covers the second half of the Algebra course 0025 and must be preceeded by math 0010. It uses the same text as math 0010 with the same course organization (see above). Topics include exponential, power, logarithmic, quadratic and polynomial functions. A graphing calculator, TI-83 or equivalent, is required. One-letter grade rule applies if there is a common final exam. This is a Self–Paced course. Workshop attendance is strongly advised. Workshop dates: 9/8, 10/13, 11/10/2012.

Prerequisite(s): PREQ: MATH 0010

Check with the department on how often this course is offered.

This course is designed for non-science or non-math majors who want to fulfill the algebra requirement. It provides mathematical skill needed for entry-level Statistics courses (Stat 0200 and Stat 0800). Students intending to take calculus or upper level Statistics courses (numbered 1000 or above) or Physics 0110 should choose Math 0031 instead. The course will stress the use of real life data and statistics that model algebraic functions. The course will cover linear, quadratic, polynomial, exponential and logarithmic functions, with emphasis placed on problem solving skills. Recitations are an integral part of this course. A graphing calculator is required and some assignments will use the computer. This course parallels Math 0031, and students need the appropriate placement score to enter this course. The instructor lectures three times a week and undergraduate teaching assistants lead problem-solving recitations once a week. A graphing calculator, TI-83 or equivalent, is required.

Prerequisite(s): none

Check with the department on how often this course is offered.

This course will help students master basic algebra skills. It will provide the mathematical foundations for courses such as Business Calculus (Math 0120). Students can prepare for Science and Engineering Calculus by combining this course with Trigonometry (Math 0032), or by taking the faster paced Prep for Scientific Calculus (Math 0200). Emphasis will be placed on the visualization of algebra concepts with a graphing calculator or computer. Topics include: linear, quadratic, polynomial, rational, exponential and logarithmic functions. The instructor lectures three time each week and undergraduate teaching assistants lead problem solving recitation sessions once each week. A graphing calculator is required.

Prerequisite(s): none

Check with the department on how often this course is offered.

This course is designed to help students with weak mathematical backgrounds review algebra. For faster paced or more advanced courses, students who know algebra should consider either Math 0032, 0100, or 0200, which can be used to satisfy the algebra requirement and/or prepare students for calculus. In this course emphasis will be placed on the visualization of algebra concepts with a graphing calculator . Topics include: linear, quadratic, polynomial, rational, exponential and logarithmic functions. The instructor lectures three time each week and undergraduate teaching assistants lead problem solving recitation sessions once each week. This course serves as a prerequisite for Math 0120 and, together with 0032, as a prerequisite for Math 0220. A graphing calculator, TI-83 or equivalent, is required. One-letter grade rule applies if there is a common final exam.

Prerequisite(s): none

Check with the department on how often this course is offered.

This course is designed to enable students who have mastered algebra to LEARN trigonometry and functions; it is the sequel to Math 0031, College Algebra. Students who score high on the algebra placement test without passing, may fulfill their algebra requirement with this course. In addition to trigonometric functions, material on graphing, functions, matrices and conic sections is included. An emphasis will be placed on the visualization of algebraic and trig topics using a TI-83 graphing calculator (or equivalent). Students who have not had trigonometry and intend to take Math 0220 should take Math 0032 BEFORE taking Math 0220. For a faster paced course take Math 0200. The instructor lectures twice a week and undergraduate teaching assistants lead problem solving recitations twice a week. A graphing calculator, TI-83 or equivalent, is required. One-letter grade rule applies if there is a common final exam.

Prerequisite(s): none

Check with the department on how often this course is offered.

This course is designed for students in business, economics, and other social sciences. It introduces the basic concepts of limits, continuity, differentiation, integration, maximization, minimization and partial derivatives. The Calculus of Trigonometric Functions is NOT covered. Applications to the social sciences, especially business and economics, are stressed. A graphing calculator, TI-83 or equivalent, is required. One-letter grade rule applies if there is a common final exam.

