Math 6302 - Section: 24304 - Modern Algebra - by Kaiser
MATH 6302: Modern Algebra(section# 24304 ) |
Time: |
MoWeFr 11:00AM - 12:00PM - Room: SEC 105 |
Instructor: |
Kaiser |
Prerequisites: |
Graduate Standing. |
Text(s): |
Thomas W. Hungerford, Algebra, Springer Verlag (required). I will also circulate my own classroom notes. |
Description: |
During the first semester
we will cover the basic theory of groups, rings
and fields with strong emphasis on principal ideal domains. We will also discuss the
most important
algebraic constructions from a universal algebraic
as well as from a categorical point of view.
The second semester will be mainly on modules
over principal ideal domains, Sylow theory, free
algebras and co-products and ultraproducts.
Course Organization and Grading: Students will
receive on a regular basis homework assignments.
There will be a midterm and a final. Homework
counts for 30%, midterm for 30% and final for
40% of the final grade. |
Math 6304 - Section: 32305 - Theory of Matrices - by Paulsen
MATH 6304: Theory of Matrices ( section# 32305 ) |
Time: |
MoWeFr 12:00PM - 1:00PM - Room: SR 138 |
Instructor: |
Vern Paulsen |
Prerequisites: |
Math 4377 and 4331 or Math 6377. |
Text(s): |
"Matrix Analysis", by Roger A. Horn and Charles R. Johnson, Cambridge University Press ISBN 0-521-38632-2
NOTE: This book is available in paperback. |
Description: |
We will present topics in linear algebra and matrix theory that have
proven to be important in analysis and applied mathematics. We assume that the student
is familiar with standard concepts and results from linear algebra and basic analysis.
We will study canonical factorizations of matrices, including the QR, triangular and
Cholesky factorizations. We will develop ways to achieve the Jordan canonical form. We
will study eigenvalue perturbation and estimation results and we will study special
families of matrices such as positive definite, Hermitian, Hankel, and Toeplitz. Matrix
analysis is in a sense an approach to linear algebra that is willing to use concepts
from analysis, such as limits, continuity and power series to get results in linear
algebra. |
Math 6308 - Section: 33227 or 33228 - Advanced linear algebra I - by J.Johnson & Friedberg
MATH 6308: Advanced linear algebra I ( section# 33227 ) |
Time: |
TuTh 1:00PM - 2:30PM - Room: F 162 |
Instructor: |
Michael Friedberg |
Prerequisites: |
Math 2331 and a minimum of three semester hours of 3000-level mathematics. |
Text(s): |
Linear Algebra,2nd edition, by Hoffman and Kunze, Prentice Hall. |
Description: |
Topics covered include linear systems of equations, vector spaces, linear transformation, and matrices. |
Remark: |
There is a limitation for counting graduate
credits for Math 6308, 6309, 6312, or 6313. For detailed
information, see Masters Degree Options. |
MATH 6308: Advanced linear algebra I ( section# 33228 ) |
Time: |
MoWe 1:00PM - 2:30PM - Room:PGH 350 |
Instructor: |
Johnny Johnson |
Prerequisites: |
Math 2331 and a minimum of three semester hours of 3000-level mathematics. |
Text(s): |
Linear Algebra,2nd edition, by Hoffman and Kunze, Prentice Hall. |
Description: |
Topics covered include linear systems of equations, vector spaces, linear transformation, and matrices. |
Remark: |
There is a limitation for counting graduate
credits for Math 6308, 6309, 6312, or 6313. For detailed
information, see Masters Degree Options. |
Math 6312 - Section: 33229 - Introduction to Real Analysis - by Field
MATH 6312: Introduction to Real Analysis(section# 33229 ) |
Time: |
MoWe 4:00PM - 5:30PM - Room: SEC 203 |
Instructor: |
Mike Field |
Prerequisites: |
MATH 3333 and preferably MATH 3334. Otherwise, consent of instructor. MATH 3334 is not required for MATH 4331 alone |
Text(s): |
Set Theory and Metric Spaces, IRVING KAPLANSKY, University of Chicago AMS CHELSEA PUBLISHING, American Mathematical Society. |
Description: |
An introduction to real analysis.
The first semester MATH 6312 will focus mainly on 1-variable analysis and will include topics from
(a) basic properties of the real number system,
(b) series, sequences and uniform convergence,
(c) properties of
functions (analyticity, smoothness, nowhere differentiability, Weierstrass approximation theorem),
(d) classical function theory (Gamma function, infinite products),
(e) Euler-Maclaurin formula and asymptotics.
