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Graduate Course Outlines, Summer 2017-Spring 2018
Contact
Summer 2017
- 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 2017
- MATH 4997-1: Vertically Integrated Research: Combinatorics and geometry of flag manifolds
- 12:00-1:20 T Th
- Instructor: Profs. Achar and Sage
- Prerequisites: Math 4200 and 4153
- Text: notes to be distributed in class
- A flag manifold is a generalization of projective space (the set of lines in a vector space) and of Grassmannians (the set of linear subspaces of some fixed dimension). Flag manifolds have rich connections to representation theory and combinatorics, especially through classical 19th century problems of enumerative geometry that are collectively known as "Schubert calculus." In this course, we will study both classical and modern aspects of this theory, including recent work on invariants of Schubert varieties.
- MATH 7001: Communicating Mathematics I
- 3:00-4:50 T Th
- Instructor: Prof. Oporowski.
- 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
- 10:30-11:20 MWF
- Instructor: Prof. Long.
- Prerequisites: Math 4200 or equivalent
- 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. Topics will include group actions and Sylow Theorems, finitely generated abelian groups; rings and modules: PIDs, UFDs, finitely generated modules over a PID, applications to Jordan canonical form, exact sequences.
- MATH 7230: Modular Forms.
- 1:30-2:20 MWF
- Instructor: Prof. Tu.
- Prerequisites:
- Math 7210 (Algebra), Math 4036 (complex analysis) or equivalent are very helpful. You should be comfortable with meromorphic and analytic functions, groups, group actions and Galois groups (definitions and examples).
- Know how to use a computer.
- Text: There is no required text, but the following references and web sources are useful.
- Introduction to Elliptic Curves and Modular Forms (GTM) by Neal I. Koblitz
- Modular Forms by T. Miyake.
- Explicitly Computing Modular Forms by William A. Stein.
- A First Course in Modular Forms (GTM) by F. Diamond and J. Shurman.
- Lectures on Modular Forms and Hecke Operators by Kenneth A. Ribet and William A. Stein.
- Introduction to Arithmetic Theory of Automorphic Forms by G. Shimura.
- Classical modular forms appear naturally in connection with problems arising in many areas such as number theory, geometry, differential equations, physics and so on. Modular forms play a central role in the development of number theory. One of the reasons is because they have very tight connection with elliptic curves and Galois representations. All the arithmetic properties of elliptic curves, modular forms, and Galois representations are extreme important in Weil's proof of Fermat's Last theorem.
The aim of this course is to cover the classical theory of classical modular forms. The course will begin with basic definitions and examples. In the later of the semester, we will discuss some applications, advanced topics, and open problems, depending on audiences' interests.
- MATH 7260: Homological Algebra.
- 1:30-2:50 TTh
- Instructor: Profs. Achar. and Sage.
- Prerequisites: Math 7211 and 7512 or permission of the instructor
- Text: Gelfand and Manin, Methods of Homological Algebra.
- Homological algebra is a branch of algebra that developed in the mid-twentieth century as a way to systematize and abstract techniques from algebraic topology involving homology--a procedure in which a sequence of abelian groups or modules is associated to each object in a given category. Its influence has expanded far beyond its primarily topological origins, and it is now a fundamental tool in areas such as representation theory, algebraic geometry, and number theory.
In this course, we will develop homological algebra in a modern, algebraic way with a focus on the language of derived categories. We will discuss various applications with a particular emphasis on sheaf cohomology. This course will provide much of the background material needed for the geometric representation theory class Math 7250 that will be offered in the spring.
- MATH 7290: Hopf Algebras and their representations
- 11:30-12:20 MWF
- Instructor: Prof. Ng.
- Prerequisites: Some notions on finite groups and rings, and a good knowledge of linear algebra covered in Math 7210 or equivalent.
- Text:
- Hopf algebras, sometimes also known as quantum groups, are generalizations of groups and Lie algebras. Their representation categories have many applications to other areas of mathematics such as mathematical physics and low dimensional topology. In this course, we present an introduction to the theory of Hopf algebras and their representations. The basic theory of representations of finite groups will be introduced as motivation at the beginning of the course.
- MATH 7311: Real Analysis I.
- 1:30-2:50 TTh
- Instructor: Prof. Walker.
- Prerequisites: Math 4032 or equivalent.