Prerequisite(s): none

Check with the department on how often this course is offered.

This course provides an introduction to differential calculus for students in business and the social sciences. It covers the first half of the material presented in MATH 0120.

Prerequisite(s): none

Check with the department on how often this course is offered.

This course is designed for those students who have had high school courses, Algebra I, II, Geometry, and Trigonometry, but feel that more preparation is prudent or necessary to enable them to tackle calculus successfully. The following topics are studied: polynomial functions, rational functions, logarithmic and exponential functions, inverse functions, graphs, asymptotes, conic sections, translation and rotation of axes, trigonometric identities and equations, polar coordinates, systems of equations and limits. Emphasis will be placed on visualization of functions with graphing calculators. Students who want a slower paced precalculus course should take Math 0031 followed by Math 0032. A graphing calculator TI-83 or equivalent is required. The instructor lectures three time a week and graduate teaching assistant leads a problem solving recitation twice a week. One-letter grade rule applies if there is a common final exam.

Prerequisite(s): none

Check with the department on how often this course is offered.

This is the first course in the basic calculus sequence and is intended for all mathematics, engineering, science, and statistics students. Math 0220 covers the derivative and integral of functions of a single variable. A lab component in which students apply numeric, algebraic, and graphing technologies to calculus problems is an integral part of the course. For addition information refer to the web page http://calculus.math.pitt.edu. A scientific calculator is required, preferably a graphing calculator. One-letter grade rule applies if there is a common final exam.

Prerequisite(s): none

Check with the department on how often this course is offered.

This is the second course in the basic calculus sequence and is intended for all mathematics, engineering, science, and statistics students. Math 0230 covers symbolic and numerical integration techniques and applications, modeling, differential equations, and Taylor series. A lab component in which students apply numeric, algebraic, and graphing technologies to calculus problems is an integral part of the course. For addition information, refer to the web page http://calculus.math.pitt.edu. A scientific calculator is required, preferably a graphing calculator. One-letter grade rule applies if there is a common final exam.

Prerequisite(s): PREQ: MATH 0220; MIN GRADE: 'C'

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This course covers the derivative and integral of functions of a single variable. Students are introduced to the calculus of log, exponential and inverse trig-functions. Model building using differential equations is introduced and explored from numerical as well as a theoretical viewpoint. Approximations with Taylor Polynomials and Taylor Series are introduced. The course will be accepted by the Department of Mathematics as a more demanding enriched alternative to the regular sections of Math 0220 and Math 0230. There will be more emphasis on computer experimentation and computer applications using Maple. The course is intended for honors students majoring in Mathematics, Science, Engineering, Computer Science, and Statistics. Familiarity with high school calculus will be assumed.

Prerequisite(s): none

Check with the department on how often this course is offered.

UHC-designated courses: Students below a 3.25 GPA must go to the Honors College for permission. 

Math 0240 is the third course in the calculus sequence and is intended for honors students majoring in mathematics, engineering, science and statistics. It covers vectors, parameterized curves and surfaces, differentiation of functions of several variables, optimization, integration of functions of two and three variables, line integrals, flux integrals, and calculus of vector fields. The one letter grade rule applies if there is a common final exam.

Prerequisite(s): PREQ: MATH 0230 or 0235; MIN GRADE: 'C'

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This course is designed primarily for engineering students. The main subject of the course is ordinary differential equations. Topics include first order differential equations, higher order linear differential equations and systems of first order linear and nonlinear differential equations. Matrix methods will be introduced and used to solve systems of linear equations. The computer package Matlab will be used to assist in computations. One-letter grade rule applies if there is a common final exam.

Prerequisite(s): PREQ: MATH 0120 or 0220

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This course presents an introduction to the theory of differential equations from an applied perspective. Topics include linear and nonlinear ordinary differential equations, Laplace transform, and introduction to partial differential equations.

