Graduate Course Outlines, Summer 2014-Spring 2015

Contact

Please direct inquiries about our graduate program to:
grad@math.lsu.edu

Summer 2014

  • MATH 7999-1: Problem Lab in Algebra —practice for PhD Qualifying Exam in Algebra.
  • Instructor:
  • Prerequisite: Math 7210.
  • Text: Online Test Bank.
  • MATH 7999-2: Problem Lab in Real Analysis—practice for PhD Qualifying Exam in Analysis.
  • Instructor:
  • Prerequisite: Math 7311.
  • Text: Online Test Bank.
  • MATH 7999-3: Problem Lab in Topology—practice for PhD Qualifying Exam in Topology.
  • Instructor:
  • Prerequisite: Math 7510.
  • Text: Online Test Bank.

Fall 2014

  • MATH 4997-2: Vertically Integrated Research: Curves & Surfaces in 3D
  • Instructor: Profs. Dasbach and Stoltzfus
  • Prerequisites: Math 2057, Math 2085 (or Math 2090) or equivalent, or permission of instructors.
  • Text:
  • We will study the mathematical development of the geometry and manipulation of curves and surfaces in 3D with a blend of computer graphics. The course will develop the mathematical classification of surfaces by genus. As an application we will study embedded surfaces whose boundary is a given knotted circle.

    The computer representation of curves and surfaces in Mathematica (STL "Standard Tessellation Language" files) will be covered as well as options for scanning and printing of surfaces.

  • MATH 7001: Communicating Mathematics I
  • Instructor: Prof. Oxley.
  • Prerequisite: Consent of department. This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for one-semester-hour's credit.
  • This course provides practical training in the teaching of mathematics at the pre-calculus level, how to write mathematics for publication, and treats other issues relating to mathematical exposition. Communicating Mathematics I and II are designed to provide training in all aspects of communicating mathematics. Their overall goal is to teach the students how to successfully teach, write, and talk about mathematics to a wide variety of audiences. In particular, the students will receive training in teaching both pre-calculus and calculus courses. They will also receive training in issues that relate to the presentation of research results by a professional mathematician. Classes tend to be structured to maximize discussion of the relevant issues. In particular, each student presentation is analyzed and evaluated by the class.
  • MATH 7210: Algebra I
  • Instructor: Prof. Mahlburg.
  • Prerequisites: MATH 4200 and 4201, or equivalents
  • Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003.
  • This is the first semester of the first year graduate algebra sequence, and covers the material required for the Comprehensive Exam in Algebra. It will cover the basic notions of group, ring, module, and field theory. Topics will include symmetric groups, the Sylow theorems, group actions, solvable groups, Euclidean domains, principal ideal domains, unique factorization domains, polynomial rings, modules over PIDs, vector spaces, field extensions, and finite fields.
  • MATH 7230: Algebraic Number Theory.
  • Instructor: Prof. Morales.
  • Prerequisites: The standard graduate sequence in algebra (MATH 7210-11 or equivalent).
  • Text: A Course in Algebraic Number Theory by Robert B. Ash, Dover Books on Mathematics, 2010. (Also available from the author's website.)
  • We will explore the general theory of factorization of ideals in Dedekind domains and in number fields. We will illustrate with explicit calculations the use of Kummer's theorem on lifting of prime ideals in extension fields and the factorization of prime ideals in Galois extensions. We will also discuss the Dirichlet unit theorem, the finiteness of the ideal class group of a number field and Minkowski's bounds. We will cover local fields and standard theorems such as the Artin-Whaples approximation theorem and Hensel's lemma. Time permitting, we will discuss relations with the theory of algebraic curves.
  • MATH 7240: Algebraic Geometry.
  • Instructor: Prof. Hoffman.
  • Prerequisites: Algebra at the level of Math 7210, and topology at the level of Math 7510.
  • Text: (recommended, but not required):
    1. Hartshorne, Algebraic Geometry, (Graduate Texts in Math, 52), Springer; 1st ed. 1977. Corr. 8th printing 1997 edition (April 1, 1997).
    2. Atiyah and Macdonald, Introduction to Commutative Algebra, Addison-Wesley Series in Mathematics.
  • This course will be an introduction to schemes. Along the way I will cover the necessary commutative algebra. We will start with the spectrum of a commutative ring and then proceed to sheaf theory. Affine and projective varieties over an algebraically closed field will provide key examples, and will be discussed. There is no required text, but rather there will be online resources, such as MIT’s open course. The student will be expected to do outside reading of this material.
  • MATH 7290: Cluster Algebras.
  • Instructor: Prof. Yakimov.
  • Prerequisites:
  • Text:
  • Cluster Algebras are an axiomatic family of algebras that was invented by Fomin and Zelevinsky in 2000. Within a few years this area grew into a very dynamic area of pure mathematics with many relations and applications to algebra, representation theory, combinatorics, topology, Poisson geometry and mathematical physics. In this course we will introduce some of the main methods for studying cluster algebras and some of their applications. Many examples will be supplied throughout the course.