- Text: Measure and Integral: An Introduction to Real Analysis, Second Edition, by Richard L. Wheeden and Antoni Zygmund, Chapman & Hall/CRC Pure and Applied Mathematics, ISBN-13: 978-1498702898.
- The course covers measure theory, starting from the Lebesgue measure in Euclidean space. An outline of the course is:
- Basic background: compact sets, transformations, Riemann integral.
- Functions of bounded variation; Riemann-Stieltjes integral.
- Lebesgue measurable sets, functions (Egorov's and Lusin's Theorem), the Lebesgue integral.
- Fubini's Theorem, convergence theorems (e.g. Lebesgue's Dominated Convergence Theorem), etc.
- Lebesgue's differentiation theorem; Vitali covering lemma.
- L^{p} classes.
- Abstract integration theory.
- MATH 7350: Complex Analysis.
- 10:30-11:50 TTh
- Instructor: Prof. Olafsson.
- Prerequisites: Math 7311 or its equivalent.
- Text: Narasimhan, R. and Nievergelt, Y., Complex Analysis in One Variable, second edition, Birkhauser, Boston, 2001.
- This is the first graduate level course in complex analysis. We expect that the students are familiar with the basic notation of complex differentiable functions. But that material will be reviewed. We will then move into more advanced topics including analytic extension, covering spaces and the monodromy theorem, winding numbers, residues, Runge's theorem, Riemann mapping theorem, and harmonic functions. We will also include some material on Hilbert spaces of holomorphic functions. That material is not covered in the book, but we will make notes available. There will be homework problems every second week and a take-home midterm.
- MATH 7360: Probability Theory.
- 12:30-1:20 MWF
- Instructor: Prof. Ganguly.
- Prerequisites: Familiarity with Measure theory (Math 7311) is desirable but not required. However a good mathematical maturity and a solid understanding of undergraduate analysis at the level of Math 4027-4032 is important.
- Text: None
- The course is a self-contained introduction to modern probability theory. It starts from the concept of probability measures,random variables, and independence. Several important notions of convergence including almost sure convergence, convergence in probability, convergence in distribution will be introduced. Well-known limit theorems for sums of independent random variables such as the Kolmogorov's law of large numbers, Central Limit Theorem and their generalizations 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 in probaility and other areas of mathematics. A main part of the course will be devoted to understanding conditional probability and building the basic theory of martingales. Brownian motion is an important example of a continuous-time martingale. Its basic features will be briefly discussed. The course will not have any fixed text book, but some books will be cited for reference. It is important that the students take and read the class notes carefully.
- MATH 7365: Applied Stochastic Analysis.
- 10:30-11:20 M W F
- Instructor: Prof. Kuo.
- Prerequisites: Math 7311 (Real Analysis I) or equivalent
- Text: Kuo, H.-H.: Introduction to Stochastic Integration.
Universitext, Springer, 2006.
References:- Kuo, H.-H.: Gaussian Measures in Banach Spaces. Lecture Notes in Math., Vol. 463, Springer, 1975. (Reprinted by BookSurge Publishing, 2006)
- Kuo, H.-H.: White Noise Distribution Theory, CRC Press, 1996.
- In this course we will study the following applied topics of stochastic analysis:
- Stochastic differential equations.
- Constructions of diffusion processes.
- Mathematical finance.
- Filtering theory.
- White noise theory.
- A new theory of stochastic integration.
- MATH 7386: Partial Differential Equations.
- 2:30-3:20 MWF
- Instructor: Prof. Zhu.
- Prerequisites: Math 7311( Real Analysis I) or the equivalent.
- Text: Primary Text: Partial Differential Equations by Lawrence C. Evans; Some material may also come from: Elliptic partial differential equations by Qing Han and Fanghua Lin.
- This course provides an introduction to the theory of partial differential equations. Topics to be covered include:
- Introduction of Laplace's equation, the heat equation, and the wave equation
- Introduction of Sobolev spaces on weak deriatives, traces and Sobolev embedding
- Elliptic equations on existence, regularity and maximum principle
- Introduction of Calculus of Variations on Euler-Lagrange equations, Existence of minimizers and eigenvalue of self-adjoint elliptic operators
- MATH 7390: Topics in Numerical Analysis.
- 1:30-2:20 MWF
- Instructor: Prof. Wan.
- Prerequisites:
- Text: Instructors Notes.