The second semester MATH 6313 will provide an introduction to metric spaces and
compactness that develops from results proved in the first semester (for example, the
compactness of a closed interval). Also included will be a short review of multivariable
calculus and applications of metric space techniques (the contraction mapping lemma) to
the inverse and implicit function theorems and the existence theorem for ordinary
differential equations. If there is time (in either semester 1 or 2), there will also be
some introductory lectures on Fourier series - in particular, the verification that
under appropriate assumptions, the Fourier series does converge to the function.
Overall the aim of MATH 6312/3 is to provide an introduction to some of the powerful
techniques, results and ideas of analysis as developed over the past three hundred years
|
Remark: |
There is a limitation for counting graduate
credits for Math 6308, 6309, 6312, or 6313. For detailed
information, see Masters Degree
Options. |
Math 6320 - Section: 24374 - Theory of Functions of a Real Variable - by Papadakis
MATH 6320: Theory of Functions of a Real Variable(section# 24374 ) |
Time: |
MoWeFr 10:00AM - 11:00AM - Room: PGH 348 |
Instructor: |
Papadakis |
Prerequisites: |
Math 4332 - or equivalent course with metric space topology |
Text(s): |
Donald Cohn, Measure Theory, Birkhauser
Recommended text: Gerald Folland, Real Analysis; Modern Techniques and their Applications, published by Wiley-Interscience. |
Description: |
This course covers abstract measure theory and the theory of integration. Topics: Sigma algebras, Outer measures, measures and measure spaces, measurable functions, integration on abstract measure spaces, Lebesgue measure and integration, Modes of Convergence, L^p spaces, Signed and Complex measures, Absolute continuity, The Radon-Nikodym theorem. |
Math 6324 - Section: 32306 - Differential Equations - by Nicol
MATH 6324: Differential Equations(section# 32306 ) |
Time: |
TuTh 11:30AM - 1:00PM - Room: PGH 350 |
Instructor: |
Matthew Nicol |
Prerequisites: |
Math 4331 or consent of instructor. |
Text(s): |
Recommended Texts:
- Differential Equations, Dynamical Systems and Linear Algebra by M. Hirsch and S. Smale (available at Amazon or in the library)
- Ordinary Differential Equations by V. I. Arnold, M.I.T press, 1998 (paperback)
- Geometrical Methods in the Theory of Ordinary Differential Equations by V. I. Arnold, Springer Verlag, 2nd Edition 1988.
- Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields by J. Guckenheimer and P. Holmes (Applied Mathematical Sciences Vol 42) Springer Verlag.
- Mathematical Methods of Classical Mechanics by V. I. Arnold, Springer Verlag, 2nd Edition.
It is not neceesary to buy anything of the references, though Ordinary Differential Equations by V. I. Arnold and Differential Equations, Dynamical Systems and Linear Algebra by M. Hirsch and S. Smale would be most useful to own. The books are useful for reference and lecture notes will be based on these texts and a variety of sources. |
Description: |
This course is an introduction to differential equations. We cover linear theory: existence and uniqueness for autonomous and non-autonomous equations; stability analysis; stable and unstable manifolds; floquet theory and elementary bifurcation theory. We will also cover topics such as quasiperiodic motion; normal form theory; perturbation theory and classical mechanics. |
Math 6342 - Section: 24380 - Topology - by Tomforde
MATH 6342: Topology (section# 24380 ) |
Time: |
MoWeFr 1:00PM - 2:00PM - Room: PGH 345 |
Instructor: |
Mark Tomforde |
Prerequisites: |
A course in real analysis at the Baby Rudin level, e.g., MATH 4331. |
Text(s): |
"Topology" (2nd Edition) by James Munkres |
Description: |
We will cover the basics of point-set topology. |
Math 6360 - Section: 32307 - Applicable analysis - by Auchmuty
MATH 6360: Applicable analysis (section# 32307 ) |
Time: |
TuTh 5:30PM - 7:00PM - Room: AH 301 |
Instructor: |
Giles Auchmuty |
Prerequisites: |
Math 4332 or consent of instructor |
Text(s): |
Reference (not required) Naylor and Sell, Linear Operator theory in
Science and engineering. Springer Verlag. |
Description: |
The course studies the construction and analysis of solutions of various classes of equations. It assumes that students have a working knowledge of metric space topology and linear algebra. Topics to be studied include the contraction mapping principle and its application to finite dimensional equations, the implicit function theorem, the existence of solutions of initial value problems for ordinary differential equations and of integral equations, and the theory of solvability of linear equations on Hilbert spaces. |
Math 6366 - Section: 24382 - Optimization and Variational Methods - by Dean
MATH 6366: Optimization and Variational Methods (section# 24382 ) |
Time: |
TuTh 4:00PM - 5:30PM - Room: AH 301 |
Instructor: |
Edward Dean |
Prerequisites: |
Math 4331 and 4377 or consent of instructor. |
Text(s): |
Numerical Optimization, by J. Nocedal and S.J. Wright. (2nd edition) |
Description: |
This is the first semester of a two semester course. The topics for this first semester will include the theory of finite dimensional linear and nonlinear optimization and numerical methods. This course will be a mix of mathematics and practicalities.