    The list of topics that will be covered include the following: quiver mutation, Laurent phenomenon, finite type classification and other examples, dependence on coefficients, additive categorifications, quivers with potentials, cluster algebras in Teichmüller theory and Poisson geometry, Zamolodchikov periodicity, and dilogarithm identities.

  • MATH 7311: Real Analysis I.
  • Instructor: Prof. Lipton.
  • Prerequisite: Math 4032 or 4035 or the equivalent.
  • Required Text: L. Richardson, Measure and Integration: A Concise Introduction to Real Analysis, John Wiley & Sons, June 2009. ISBN: 978-0-470-25954-2;
  • Recommended Reference: H. L. Royden, Real Analysis, 3rd ed., Macmillan, ISBN 0024041513.
  • We will treat measure theory and integration on measure spaces. The examples of the real line and of Euclidean space will be emphasized throughout. Topics will include the Hopf extension theorem, completion of the Borel measure space, Egoroff's theorem, Lusin's theorem, Lebesgue dominated convergence, Fatou's lemma, product measures, Fubini's theorem, absolute continuity, bounded variation, Vitali's covering theorem, Lebesgue differentiation theorems, and the Radon-Nikodym theorem. Applications to Lp and its dual, and the Riesz-Markov-Saks-Kakutani theorem may be presented if there is sufficient time.
  • MATH 7325: Finite Element Method.
  • Instructor: Prof. Sung.
  • Prerequisites: Advanced Calculus and/or Real Analysis
  • Text: The Mathematical Theory of Finite Element Methods (Third Edition) by Susanne Brenner and Ridgway Scott
  • This course provides an introduction to the theory of finite element methods. The following topics will be covered: Hilbert Spaces, Sobolev Spaces, Variational Formulation of Elliptic Boundary Value Problems, Finite Element Methods, Interpolation Error Estimates, a priori and a posteriori Error Estimates, Nonconforming Finite Element Methods and Mixed Finite Element Methods.
  • MATH 7350: Complex Analysis.
  • Instructor: Prof. Shipman.
  • Prerequisites: Math 7311 or equivalent
  • Text: Complex Analysis by Elias Stein and Rami Shakarchi, Princeton Lectures in Analysis II.
  • Theory of holomorphic functions of one complex variable; path integrals, power series, singularities, mapping properties, normal families, other topics.
  • MATH 7360: Probability Theory.
  • Instructor: Prof. Sundar.
  • Prerequisites: Mathematics 7311 or its equivalent
  • Text: Probability: A Survey of the Mathematical Theory by J. W. Lamperti
  • The course is a self-contained introduction to modern probability theory. It starts from the concept of probability measures, random variables, and independence. Well-known limit theorems for sums of independent random variables such as the Kolmogorov's law of large numbers will be studied. Sums of independent, centered random variables form the prototype for an important class of stochastic processes known as martingales, and therefore play a major role. Next, weak convergence of probability measures will be introduced and the central limit theorem will be established. A main part of the course is to understand conditional probability and build the basic theory of martingales. Brownian motion is an important example of a continuous-time martingale. Its basic features will be briefly discussed.
  • MATH 7375: Fourier Analysis and Wavelets.
  • Instructor: Prof. Olafsson.
  • Prerequisites: Math 7311
  • Text: : No final decision has been made but we will probably use Introduction to Fourier Analysis and Wavelets by M. A. Pinsky (AMS) and also our own notes.
  • This is a basic introductory course in Fourier analysis and wavelets. We start with Fourier series of smooth and square integrable periodic functions..We then discuss the Fourier transform on the n-dimensional Euclidean space. Topics includes convolution of functions, approximate identity, the Fourier transform of rapidly decreasing functions and tempered distributions. Applications includes the Shannon sampling theorem, Heisenberg uncertainty principle, and the windowed/short time Fourier transform. We then introduce the basic idea behind the wavelet theory and,as a simple example, the Haar wavelet. We then introduce multiresolution analysis, compactly supported wavelets, and wavelets in higher dimensions. We then finish off by given some basic ideas in constructing wavelets using tools from abstract harmonic analysis.
  • MATH 7384: Topics in Material Science: Riemann-Hilbert Problem
  • Instructor: Prof. Antipov.
  • Prerequisites: at least one of 7350 Complex Analysis, 7386 Theory of PDEs, 7311 Real Analysis.
  • Text: Lecture notes, Applied and Computational Complex Analysis, Vol. 3 by P. Henrichi and Boundary value problems by F.D. Gakhov.
  • The Riemann-Hilbert problem (RHP) is a powerful tool for the solution of boundary value problems for PDEs, singular integral equations, construction of conformal mappings, analysis of nonlinear integrable differential equations, and the Wiener-Hopf matrix factorization. It is widely used in fluid mechanics, materials science, and electromagnetics.