Prerequisite(s): PREQ: MATH 0230 or 0235; MIN GRADE: 'C'

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This course is designed to introduce students from various disciplines to the applied world of mathematics within a discrete context. In particular, students can expect to work with linear models including a basic algebraic introduction, solution of linear systems, and matrix algebra techniques. Additionally, a substantial part of the course will be devoted to the development and application of the basic laws of probability, probability models, and combinatorial techniques. Logic, Markov Chains, and game theory will be considered as time permits. Students from a variety of academic disciplines within the social, life, and physical sciences as well as the humanities looking to strengthen their mathematical background will find this course to be interesting and useful as real-world applications will be stressed. One-letter grade rule applies if there is a common final exam. This course is designed to introduce students from various disciplines to the applied world of mathematics within a discrete context. In particular, students can expect to work with linear models including a basic algebraic introduction, solution of linear systems, and matrix algebra techniques. Additionally, a substantial part of the course will be devoted to the development and application of the basic laws of probability, probability models, and combinatorial techniques. Logic, Markov Chains, and game theory will be considered as time permits. Students from a variety of academic disciplines within the social, life, and physical sciences as well as the humanities looking to strengthen their mathematical background will find this course to be interesting and useful as real-world applications will be stressed. One-letter grade rule applies if there is a common final exam.

Prerequisite(s): none

Check with the department on how often this course is offered.

This "W" course is an introduction to the theoretical treatment of the real numbers, sets, functions, sequences, and limits. Classwork and homework will concentrate on the writing and understanding of proofs of theorems centered on these topics. For most students, the techniques of reading and doing proofs will be new. Two recitations per week, therefore, are required to practice this skill as well as to rehearse the many definitions of mathematical concepts in the course. It is required for all Mathematics and Statistics majors except for Departmental Honors majors who may wish to accelerate by substituting Mathematics 0450 for 0413 and 0420, and then taking Advanced Calculus. One-letter grade rule applies if there is a common final exam.

Prerequisite(s): PREQ: MATH 0230 or 0235

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The course provides a careful treatment of the theoretical concepts of limit, continuity, derivative and integral, including The Fundamental Theorem of Calculus. One-letter grade rule applies if there is a common final exam.

Prerequisite(s): PREQ: MATH 0413

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This course introduces the student to abstract algebraic concepts: groups, rings, integral domains, and fields. Many examples will be presented during class and in the homework. The students are expected to enhance their proof writing techniques.

Prerequisite(s): PREQ: MATH 0413 or 0450 or 1185

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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.

Prerequisite(s): none

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This course will focus on the role played by Number Theory in mathematics and will include Divisibility, Congruences, Diophantic Equations, and Algebraic Numbers among other topics.

Prerequisite(s): PREQ: MATH 0430

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This course is an 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 of 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.

Prerequisite(s): PREQ: MATH 0280

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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, amortization 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.

Prerequisite(s): PREQ: MATH 0230 or 0235

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This course stresses the theoretical and rigorous development of linear algebra. Major topics include the theory of vector spaces, linear transformations, matrices, eigenvalues and vectors, bases and canonical forms. Other topics may be covered as time permits. Math 1180 is suitable for mathematics majors, joint mathematics majors, statistics majors, and others interested in a theoretical treatment. Students interested only in the techniques of matrix theory should enroll in Math 0280.

Prerequisite(s): CREQ: MATH 0413 or MATH 0450

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This UHC course provides an introduction to both computational and theoretical aspects of linear algebra, and is suitable for those wanting both theory and applications. Linear algebra is a combination of algebra and geometry. The geometry is relatively simple, as it only deals with lines, planes, and high-dimensional analogues. The algebra is also simple in concept, as it mostly involves adding columns and rows of numbers or multiplying a whole column or row of numbers by the same constant. In this course the geometry will be stressed by many examples in two and three dimensions, while the algebraic computations for more complicated examples will be done by computer. Use will be made of the computer algebra program Matlab, which will be introduced from scratch in this course. Majors in mathematics or any of the sciences, engineering, economics, or business will find this a valuable tool in their discipline. Among the specific applications which may be covered are computer graphics, game theory, and Leontief models in economics.