- This is an introductory course on uncertainty quantification (UQ). Uncertainty quantification is interested in how to deal with uncertainty in realistic applications, which has received much attention in both academia and industry. The main mathematical model will be partial differential equations subject to uncertainty, where the source of uncertainty includes physical coefficients, initial/boundary conditions, forcing, etc. In this course, we will introduce numerical strategies to deal with typical problems in uncertainty quantification. Topics include approximation of random field, polynomial chaos expansion, multi-level Monte Carlo, Bayesian inverse problem and numerical large deviation principle.
- MATH 7490: Matroid Theory
- 2:30-3:20 MWF
- 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 7510: Topology I
- 9:00-10:30 TTh
- 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 very 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.
- 9:30-10:20 MWF
- Instructor: Prof. Dasbach.
- Prerequisites: Math 7510 and 7512
- Text:
- This course continues the study of algebraic topology begun in MATH 7510 and
MATH 7512.
While MATH 7510 developed the theory of fundamental groups and MATH 7512
developed homology theories for topological spaces the focus of this course
will be cohomology theory which is dual to homology theory. However, one of
the advantages of developing cohomology theory of spaces is that they are
naturally equipped with a ring structure.
The course will be mainly based on Hatcher's book Algebraic topology, and additional material that we will hand out.
- MATH 7590: Geometry of non-positively closed manifolds.
- 10:30-11_50 TTh
- Instructor: Prof. Dani.
- Prerequisites: 7210, 7512, 7550
- Text:
- Manifolds of non-positive curvature exhibit some surprising rigidity phenomena, that is, statements which say that under appropriate (often algebraic) conditions, two manifolds are actually isometric. Probably the most well-known of these is the Mostow Rigidity Theorem, which says that if two complete finite-volume hyperbolic manifolds of dimension at least 3 have isomorphic fundamental groups, then they are isometric.
In this course, we will begin with the necessary prerequisites about Riemannian manifolds and Lie groups, and then spend a large part of the semester getting comfortable with symmetric spaces of noncompact type (which are the simply-connected models for manifolds of non-positive curvature). We will end with the Mostow Rigidity Theorem and (hopefully) generalizations by Gromov, Ballman and Burns--Spatzier. A good reference for this material is the book Geometry of Nonpositively Curved Manifolds by Patrick Eberlein.
Spring 2018
- MATH 4997-1: Vertically Integrated Research: Combinatorics and geometry of flag manifolds
- 12:00-1:20 TTh
- Instructor: Profs. Achar. and Sage.
- Prerequisites: Math 4200 and Math 4153
- Text:
- A flag manifold is a generalization of projective space (the set of lines in a vector space) and of Grassmannians (the set of linear subspaces of some fixed dimension). Flag manifolds have rich connections to representation theory and combinatorics, especially through classical 19th century problems of enumerative geometry that are collectively known as "Schubert calculus." In this course, we will study both classical and modern aspects of this theory, including recent work on invariants of Schubert varieties. (This course continues the theme of the Fall 2017 semester, but that semester is not a prerequisite for this course.)
- 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.
- 10:30-11:20 MWF
- Instructor: Prof. Adkins.
- Prerequisites: Math 7210 Algebra I.
- Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003.
- This is the second semester of the first-year graduate algebra sequence. In this course, we will further develop the topics introduced in the first semester. Specific topics include: normal and separable field extensions; Galois theory and applications; solvable groups, normal series, and the Jordan-Holder theorem; tensor products and Hom for modules; noetherian rings; the Hilbert Basis Theorem; and algebras over a field, including Wedderburn's and Maschke's Theorems.
- MATH 7220: Commutative Algebra
- 11:30-12:20 MWF
- Instructor: Prof. Hoffman.
- Prerequisites: Algebra at the level of Math 7210 (familiarity with groups, rings, ideals, modules). Some beginning topology (topological spaces, continuous maps).
- Text: Introduction To Commutative Algebra, Student Economy Edition By Michael Atiyah, Westview Press; 1 edition (December 8, 2015), ISBN-10: 0813350182 ISBN-13: 978-0813350189
- We will do all the topics in the book. Commutative rings, ideals, modules, localization and completion. Noetherian rings. Dimension theory. This course is preparation for a course in Algebraic Geometry and also for Algebraic Number Theory. There will be an introduction to the use of software systems such as Singular and Macaulay2. There are many exercises in the book. The student will do these and turn in a certain number for grading.
- MATH 7230: Analytic Number Theory
- 10:30-11:50 TTh
- Instructor: Prof. Mahlburg.
