# Graduate classes, Fall 2013, Mathematics

 MATH 500: Probability Credits: 3 − Description − Sections
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000MSC: E406TuTh 10:00am - 11:15amZoltan Furedimax 16
 MATH 511: Analysis I Credits: 3 − 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.
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000MSC: W306MWF 10:00am - 10:50amDavid Borthwickmax 20
 MATH 515: Numerical Analysis I Credits: 3 − 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.
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000MSC: W304TuTh 11:30am - 12:45pmMichele Benzimax 20
 MATH 521: Algebra I Credits: 3 − Description − Sections
Content: An introduction to fundamental algebraic concepts including: Groups, homomorphisms, the action of a group on a set, Sylow theorems, group representations and characters, rings, ring homomorphisms, and basics on commutative rings.
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000MSC: E408MW 2:30pm - 3:45pmRaman Parimalamax 16
 MATH 523: Commutative Algebra and Geometry Credits: 3 − Description − Sections
Content: This is an introductory course covering topics like affine and projective varieties, Hilbert's Nullstellensatz, morphism and rational maps of varieties, dimension, smoothness, algebraic curves, intersection multiplicity and Bezout's theorem. Topics from commutative algebra like integral extensions, Noether's normalisation lemma and primary decomposition will also be treated simultaneously.
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000MSC: E408MW 11:30am - 12:45pmSuresh Venapallymax 16
 MATH 528: Algebraic Number Theory Credits: 3 − Description − Sections
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000MSC: E408MW 1:00pm - 2:15pmKen Onomax 16
 MATH 531: Graph Theory I Credits: 3 − Description − Sections
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000MSC: E408TuTh 8:30am - 9:45amZoltan Furedimax 16
 MATH 543: Algebraic Topology I Credits: 3 − Description − Sections
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000MSC: E408TuTh 11:30am - 12:45pmRobin Formanmax 16
 MATH 545: Introduction to Differential Geometry I Credits: 3 − Description − Sections
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000MSC: W304TuTh 2:30pm - 3:45pmVladimir Olikermax 16
 MATH 577R: Seminar in Combinatorics Credits: 3 − 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.
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000MSC: W306F 4:00pm - 4:50pmDwight Duffusmax 20
 MATH 577R: Seminar in Combinatorics: Introduction to Complex Networks Credits: 3 − Description − Sections
Content:  Mathematical background. A brief on matrix algebra. A brief on data analysis.  Introduction to networks. History; Basic Definitions; Walks and Paths: Eulerian and Hamiltonian networks, random walks, shortest path distance, Degree Distributions: Poisson, power law, identifying correlations; Graph Connectivity: connected components, cliques; Graph Structures: trees, bipartivity and planarity; Modern Applications.  Adjacency relation in networks. Degree distributions; degree-degree correlations; Spectral properties of networks. Spectrum of the Adjacency Matrix, the Graph Laplacian.  Fragments in complex networks. subgraphs, network motifs, graphlets, closed walks, combinatorics of subgraphs, clustering coefficients, combinatorics of assortativity, network hierarchy, network reciprocity  Matrix functions on networks. exponential adjacency matrix, hyperbolic functions of the adjacency matrix, network entropy and free energy, network communicability, returnability  Centrality measure in networks. A description of the definitions and applications of centralities like degree, Katz, eigenvector, PageRank, subgraph, closeness and betweenness.  Global measures of networks. A description of a few global measures characterising the degree heterogeneity, fractality, and topological classes of networks. Adjacency and distance-based invariants; expansion and network classes; spectral scaling method; network bipartivity  Community structure in networks. Network partition methods; local improvement methods; spectral partittioning; methods based on link centrality; quality criteria; methods based on modularity; the problem of resolution; clustering based on similarity; communities based on communicability; how to find bipartitions in networks.  Random models of networks. Classical random networks; small-world random networks; scale-free networks; random geometric networks.
Texts: Estrada, E. The Structure of Complex Networks. Theory and Applications. Oxford University Press, 2011. ISBN: 978-0-19-959175-6
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001MSC: W302TuTh 10:00am - 11:15amErnesto Estradamax 20
 MATH 578R: Seminar in Algebra Credits: 3 − Description − Sections
Content: Research topics in algebra of current interest to faculty and students.
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000MSC: W306W 4:00pm - 4:50pmKen Onomax 20
001MSC: E406F 1:00pm - 2:00pmDavid Zureick-Brownmax 16
 MATH 579R: Seminar in Analysis Credits: 3 − Description − Sections
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's
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000MSC: W301Tu 4:00pm - 4:50pmVladimir Olikermax 20
001MSC: W302F 12:00pm - 12:50pmAlessandro Venezianimax 25
 MATH 597R: Directed Study Credits: 1 - 9 − Description − Sections
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 MATH 599R: Master's Thesis Research Credits: 1 - 9 − Description − Sections
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 MATH 797R: Directed Study Credits: 1 - 9 − Description − Sections
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 MATH 799R: Dissertation Research Credits: 1 - 9 − Description − Sections
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