Graduate classes, Fall 2010, Mathematics
MATH 511: Analysis I  Credits: 4  − Description  − Sections 
Content: An introduction to fundamental analytic concepts including: The complex number system, geometry and topology of the complex plane, analytic functions, conformal mappings, complex integration, and singularities. Texts: Functions of One Complex Variables I, by John B. Conway. Springer Publishing, ISBN: 3540903283. Assessments: TBA Prerequisites: TBA  000  MSC: E408  TuTh 10:00am  11:15am  Shanshuang Yang  max 16  MATH 515: Numerical Analysis I  Credits: 4  − Description  − Sections 
Content: This course covers topics in numerical linear algebra, including: matrix factorizations such as LU, QR and SVD; direct and iterative methods for solving linear systems and least squares problems; numerical approaches to solving eigenvalue problems; problem sensitivity and algorithm stability. A solid theoretical background of algorithms will be balanced with practical implementation issues. Texts: Matrix Computations by Gene H. Golub & Charles F. Van Loan. John Hopkins University Press. ISBN# 0801854148. Assessments: TBA Prerequisites: TBA  000  MSC: W306  TuTh 2:30pm  3:45pm  James Nagy  max 16  MATH 521: Algebra I  Credits: 4  − Description  − Sections 
Content: Linear algebra, including canonical forms, infinitedimensional vector spaces, tensor products, and multilinear algebra. Group theory including group actions and representations. Texts: Abstract Algebra, by Dummit adn Foote. Wiley Publishing. ISBN # 97804711433347 Assessments: TBA Prerequisites: TBA  000  MSC: E408  TuTh 11:30am  12:45pm  Raman Parimala  max 16  MATH 528: Algebraic Number Theory  Credits: 4  − Description  − Sections 
Content: Topics will include: Algebraic numbers and integers, Dedekind domains, discriminants, norm and trace, cyclotomic
fields, factorization of ideals in Dedekind domains, decomposition of prime ideals in integral extensions, Hilbert ramification theory, lattice methods, class group, finiteness of class number, Dirichlet’s unit theorem, valuations, completions, local fields, and the ramification theory of local fields. Texts: (1) Number Fields, by Daniel Marcus. Springer Publishing. ISBN # 1003787902791.
(2) Introductory Algebraic Number Theory, by Saban Alaca and Kenneth Williams. Cabridge University Press. ISBN 1005211540119. Assessments: TBA Prerequisites: Math 521 and 522, or consent of instructor.  000  MSC: E406  MWF 12:50pm  1:40pm  Ken Ono  max 16  MATH 535: Combinatorics I  Credits: 4  − Description  − Sections 
Content: This is the first course of a 2course sequence on combinatorial mathematics.
We shall focus on mostly introductory material on enumeration, relational structures [graphs, directed graphs, partially ordered sets, hypergraphs, matroids], and configurations [designs, codes, permutation groups]. Texts: The main reference will be P. J. Cameron, "Combinatorics: Topics, Techniques, Algorithms" [ISBN 0 521 45761 0] Assessments: The course grade will be determined by written assignments, collected throughout
the term, and a combination takehome and inclass final exam. We may replace some assignments and exams with student presentations. Prerequisites: The course does not assume background in combinatorics. It is assumed that
all students have good undergraduate background in linear algebra, abstract algebra, and analysis.  000  MSC: E406  MWF 10:40am  11:30am  Dwight Duffus  max 16  MATH 543: Algebraic Topology I  Credits: 4  − Description  − Sections 
Content: Homotopy theory, the fundamental group, free products of groups with amalgamation, Van Kampen's Theorem, covering spaces, classification of surfaces, classifying spaces, higher homotopy groups Texts: Introduction to Manifolds, by John Lee. Springer Publishing. ISBN# 0387950265. Assessments: TBA Prerequisites: TBA  000  MSC: E408  TuTh 1:00pm  2:15pm  Aaron Abrams  max 16  MATH 545: Introduction to Differential Geometry I  Credits: 4  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E406  MWF 11:45am  12:35pm  David Borthwick  max 16  MATH 561: Matrix Analysis  Credits: 4  − Description  − Sections 
Content: Main topics:
Eigenvalues and eigenvectors of matrices, invariant subspaces, Schur triangular form, diagonalizable matrices, minimal polynomial, characteristic polynomial, HamiltonCayley Theorem, localization of eigenvalues, Gerschgorin's Theorem. Unitary, Hermitian and skewHermitian matrices. Normal matrices and the Spectral Theorem. Orthogonalization. Householder matrices and the QR factorization.