This will include:
Theory and algorithms for unconstrained optimization:
(i) Newton and Quasi-Newton methods,
(ii) Linesearch and Trust Region methods.
Theory and algorithms for constrained optimization:
(i) Linear Programming (Interior Point methods, Simplex methods),
(ii) Quadratic Programming,
(iii) Equality and Inequality linear constraints,
(iv) Barrier and Augmented Lagrangian methods,
(v) Sequential Quadratic Programming. |
Math 6370 - Section: 24384 - Numerical analysis - by Pan
MATH 6370: Numerical analysis(section# 24384 ) |
Time: |
MoWe 4:00PM - 5:30PM - Room: PGH 350 |
Instructor: |
Tsorng-Whay Pan |
Prerequisites: |
Graduate standing or consent of instructor.
Students should have had a course in Linear Algebra (for instance Math 4377-4378) and an introductory course in Analysis (for instance Math 4331-4332). |
Text(s): |
J. Stoer and R. Bulirsch: Introduction to Numerical Analysis, 3rd ed., Springer-Verlag, New York, 2002.
Further reference: Numerical Mathematics, by A. Quarteroni, R. Sacco, and F. Saleri. Springer-Verlag, 2000 |
Description: |
We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. The course will focus on interpolation, numerical differentiation and integration, numerical quadrature, solving nonlinear equations, and numericalsolutions of ordinary differential equations.
Note: This is the first semester of a two semester course. |
Math 6376 - Section: 31826 - Numerical linear algebra - by Kuznetsov
MATH 6376: Numerical linear algebra(section# 31826 ) |
Time: |
MoWe 1:00PM - 2:30PM - Room: PGH 348 |
Instructor: |
Yuri Kuznetsov |
Prerequisites: |
Senior Undergraduate Courses on Advanced Linear Algebra and Numerical Analysis are highly recommended |
Text(s): |
G.H.Golub and C.F.Van Loan, Matrix Computations |
Description: |
In this course,we consider the basic numerical methods for the numerical solution of linear algebraic systems and eigenvalue problems with symmetric and nonsymmetric matrices. Special attention will be paid to large scale algebraic problems. The list of methods includes Gaussian elimination, orthogonal decompositions,relaxation and GMRES methods for algebraic sytems as well as the QR-algorithm and the Lanczos method for eigenvalue problems.We also will consider methods for the least square problems and constrained minimization problems which results in algebraic systems with saddle point matrices.The methods and algotithms will be illustrated by examples from advanced practical applications. |
Math 6377 - Section: 24386 - Basic Tools for the Applied Mathematician - by Sanders
MATH 6377: Basic Tools for the Applied Mathematician (section# 24386 ) |
Time: |
TuTh 4:00PM - 5:30PM - Room: SR 121 |
Instructor: |
Sanders |
Prerequisites: |
Second year Calculus. Elementary Matrix Theory. Graduate standing or consent of instructor. |
Text(s): |
Lecture notes will be supplied by the instructor. |
Description: |
Finite dimensional vector spaces, linear operators, inner products, eigenvalues, metric spaces and norms, continuity, differentiation, integration of continuous functions, sequences and limits, compactness, fixed-point theorems, applications to initial value problems. |
Math 6382 - Section: 24388 - Probability Models and Mathematical Statistics - by Josic
MATH 6382: Probability Models and Mathematical Statistics (section# 24388 ) |
Time: |
TuTh 1:00PM - 2:30PM - Room: PGH 347 |
Instructor: |
Kresimir Josic |
Prerequisites: |
Undergraduate course in probability. |
Text(s): |
Jeffrey Rosenthal: A first look at rigorous probability
Publisher: World Scientific Publishing Company; 2 edition (November 14, 2006)
Language: English
ISBN-10: 9812703713
ISBN-13: 978-9812703712 |
Description: |
Emphasis will be placed on a thorough understanding of the basic concepts as well as developing problem solving skills. Topics covered include: combinatorial analysis, independence and the Markov property, Markov chains, the major discrete and continuous distributions, joint distributions and conditional probability, modes of convergence. These notions will be examined through examples and applications. |
Math 6384 - Section: 24390 - Discrete - Time Models in Finance - by Weiwei Xie