    The RHP in its modern formulation is a boundary-value problem of the theory of analytic, automorphic, algebraic, and generalized analytic functions. First this course will give an introduction to the classical RHP of the theory of analytic functions. Then more advanced topics will be considered. They include the RHP of symmetric and automorphic functions (the cases of finite and infinite discontinuous groups) and the RHP on an algebraic Riemann surface. The elements of the theory of abelian integrals and differentials, analogues of the Cauchy integral, the Riemann theta function and the Jacobi problem will also be discussed. The course will give applications of the RHP to the theory of conformal mappings, fracture mechanics, materials science (the problem on multiple neutral inclusions), and diffraction theory.

  • MATH 7386: Theory of Partial Differential Equations.
  • Instructor: Prof. Almog.
  • Prerequisites:
  • Text: Partial Differential Equations by Lawrence C. Evans
  • Introduction
    • A variational approach to the Dirichlet problem
    Sobolev spaces
    • Weak derivatives
    • One-dimensional Sobolev spaces
    • Approximation by smooth functions, extensions and the trace operator
    • Sobolev embeddings and the Rellich-Kondrachov theorem
    • additional topics: Poincare inequalities etc.
    Elliptic equations
    • Existence: the Lax-Milgram theorem
    • Regularity
    • Maximum principles and uniqueness
    Introduction to the calculus of variations
    • The Euler-Lagrange equations
    • Existence of minimizers
    • Eigenvalues of self-adjoint elliptic operators
  • MATH 7390: Calculus of Variations and Optimal Control.
  • Instructor: Prof. Wolenski.
  • Prerequisites:
  • Text:
  • This course is an introduction to the Calculus of Variations and Optimal Control Theory, and is aimed at Mathematics, Engineering, and Physics graduate students. The first half of the course will be devoted to the Calculus of Variations. We shall derive the classical necessary conditions of Euler-Lagrange, Weierstrass, Legendre, and Jacobi, and delve briefly into Hamilton--Jacobi Theory. Optimal Control Theory will be highlighted in the second half of the course. Various examples will be studied, and elements of the linear case will be thoroughly explored. In the nonlinear theory, the Pontryagin Maximum Principle will be featured with a variety of its applications. Topics can be added or deleted depending on available time and interests of the class.

    More specific topics include:

    • The Euler--Lagrange equation and Erdmann corner conditions;
    • The conditions of Weierstrass, Legendre, and Jacobi;
    • Sufficient conditions;
    • Tonelli's Existence Theorem;
    • Canonical transformations and the Principle of least action;
    • Hamilton--Jacobi Theory;
    • Linear and Linear--Quadratic Optimal Control Theory;
    • Bang--Bang theorem;
    • Kalman's Controllability condition;
    • The Pontryagin Maximum Principle;
    • Duality;
    • Applications.
  • MATH 7510: Topology I
  • Instructor: Prof. Vela-Vick.
  • Prerequisites: Advanced Calculus (Math 4031)
  • Text:Topology (2nd Edition), By James Munkres
  • This course is a preparation course for the Core I examination in topology. The course will quickly discuss some elementary notions in general topology before moving on to (basic) homotopy theory, the fundamental group, and covering spaces. Topological manifolds, simplicial complexes and CW complexes will also be discussed.
  • MATH 7520: Algebraic Topology.
  • Instructor: Prof. Stoltzfus.
  • Prerequisites: MATH 7510 and 7512, or equivalent.
  • Text: Algebraic Topology by James R. Munkres; (optional) Algebraic Topology by A. Hatcher
  • The cohomology ring of a space provides a finer invariants capable of distinguishing between more topological spaces: two spaces with inequivalent invariants cannot be topologically equivalent. Applications will be made in the topology of manifolds and duality theorems also relating homology and cohomology; Blanchfield Duality of knot modules; the cohomology rings of Lie groups, fibre bundles as well as associated classifying spaces, the Stiefel and Grassmann manifolds.
  • MATH 7590: Skein Theory.
  • Instructor: Prof. Gilmer.
  • Prerequisites: Math 7510; Math 7512
  • Text:
  • Skeins in a 3-dimensional manifold are linear combinations of 1-dimensional submanifolds modulo local relations. We will discuss skein modules of 3-dimensional manifolds. as well as the skein approach to knot polynomials and quantum invariants of 3-manifolds. You don’t need to know any knot theory or 3-manifold theory before you begin as results needed from these areas will be described (but not all proved).