Prerequisite(s): none

Check with the department on how often this course is offered.

UHC-designated courses: Students below a 3.25 GPA must go to the Honors College for permission. 

Math 1230 will provide a capstone experience for undergraduate math or science students, especially those who have an interest in the history of math, the psychology of math learning, and/or in mathematics education. This course will concentrate on some of the Big Ideas of Mathematics from an historical perspective. It will demonstrate how the various strands of mathematics developed and are integrated; and how the characteristic thinking processes of mathematics (abstraction, axiomatization, proof etc) arose. It will give insight into how mathematicians create new mathematics and apply their discoveries to practical problems. It will examine how the available tools (number systems, sine and cosine tables, computers etc) have influenced the shape of mathematics.

Prerequisite(s): PREQ: MATH 0430

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Math 1231, Math Education Internship, provides an opportunity for undergraduate students to interact with high school students, and explore their potential interest in math education. The internship has two components. The first a workshop in which the connections between the content of Math 1230, `Big Ideas of Mathematics' and secondary teaching will be explored. The second mentoring a school student through a math research project.

Prerequisite(s): CREQ: MATH 1230

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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 linear equations. An introduction will be given to the qualitative theory of first-order nonlinear systems. This will include phase plane methods and stability analysis. Computer experimentation may be used to illustrate the behavior of solutions of various equations.

Prerequisite(s): PREQ: MATH 0280 or 1180 or 1185

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This course will introduce students to graphs, multigraphs, digraphs, vertex degree, isomorphic graphs, trees, connectivity, traversability and planar graphs. Time permitting, we may include: matchings, coloring and Ramsey numbers, distance and domination and some combinatorial optimization Students without the required prerequisites may be able to enroll with the permission of the instructor.

Prerequisite(s): PREQ: MATH 0413

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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, curvature), global properties of surfaces, Gauss-Bonnet formula and applications.

Prerequisite(s): PREQ: MATH 0240 or 0280 or 1180 or 1185

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This course will introduce the students to mathematical modeling via an introduction to the mathematics of nonlinear dynamical systems. It will show by examples how important questions about the observed world may be framed in terms of linear and nonlinear differential equations. By considering concrete problems from physics, biology, chemistry and engineering, the course will illustrate such concepts as equilibrium and stability, bifurcation, limit cycles, and chaos. The students will also learn important analytical techniques such as linearization, phase plane analysis and rescaling. Advanced topics treated in an elementary way will be hysteresis, coupled oscillators, Hopf bifurcations, and strange attractors. Applications will include mechanical vibrations, dynamics of interacting populations, biological rhytms, lasers, and if time permits, even a technique for using chaos to send secret messages. This course will introduce the students to mathematical modeling via an introduction to the mathematics of nonlinear dynamical systems. It will show by examples how important questions about the observed world may be framed in terms of linear and nonlinear differential equations. By considering concrete problems from physics, biology, chemistry and engineering, the course will illustrate such concepts as equilibrium and stability, bifurcation, limit cycles, and chaos. The students will also learn important analytical techniques such as linearization, phase plane analysis and rescaling. Advanced topics treated in an elementary way will be hysteresis, coupled oscillators, Hopf bifurcations, and strange attractors. Applications will include mechanical vibrations, dynamics of interacting populations, biological rhytms, lasers, and if time permits, even a technique for using chaos to send secret messages.

Prerequisite(s): PREQ: MATH 0290 or 1270 or 1275

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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 students 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.

Prerequisite(s): PREQ: MATH 0240 and {[(MATH 0280 or 1180 or 1185) and (0290 or 1270)] or 0250}

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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.

Prerequisite(s): PREQ: MATH 0420 or 0450

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Topics which will be 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 and either Math 0250 or Math 0280 or Math 1180.