MATH 6384: Discrete - Time Models in Finance(section# 24390 ) |
Time: |
TuTh 2:30PM - 4:00PM - Room: PGH 350 |
Instructor: |
Weiwei Xie |
Prerequisites: |
Math 6382, or equivalent background in probability. |
Text(s): |
Introduction to Mathematical Finance: Discrete-Time Models, by Stanley Pliska, Blackwell Publishing, 1997 ISBN 1-55786-945-6 |
Description: |
This course is for students who seek a rigorous introduction to the modern financial theory of security markets. The course starts with single-periods and then moves to multiperiod models within the framework of a discrete-time paradigm. We study the valuation of financial,interest-rate, and energy derivatives and optimal consumption and investment problems. The notions of risk neutral valuation and martingale will play a central role in our study of valuation of derivative securities. The discrete time stochastic processes relating to the subject will also be examined. The course serves as a prelude to a subsequent course entitled Continuous-Time Models in Finance. |
Math 6395 - Section: 32297 - Complex geometry and analysis - by Ji
MATH 6395: Complex geometry and analysis (section# 32297 ) |
Time: |
MoWeFr 9:00AM - 10:00AM - Room: PGH 350 |
Instructor: |
Shanyu Ji |
Prerequisites: |
Math 6322-6323, or equivalent. |
Text(s): |
Lecture Note ( No required textbook) |
Description: |
L^2 Estimates,
Coherent Sheaves, Complex Analytic Spaces,
Positive Currents and Potential Theory,
Sheaf Cohomology, Positive Vector Bundles and Vanishing Theorems. |
Math 6397 - Section: 32299 - Automatic learning applied to proteomics and genomics - by Azencott
MATH 6397: Automatic learning applied to proteomics and genomics (section# 32299 ) |
Time: |
TuTh 2:30PM - 4:00PM - Room: AH 301 |
Instructor: |
Robert Azencott |
Prerequisites: |
Basic notions of probability theory will be redefined in the course; previous familiarity with random vectors and standard probability distributions at the undergraduate level will be assumed; basic definitions concerning Hilbert spaces and the Fourier transform, will be assumed to be known. |
Text(s): |
Kernel Methods in Computational Biology, B. Schölkopf, K. Tsuda, J.-P. Vert.
Reference book : selected chapters in Introduction to Support Vector Machines N. Cristianini and J. Shawe-Taylor Cambridge University Press, 2000,ISBN: 0 521 78019 5. |
Description: |
Automatic Learning of unknown functional relationships Y = f(X) between multidimensional inputs X and outputs Y, involves algorithms dedicated to the intensive analysis of large finite "training sets" of "examples" of inputs/outputs pairs (X,Y). This formalism applies to perceptive tasks involving automatic classification / identification of complex patterns such as shapes, sounds and speech, handwriting, texts, and has more recently been widely extended to classification / diagnosis in proteomics and genomics.
New machine learning algorithms have emerged in the last 10 years, as well as powerful mathematical concepts, focused on key learning features : generalisation, accuracy, speed, robustness. The mathematics involved rely on the use of positive definite kernels associated to Hilbert space representations of data, information and probability theory, quantification of complexity, functional approximation. The course will present major kernel based learning architectures : Support Vector Machines and selected applications to proteomics and genomics. |
Math 6397 - Section: 32300 - Information theory with applications - by Bodmann
MATH 6397: Information theory with applications (section# 32300 ) |
Time: |
TuTh 10:00AM - 11:30AM - Room: PGH 350 |
Instructor: |
Bernhard Bodmann |
Prerequisites: |
MATH 4320 or 5382 or 5385 or 6382 or 6388 or equivalent. Knowledge of Matlab useful, but not a strict prerequisite. |
Text(s): |
T.-S. Han and K. Kobayashi, Mathematics of Information and Coding, AMS, 2001 (approx \$100);
R. Gray, Entropy and Information Theory, Springer, 1991, available online (free). |
Description: |
Source and channel coding from the very basics to currently active research in an intuitive, but mathematically sound manner. The material will be interspersed with simple programming projects and experiments.