Spring 2015

  • MATH 4997-1: Vertically Integrated Research: Quantum Information
  • 10:30-11:50 T Th
  • Instructor: Prof. Lawson.
  • Prerequisites: A reasonably good background in linear algebra (e.g.Math 2085) should be considered minimal background for the course. Some very basic knowledge of quantum theory and elementary probability theory would also be helpful.
  • Text: Quantum Computing by Nakahara and Ohmi (supplemented by class notes)
  • This introductory course on the recently emerging topics of quantum information theory and quantum computing will introduce students to major recent developments such as quantum encoding and cryptography, teleportation, error correction, and quantum computing. Basic concepts of quantum theory such as quantum states, qubits, entanglement, measurement, quantum gates etc. will be incorporated into the course. The mathematical content will center on a linear algebra approach to the subject through basic matrix theory (unitary and Hermitian matrices, inner products and orthonormal bases, tensors, positive and completely positive operators, etc.) together with some elementary probabilistic content.
  • MATH 4997-4: Vertically Integrated Research: Combinatorial Models in Representation Theory
  • 10:30-11:50 T Th
  • Instructor: Profs. Achar and Sage
  • Prerequisites: Familiarity with basic group theory and linear algebra, such as from Math 4200 and Math 4153.
  • Text: None
  • A basic problem in representation theory is that of understanding how to decompose tensor products. In 1999, Knutson and Tao proved the "saturation conjecture" for tensor products of representations of GL(n, C), by inventing and using a remarkable combinatorial tool called the "honeycomb model." This semester, we will recall how to construct the irreducible representations of GL(n, C). We will then learn what honeycombs are, and we will focus on reading the Knutson-Tao paper. If there is time later in the semester, we may look at other combinatorial models in representation theory.
  • MATH 4997-3: Vertically Integrated Research: Big Data and Topology
  • 3:30-4:20 M W F
  • Instructor: Profs. Dasbach and Stoltzfus
  • Prerequisites: Multivariate Calculus and Linear Algebra
  • Text: Computational Topology by Herbert Edelsbrunner and John L. Harer
  • In many applications data (of numerical information) can be represented by a point set in an n-dimensional space. Computational topology focuses on the computational aspects of topology, and applies topological tools to analysis of those data sets. Topological tools of simplicial homology, shape and discrete morse theory will be presented. Algorithms and software for the analysis of the topology, connectivity and shape of point sets will be explored. Projects will be either a software exploration implementation on a point data set of interest to you or a presentation of recent papers in the area.
  • MATH 7002: Communicating Mathematics II
  • 3:00-4:50 T Th
  • Instructor: Prof. Oxley .
  • Prerequisite: Consent of department. This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for one-semester-hour's credit.
  • MATH 7211: Algebra II.
  • 8:30-9:20 M W F
  • Instructor: Prof. Morales.
  • Prerequisites: Math 7210, or equivalent.
  • Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003.
  • This is the second semester of the first year graduate algebra sequence. Topics will include Galois theory, basics of commutative algebra, algebraic geometry, and homological algebra. Time permitting, we will also include an introduction to semi-simple associative algebras and linear representations of finite groups.
  • MATH 7250: Geometric Representation Theory
  • 12:00-1:20 T Th
  • Instructor: Prof. Sage.
  • Prerequisites: Math 7211 or permission of the instructor.
  • Text: Flag Varieties: An Interplay of Geometry, Combinatorics, and Representation Theory by Lakshmibai and Brown
  • Geometric methods play a central role in modern representation theory. Indeed, geometry often reveals unexpected connections between disparate phenomena. For example, the irreducible representations of the Weyl group of SL(n, C) are parametrized by partitions of n. Nilpotent conjugacy classes in the Lie algebra of SL(n, C) are also parametrized by partitions of n. Is this just a combinatorial coincidence? No--it is possible to define a natural bijection between these two sets via a geometric construction, and this bijection (the Springer correspondence) has been of fundamental importance in representation theory.