Prerequisite(s): PREQ: MATH 0240 and (0250 or 0280 or 1180 or 1185)

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An academic internship is an approved and monitored work experience that meets specific learning goals related to academic field of study. For information on internships, visit www.pitt.edu/~intern/.

Prerequisite(s): none

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Under the direction of a faculty member a student studies a mutually agreed upon topic in Mathematics.

Prerequisite(s): none

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

Prerequisite(s): none

This course is offered every term.

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.

Prerequisite(s): none

This course is offered at least once a year.

This course will help students gain the skills and experience needed to do research in mathematics. The focus will be on all forms of communicating math: writing a journal article, how to submit a paper, writing a research proposal, refereeing papers, and giving presentations, along with the tools needed for communication (especially advanced latex skills). Other topics covered may include: finding an advisor and research topic, developing career portfolios, expectations for Comprehensive exams, Overview and Dissertation.

Prerequisite(s): none

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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 where you learn to implement your own solution methods. Topics for the first term include: roundoff errors and stability analysis,rootfinding for nonlinear equations, interpolation, approximation of functions and numerical integration. The presented approximation methods will be combined for the solution of ordinary and partial differential equations. 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. The laboratory makes use of the Campus VMS and Unix machines. There are two lab sections per week; registration in one of the lab sections is required. Lab. schedule reassignment may be done to optimize equipment use. Students should keep both lab periods available if possible. The class number for the lab sessions is 11844.

Prerequisite(s): none

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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. We will discuss the principal classes of numerical methods and of their theory, including convergence and stability considerations, order and step size selection and the effects of stiffness. Numerical methods for stochastic differential equations will also be discussed.

Prerequisite(s): none

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This course is an introduction to Real Analysis/Measure Theory and Functional Analysis. Topics include: Lebesgue Measure and Integral, Monotone Convergence Theorem, Lebesgue's Dominated Convergence Theorem, Fatou's Lemma, Jensen, Holder and Minkowski Inequalities, Signed Measures, Hahn Decomposition, Absolutely continuous measures, Radon-Nikodym Theorem; Baire Category Theorem, Banach Spaces, The Big Theorems of Functional Analysis (Open Mapping Theorem/Closed Graph Theorem, Uniform Boundedness Principle and Hahn-Banach Theorem), Riesz Representation Theorem, Fejer's Theorem, Hilbert Spaces and Sobolev Spaces.

Prerequisite(s): none

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This is a course in the Graduate Analysis sequel, directed at students who have taken 2301 Analysis 1; so that knowledge of Lebesgue measure and integration, Lebesgue spaces L^p and basic knowledge of Sobolev spaces W^{1,p} is assumed. The analysis and linear algebra material in the Math Preliminary Exams syllabus is also assumed. In addition to the standard theorems on linear operators on Hilbert spaces, and topics in spectral theory, the course will touch upon some more recent subjects in the modern Mathematical Analysis, such as: the Euler equations as a differential inclusion, the h-principle in convex integration and relation to construction and uniqueness of weak solutions to the equations of fluid dynamics, the distributional Jacobian and the Bourgain-Brezis-Nguyen formula, stochastic homogenization and the game-theoretical view of some estimates in PDEs.

Prerequisite(s): none

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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. Major topics of the course include criteria for the diagonalization of a linear operator the spectral theorem for linear operators, cannonicl forms, and spectral theory. Related material about the dual and double duals of vector space.

Prerequisite(s): none

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The course is the second term of a two-term graduate algebra sequence. It covers rings, modules, and elements of homological algebra and commutative algebra. Highlights of the course will include: the structure of modules over principal ideal domains, Jordan canonical form, Hilbert's Nullstellensatz.