- Prerequisites: Graduate Abstract Algebra (MATH 7210), Elementary Number Theory (MATH 4181), and Real Analysis / Advanced Calculus (MATH 4031). Complex Analysis (MATH 4036) is recommended.
- Text: Multiplicative Number Theory I: Classical Theory (Cambridge Studies in Advanced Mathematics) 1st Edition by Hugh L. Montgomery and Robert C. Vaughan ISBN-13: 978-0521849036 Cambridge University Press, 2006
- This is an introductory graduate course in Analytic Number Theory, which is the quantitative study of the arithmetic properties of the integers. Topics may include arithmetic functions; primes in arithmetic progression and Dirichlet’s theorem; Dirichlet characters; the Prime Number Theorem; L-functions, zeta functions, and the Riemann Hypothesis; sieve techniques, particularly for prime gaps; quadratic forms; Tauberian theorems; combinatorial applications, Hardy-Ramanujan’s formula for integer partitions.
- MATH 7250: Geometric Representation Theory
- 1:30-2:50 TTh
- Instructor: Profs. Achar. and Sage.
- Prerequisites: Math 7211 and Math 7512 or permission of the instructor
- Text:
- This course will be an introduction to the use of geometric methods (topology, sheaf theory) to answer questions in representation theory. Possible topics include: flag manifolds and Kazhdan-Lusztig theory; nilpotent cones and Springer theory; moduli spaces of bundles; geometric Satake equivalence. No background on algebraic groups is necessary.
- MATH 7320: Ordinary Differential Equations.
- 10:30-11:50 TTh
- Instructor: Prof. Shipman
- Prerequisites:
- Text: Ordinary Differential Equations by V. I. Arnold
- Courses in ODEs are often perceived as boring, or at least tedious, because of the expected nature of some of the theorems and the technical nature of some of the proofs. My intention is that this course will deviate from that stereotype, as the field is in fact fascinating, beautiful, and pervasive in science and mathematics. As a foundation for the material, I will use the classic textbook by Vladimir I. Arnol'd, which is universally recognized as a mathematical gem. We will do both linear and nonlinear ODEs. These are genuinely different areas of mathematics, as the questions that are investigated are very different in nature.
The linear theory is sometimes considered part of linear algebra. The overarching concept is the matrix exponential and how its structure reflects canonical forms of linear operators. I will introduce the spectral theory of linear differential operators and its intimate connection with complex analysis. I also want to include some more specialized topics, such as linear systems governed by indefinite quadratic forms.
The overarching concepts for the nonlinear theory are flows of vector fields and dynamical systems. Upon that basis, one studies diverse phenomena such as bifurcations, separation of time scales, bursting (such as in neurobiology), hysteresis, stability (this is the connection between linear and nonlinear), control systems, chaos, and strange attractors.
Obviously, a course cannot come close to doing justice to all of these topics, and neither can one person. My goal is twofold: (1) to present the foundational rigorous theory of ODEs, and (2) to introduce a broad variety of topics in ODEs that highlight what makes the field interesting.
- MATH 7330: Functional Analysis.
- 9:30-10:20 MWF
- Instructor: Prof. He.
- Prerequisites: Math 7311
- Text: Banach Algebra Techniques in Operator theory GTM, 2nd edition by Ron Douglas
- In this course, we shall discuss the basics of Banach space, Banach algebra, operators on Hilbert spaces. We will then discuss the Hardy space and Toeplitz operators. We will follow loosely the GTM book by Ron Douglas.
- MATH 7380: Singular Integrals
- 2:30-3:20 MWF
- Instructor: Prof. Nguyen.
- Prerequisites: MATH 7311 and MATH 7350.
- Text: Loukas Grafakos, Classical Fourier Analysis, Third Edition, GTM 249, Springer, New York, 2014. xviii+638 pp. ISBN: 978-1-4939-1193-6.
Recommended Reference: Elias Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. - This is an introductory course on Fourier Analysis with an emphasis on the boundedness of Singular Integral Operators of convolution type. Such a boundedness property plays a fundamental role in various applications in pure and applied analysis. The course also covers such classical topics as Interpolation, Maximal Functions, Fourier Series, and possibly Littlewood-Paley Theory if time permits.
- MATH 7384: Topics in Material Science: Variational Phase-Field Models of Fracture
- 3:30-5:00 MW
- Instructor: Prof. Bourdin.