    In this course, we will discuss several examples of geometric methods in representation theory. In particular, we will study flag varieties and the nilpotent cone and how they can be used to study representations (of Weyl groups, simple algebraic groups, Hecke algebras, etc) from a geometric perspective. Time permitting, we will also examine affine flag manifolds and the Bruhat-Tits building of a simple algebraic group.

  • MATH 7290: Quiver Varieties
  • 10:30-11:20 M W F
  • Instructor: Prof. Achar.
  • Prerequisites: 7240 or some familiarity with algebraic geometry. Some Lie theory will be helpful but not required
  • Text: none. I will distribute various lecture notes and research papers
  • A "quiver" is just a directed graph, and a "representation" of a quiver is an assignment of a vector space to each vertex of the quiver, and a linear map to each edge. A "quiver variety" is (roughly) a space whose points parametrize representations of a given quiver. The geometry of quiver varieties is a rich and multifaceted subject, with surprising and deep connections to Lie theory and symplectic geometry. We will cover the basic facts on Lusztig quiver varieties and Nakajima quiver varieties, but the precise selection of topics will depend to some extent on the background and interests of the students.
  • MATH 7320: Ordinary Differential Equations.
  • 11:30-12:20 M W F
  • Instructor: Prof. Wolenski.
  • Prerequisites:
  • Text:
  • MATH 7330: Functional Analysis.
  • 10:30 - 11:50 T Th
  • Instructor: Prof. Rubin.
  • Prerequisites: MATH 7311 or equivalent.
  • Text: No special textbook. I will distribute my own notes.
  • Functional Analysis is the language of modern mathematics. The course provides an introduction to the general theory of normed and metric spaces, linear operators and linear functionals, Banach and Hilbert spaces, spectral theory, and other important topics.
  • MATH 7366: Stochastic Analysis.
  • 9:30-10:20 M W F
  • Instructor: Prof. Sundar.
  • Prerequisites: Math 7360, or its equivalent
  • Text:
  • The course will begin with a study of Brownian motion, and stochastic processes related to it. Next, we will establish a fundamental connection between Brownian motion and partial differential equations. We will proceed to build stochastic integrals and prove the Itô formula. This will lead us naturally to the study of stochastic differential equations. Such equations arise in numerous applications, ranging from finance to mathematical biology and physics. The course will include a discussion of techniques that are unique to probability theory such as removal of drift by change of measure, and the Stroock-Varadhan martingale problem.
  • MATH 7384: Topics in Material Science: An introduction to High Performance Computing.
  • 2:00-5:00 W
  • Instructor: Prof. Bourdin.
  • Prerequisites: Basic computer proficiency is expected. A laptop (or image) running mac OS or linux is required for this class. Basic knowledge of numerical analysis will help but is not absolutely necessary.
  • Text: Additional material will be provided by the instructor
  • Syllabus
    • Unix command line
    • Version control systems: mercurial and GIT, bitbucket or github
    • Python scripting, ipython, ipython notebook, basic use of matplotlib
    • Basic python programming
    • Basic fortran programming
    • Basic C programming
    • Programming tools: makefiles, debuggers, etc
    • Concept of distributed programming
    • openMP
    • MPI
    • Using supercomputers: remote access using SSH, SSH keys, file transfer using globus, queuing systems.
    • PETSc
    • Visualization using python, visit and paraview
    The class will meet in Lockett 233 every Wednesday from 2PM until 5PM. Class time will be split between formal lecture and exercise sessions. The homework will consist of programming assignments.
  • MATH 7410: Graph Theory.
  • 1:30-2:20 M W F
  • Instructor: Prof. Oporowski.
  • Prerequisites:
  • Text:
  • The main theme of this course will be graph theory. We will discuss a wide range of topics, including spanning trees, eulerian trails, matching theory, connectivity, hamiltonian cycles, coloring, planarity, integer flows, surface embeddings, and graph minors. For more information see Math 7410.
  • MATH 7490-1: Graph Minors II.
  • 11:30-12:20 M W F
  • Instructor: Prof. Ding.
  • Prerequisites: Graph Minors I
  • Text: None
  • This is an extension of our first course on graph minors. We will first review main results discussed in Graph Minors I. Then we will talk about possible approaches to some open problems of the following types: 1. for a given graph H, characterize graphs that do not contain H as a minor; 2. for a given minor-closed class C of graphs, determine minimal graphs that are not contained in C; 3. the large-graph version of the last two problems. Hopefully, some of the problems could become a dissertation topic for some students.
  • MATH 7490-2: Matroid Theory
  • 2:30-3:20 M W F
  • Instructor: Prof. van Zwam.
  • Prerequisites: Consent of department
  • Text: Matroid Theory, second edition by James Oxley, Oxford University Press.
  • Matroid theory is the study of abstract properties of linear dependence. Matroids arise from finite sets of vectors, graphs, error-correcting codes, algebraic field extensions, projective geometry, optimization algorithms, and more. In its study, the influences from graph theory and from finite geometry are especially prevalent: connectivity and minors are key concepts from the former, while collinearity and span are key concepts from the latter. In this course we will introduce matroid theory, and gradually work our way up to some very recent results.
  • MATH 7512: Topology II.
  • 9:00-10:20 T Th
  • Instructor: Prof. Cohen.
  • Prerequisites: MATH 7510 (and MATH 7210), or equivalent.
  • Text: A. Hatcher, Algebraic Topology.
  • A fundamental problem in topology is that of determining whether or not two spaces are topologically equivalent. The basic idea of algebraic topology is to associate algebraic objects (groups, rings, etc.) to a topological space in such a way that topologically equivalent spaces get assigned isomorphic objects. One such object is the fundamental group, introduced in MATH 7510 Topology I. Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces. Two spaces with inequivalent invariants cannot be topologically equivalent.