Topics: Entropy, mutual information, source coding, information rate and ergodicity, arithmetic codes, channel capacity, data processing inequality, rate distortion theory, cryptography, pseudo-random number generation, vector-valued generalizations. |
Math 6397 - Section: 32301 - Stochastic Differential Equations - by Torok
MATH 6397: Stochastic Differential Equations(section# 32301 ) |
Time: |
TuTh 1:00PM - 2:30PM - Room: SEC 204 |
Instructor: |
Torok |
Prerequisites: |
Graduate (or advanced undergraduate) standing |
Text(s): |
We will mainly follow the notes of L. C. Evans (UC Berkeley), available on his web-page. Additional material will be handed out or placed on reserve in the library. |
Description: |
Stochastic differential equations arise when some randomness is allowed in the coefficients of a differential equation. They have many applications, including mathematical biology, theory of partial differential equations, differential geometry and mathematical finance.
This is an introduction to the theory and applications of stochastic differential equations. A knowledge of measure theory is strongly recommended but is not required. First we will review measure theory, probability spaces, random variables and stochastic processes. Brownian motion will be discussed in some detail. Then we will introduce the Ito integral and relevant aspects of martingale theory as a method to formulate and solve stochastic differential equations. Numerical schemes will be also discussed. Applications will include mathematical finance (arbitrage and option pricing). |
Math 7320 - Section: 32302 - Functional analysis - by Blecher
MATH 7320: Functional analysis(section# 32302 ) |
Time: |
MoWe 1:00PM - 2:30PM - Room: SR 138 |
Instructor: |
David Blecher |
Prerequisites: |
|
Text(s): |
Lecture notes
will be provided.
Recommended book: Pedersen's "Analysis Now" or Conway's "A course
in Functional Analysis". |
Description: |
Although we will be starting from scratch, if you wish to do some preliminary reading you could read the middle section in Royden's book on Real Analysis (on the Hahn Banach theorem, and so on). We will be starting with a little topology - so you might glance through any good basic book on topology to familiarize yourself with "compactness" , "locally compact", continuous functions between topological spaces, the basic theory of metric spaces. The reason I review some topology is to familiarize the students with the use of `nets' (=generalized sequences) in topology.
We will mostly avoid measure theory, so don't be too concerned if you lack that background. The tests and exam will be based on the notes given in class, and on the homework. After each chapter we will schedule a problem solving workshop, based on the homework assigned for that chapter.
Final grade is approximately based on a total score of 400 points consisting of homework (100 points), a semester test (100 points), and a final exam (200 points). The instructor may change this at his discretion.
The first semester will be a leisurely and general presentation, starting from scratch, of the basic facts in Linear Analysis, Banach spaces and Hilbert spaces. The second semester will be a more technical development of the theory of linear operators on Hilbert space; Algebras and spectral theory (focusing on the spectral theorem). Some Fourier series and transforms. Unbounded operators. We will also cover some topics which the students request. |
Math 7397 - Section: 32303 - Financial and energy time series analysis - by Weiwei Xie
MATH 7397: Financial and energy time series analysis (section# 32303 ) |
Time: |
TuTh 10:00AM - 11:30AM - Room: AH 301 |
Instructor: |
Weiwei Xie |
Prerequisites: |
MATH 6397 Continuous-Time Models in Finance
MATH 6383 Probability and Statistics.
Math 6397 Discrete-Time Models in Finance. |
Text(s): |
Analysis of Financial Time Series, by Tsay, Ruey S., Wiley, 2002. |
Description: |
This course deals with statistical analysis of financial and energy data. We discuss parameter estimation of Brownian motion and their variants (e.g., jump diffusion, mean-reversion processes etc.) We study time series models, ARCH/GARCH models for modeling volatilities and their roles in computing Value-at_Risk. The course uses S-Plus as the computing platform. |
Math 7394 - Section: 36111 - Computational methods for Newtonian & non-Newtonian incompressible viscous flows - by Glowinski
MATH 7394: Computational methods for Newtonian & non-Newtonian incompressible viscous flows (section# 36111 ) |
Time: |
TuTh 11:30AM - 1:00PM - Room: AH 301 |
Instructor: |
Roland Glowinski |
Prerequisites: |
Basic Numerical Linear Algebra; Basic Numerical Methods for Ordinary Differential Equations; some knowledge of Mechanics and/or Physics may be helpful. |
Text(s): |
Lecture note. |
Description: |
The main goal of this course is to introduce the student to basic numerical methods for the solution of the Navier-Stokes type equations modeling incompressible flow (Newtonian or not). These methods will combine finite element approximations with various types of time-discretization schemes. They will be applied first to the simulation of Newtonian viscous flow, and then motivated by applications from Oil & Gas industry, to the simulation of visco-plastic fluid flow.
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