MoorePenrose pseudoinverse. Applications to the solution of under and overdetermined systems of linear equations. Other generalized inverses. Applications to data fitting (leastsquares approximation).
The Singular Value Decomposition. Matrix norms: spectral norm and Frobenius norm. Solution to matrix nearness problems. Applications to signal processing and information retrieval. Jordan canonical form. An algorithmic proof. Powers of matrices. Matrix functions. Applications to systems of differential equations. Bilinear and quadratic forms. Hermitian forms. Congruence. Sylvester's Law of Inertia. Rayleigh's principle. CourantFischer Theorem. Positive definite and semidefinite matrices. Applications to statistics (covariance matrices) and numerical analysis (PDE's).
Additional topics:
Nonnegative matrices. The spectral radius. Positive matrices. Directed graphs. Nonnegative irreducible matrices. PerronFrobenius Theorem. Mmatrices. Applications to probability theory (Markov chains), economics (Leontiev's inputoutput model), and numerical analysis (iterative methods for linear systems). Structured matrices: circulant, Toeplitz, Hankel, Cauchy, Vandermonde, others. Block generalizations. Applications in signal processing, image processing, and numerical analysis (PDE's, interpolation). Texts: Matrix Analysis, by Horn & Johnson. Cambridge University Press. ISBN #0521386322. Assessments: TBA Prerequisites: TBA  000  MSC: W306  TuTh 1:00pm  2:15pm  Michele Benzi  max 16  MATH 577R: Seminar in Combinatorics  Credits: 4  − Description  − Sections 
Content: The seminar in combinatorics is a research seminar for students and faculty. It runs weekly, and features speakers from outside Emory who come to talk about topics of interest to the Emory faculty. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W306  F 4:00pm  4:50pm  Dwight Duffus  max 30  MATH 578R: Seminar in Algebra  Credits: 1  12  − Description  − Sections 
Content: Research topics in algebra of current interest to faculty and students. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W303  Tu 4:00pm  4:50pm  Raman Parimala  max 30  MATH 579R: Seminar in Analysis  Credits: 4  − Description  − Sections 
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W301  Tu 4:00pm  4:50pm  Vladimir Oliker  max 30  001  MSC: W306  W 12:50pm  1:40pm  James Nagy  max 30  MATH 597R: Directed Study  Credits: 1  12  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  00P  TBA   Faculty (TBA)  max 999  MATH 599R: Master's Thesis Research  Credits: 1  12  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  00P  TBA   Faculty (TBA)  max 999  MATH 772: Numerical Partial Differential Equations  Credits: 4  − Description  − Sections 
Content: Partial Differential Equations are a formidable tool for describing realworld problems ranging from fluid dynamics to economical studies. Unfortunately, in many cases an explicit solution of these equations cannot be found. Numerical techniques are mandatory for finding an approximate solution. The choice of the most appropriate method is usually problemdependent. In this course we will consider different classes of differential problems and discuss possible methods of solution, their features in terms of accuracy, computational cost, implementation.
Particular emphasis will be given to the Galerkin class of methods (Finite Elements, Spectral Methods), even if Finite Volume and Finite Differences methods will be considered as well. The course will be partially carried out in the Computer Lab, where theoretical properties of the different methods will be verified with Matlab and FreeFem codes. Texts: Numerical Models for Differential Problems, by Alfio Quarteroni. Springer Publishing. ISBN 10: 8847010705. Assessments: TBA Prerequisites: TBA  000  MSC: W304  TuTh 10:00am  11:15am  Alessandro Veneziani  max 16  MATH 787R: Topics in Combinatorics: Ramsey Theory  Credits: 4  − Description  − Sections 
Content: This course will continue the development of ramsey theory begun in Math 531532 and Math 535536. Included will be: Sets: Ramsey's theorem, the compactness principle. Progressions: Van der Waerden's Theorem, the HalesJewett Theorem, spaces  affine and vector, Roth's Theorem and Szemeredi's Theorem. Equations: Schur's Theorem, regular homogeneous equations and systems, Rado's Theorem, finite sums and unions, Folkman's Theorem. Numbers: exact ramsey numbers, asymptotics, Van der Waerden numbers. Schur and Rado numbers, higher ramsey numbers. Bipartite ramsey theory, induced ramsey theory, restricted results, Euclidean and Graph ramsey theory. Topological Dynamics, Ultrafilters, the infinite. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E406  TuTh 8:30am  9:45am  Vojtech Rodl  max 16  MATH 797R: Directed Study  Credits: 1  12  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  00P  TBA   Faculty (TBA)  max 999  MATH 799R: Dissertation Research  Credits: 1  12  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  00P  TBA   Faculty (TBA)  max 999 