    The focus of this course will be on homology and cohomology. To a topological space, we will associate sequences of abelian groups, the homology and cohomology groups. Topics (from chapters 2 and 3 of Hatcher's book) include simplicial, singular, and cellular homology, Mayer-Vietoris sequences, and universal coefficient theorems. Geometric examples, including surfaces and projective spaces, will be used to illustrate the techniques. Discussion of cohomology theory, including products and duality, will (presumably) continue in MATH 7520 Algebraic Topology.

  • MATH 7550: Differential Geometry.
  • 12:30-1:20 M W F
  • Instructor: Prof. Baldridge.
  • Prerequisites: Math 4032 (or equivalent) and Math 7510
  • Text: Glen E. Bredon, Topology and Geometry, Springer, GTM 139
  • This course gives an introduction to the theory of manifolds. Topics to be covered include: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields, differential forms, Frobenius's theorem, integration on manifolds, Stokes's theorem, de Rham cohomology, Lie groups and Lie group actions.
  • MATH 7590-1: Geometric Topology: Knot and Braid Groups.
  • 2:30-3:20 M W F
  • Instructor: Prof. Dasbach.
  • Prerequisites:
  • Text:
  • MATH 7590-2: Riemannian Geometry.
  • 1:30-2:50 T Th
  • Instructor: Prof. Dani.
  • Prerequisites: 7510, 4035 or equivalent, prior or concurrent enrollment in 7550 encouraged.
  • Text: Manfredo do Carmo, Riemannian Geometry
  • This course is an introduction to Riemannian Geometry, the study of smooth manifolds endowed with Riemannian metrics. We will cover the basics: Riemannian metrics, connections, geodesics, curvature, Jacobi fields, completeness, and spaces of constant curvature. We will then proceed to some theorems that relate curvature, topology, and analysis, such as the Hopf--Rinow Theorem, Bonnet--Myers Theorem, Preissman's Theorem and the Rauch Comparison Theorem.
  • MATH 7710: Advanced Numerical Linear Algebra.
    • 12:00-1:20 T Th
  • Instructor: Prof. Walker.
  • Prerequisites: Linear Algebra, Advanced Calculus, Some Programming Experience
  • Text: Fundamentals of Matrix Computations (3rd edition), by D. S. Watkins
  • This is a basic course in numerical linear algebra. We will cover selected topics from the text:
    1. Mathematical tools: norms, projectors, Gram-Schmidt, orthogonal matrices, spectral theorem, singular value decomposition, Gerschgorin’s circles.
    2. Error analysis: floating point arithmetic, round-off errors, IEEE standard, backward stability, conditioning.
    3. General systems: LU factorization, partial pivoting, Cholesky factorization, least squares problems, QR factorization.
    4. Iterative methods: Jacobi, Richardson, Gauss-Seidel, successive over-relaxation, steepest descent, conjugate gradient.
    5. Eigenvalue problems: power methods, Rayleigh quotient iteration, deflation, QR algorithm.