- Prerequisites: A graduate course in partial differential equations (MATH 7386 or equivalent). No prior knowledge of solid mechanics is required for this class.
- Text: Instructor's notes and research publications to be handed out.
- This course in an introduction to variational phase-field models in brittle fracture.
We will first derive variational models of fracture as an extension of Griffith's celebrated criterion.
We will then derive phase-field models through a regularization à la Ambrosio-Tortorelli, and prove they convergence in the sense of Gamma-convergence.
We will then review a more recent construction of phase-field models as "gradient-damage" models, and show how they can still be related to Griffith's criterion in this framework.
We will finally focus on fine mechanical properties of gradient damage models with a specuial emphasis on crack nucleation.
Throughout the course, we will use numerical simulations to illustrate various features of the models built in class. Course evaluation will be project-based.
- MATH 7390: Fast Solvers
- 10:30-11:50 TTh
- Instructor: Prof. Brenner.
- Prerequisites: Some background in finite element methods and numerical linear algebra
- Text: None
- Numerical methods for partial differential equations generate large ill-conditioned systems. Since direct methods are too expensive, such systems are usually solved by iterative methods. In this course we will discuss two classes of modern iterative methods for numerical PDEs: multigrid and domain decomposition.
- MATH 7410: Graph Theory.
- 1:30-2:20 MWF
- 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: Graph Minors.
- 12:00-1:20 TTh
- Instructor: Prof. Ding.
- Prerequisites: Math 4171 or equivalent
- Text: None (lecture notes will be distributed)
- This is an introduction to the theory of graph minors. We will discuss many problems of the following two types: determine all minor-minimal graphs that have a prescribed property; determine the structure of graphs that do not contain a specific graph as a minor. We will focus on connectivity and planarity.
- MATH 7512: Topology II.
- 12:30-1:20 MWF
- Instructor: Prof. Litherland.
- Prerequisites: Math 7510
- Text: Algebraic Topology by Allen Hatcher
- This course will introduce the homology theory of topological spaces. To each space X and nonnegative integer k, there is assigned an abelian group the k-th homology group of X. We will learn to calculate these groups, and use them to prove topological results such as the Brouwer Fixed point theorem (in all dimensions) and generalizations of the Jordan curve theorem. Homology theory is important in many parts of modern mathematics. If time permits, we will cover the the basic ideas of cohomology as well.
- MATH 7550: Differential Geometry.
- 1:30-2:50 TTh
- Instructor: Prof. Baldridge.
- Prerequisites: Math 7210 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 Theory.
- 1:30-2:20 MWF
- Instructor: Prof. Gilmer.
- Prerequisites: Math 7510, Math 7512
- Text: Lickorish, An Introduction to Knot Theory. Graduate Texts in Mathematics 175. Springer.
- A knot is an embedded loop in three dimensional space. A collection disjoint knots is a link. We will use algebraic topology and other tools to study topological spaces which can be constructed from knots or links. We will also study knots diagrammatically. The many techniques of modern mathematics that can be employed in the study of knots makes for a fascinating area.
- MATH 7590-2: Riemannian Geometry
- 9:00-10:20 TTh
- Instructor: Prof. Vela-Vick
- Prerequisites: Math 7550
- Text: Riemannian Geometry by Manfredo P. do Carmo
- Riemannian metrics and connections, geodesics, completeness, Hopf-Rinow theorem, sectional curvature, Ricci curvature, scalar curvature, Jacobi fields, second fundamental form and Gauss equations, manifolds of constant curvature, first and second variation formulas, Bonnet-Myers theorem, comparison theorems, Morse index theorem, Hadamard theorem, Preissmann theorem, and further topics such as sphere theorems, critical points of distance functions.
- MATH 7710: Advanced Numerical Linear Algebra.
- 9:00-10:20 TTh
- Instructor: Prof. Sung.
- Prerequisites: linear algebra, advanced calculus and some programming experience
- Text: Fundamentals of Matrix Computations, Third Edition, D.S. Watkins, Wiley, 2010
- This course will develop and analyze fundamental algorithms for the numerical solutions of problems in linear algebra. Topics include direct methods for general linear systems based on matrix factorization (LU, Cholesky and QR), iterative methods for sparse systems (Jacobi, Gauss-Seidel, SOR, steepest descent and conjugate gradient), and methods for eigenvalue problems (power methods, Rayleigh quotient iteration and QR algorithm).