Prerequisite(s): none

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TITLE: Optimal Control and Parameter Estimation ========== This course will focus on the study of optimal control and parameter estimation problems for distributed (deterministic and/or random) parameter systems, i.e., for systems described by a boundary value problem for partial differential equations. We will give an introduction to the development and analysis of several approaches for solving deterministic and stochastic optimization problems (optimal control and parameter identification), and discuss some of the many issues that arise in the practical implementation of algorithms. This course will be self-contained as far as possible. Lectures will adapt to diverse backgrounds. Please contact the instructor if you have questions about your preparation.

Prerequisite(s): none

This course is offered infrequently.

The course will cover the following topics: Geometry of curves and surfaces. First and second fundamental form. Gauss map. Curvature. Minimal surfaces. Gauss theorem. Covariant derivative. Surfaces of constant curvature. Gauss-Bonnet formula. Submanifolds of the Euclidean spaces. Abstract manifolds. Sard theorem. Brouwer fixed point theorem. Whitney embedding theorem. Degree theory. Vector fields. Frobenius theorem. Riemannian manifolds. Tensor fields. Differential forms. Stokes theorem. Prerequisite(s): The course will focus on basic material that every mathematician should know. The main prerequisite for the course is a very good understanding of Advanced Calculus

Prerequisite(s): none

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This is the first course in a two-term sequence designed to acquaint students with the basic concepts of ordinary differential equations and how they relate to other areas of mathematics. Basic existence and uniqueness of solutions as well as dependence on parameters will be covered. Linear ODES and the Jordan form and matrix exponential will be studied. Oscillations will be discussed, via an introduction to Poincare-Bendixson theory for planar systems and to Floquet theory, as will Sturm-Liouville problems. Students will also be introduced to geometric concepts of stability of fixed points and invariance. This first term will provide an excellent basis for students interested in applied mathematics

Prerequisite(s): none

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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.

Prerequisite(s): none

This course is offered every term.

This course is designed for PhD students working in their Dissertation research.

Prerequisite(s): none

This course is offered every term.

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): 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.

Prerequisite(s): none

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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.

Prerequisite(s): none

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This course covers the basic theory of options using stochastic differential equations. After a brief review of probability, we will discuss the Black-Scholes equation and various generalizations. The fundamental theory of parabolic differential equations used to solve these equations will be covered. Other topics include American options and numerical methods for solving partial differential equations. The course is mainly self-contained and requires an understanding of differential equations and related concepts at the undergraduate level.

Prerequisite(s): none

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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. We will start with some basic biophysics of cells and the Hodgkin-Huxley theory of neural activity. We then turn to synaptic dynamics and small networks. Population models will be introduced and applied to a variety of cognitive and physiological systems. We then discuss neural coding, reliability, spike triggered averages and applications of Fisher information theory Prerequisite(s): none

Prerequisite(s): none

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We will begin with classical theorems: the Banach Contraction Mapping Theorem and Brouwer's Fixed Point Theorem (and its extension to Schauder's Theorem in Banach spaces), with applications to ODE's, integral equations, and fractal image reconstruction (using the Hausdorff metric). Finding solutions of Dilation Equations is a fixed point problem in Wavelet Theory. We will discuss an iterative construction of and existence proof for Daubechies scaling function and wavelet. The course will continue with topics in metric fixed point theory and its connections to Banach space geometry and topology. The course will include some of my (joint) research in this area, and related research of other authors.

Prerequisite(s): none

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This will be a course on the "Fundamental Lemma and the Hitchin fibration". This course will be an introduction to Ngo Bao Chau's proof of the fundamental lemma. Topics to be covered include: the Hitchin fibration, affine Springer fibers, the very basic theory of stacks, some structure theory of reducive groups, the Kostant section, conjugacy and stable conjugacy, endoscopy, the Picard stack, and Tate-Nakayama duality. Prerequisites: Algebra 1, Algebra 2, and Algebraic Geometry, or the instructor's consent.

Prerequisite(s): none

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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.

Prerequisite(s): none

This course is offered every term.

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.

Prerequisite(s): none

This course is offered every term.

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