MathWiki - User contributions [en]

Math 635: Dynamical Systems

2009-02-26T23:05:24Z

Tlf2:

== Catalog Information ==

=== Title ===
Dynamical Systems.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 634]].

=== Description ===

A rigorous treatment of the theory of dynamical systems.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include topological, symbolic, and hyperbolic dynamical systems.

=== Prerequisites ===

Students are expected to have completed [[Math 634]]. This will provide the students with an understanding on the theory of ordinary differential equations.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Topological dynamical systems
#* Nonwandering set, chain recurrence
#* Topological mixing and transitivity
#* Expansive systems
#* Topological entropy
#* Topological and smooth conjugacy
# Symbolic dynamical systems
#* Subshifts of finite type
#* Perron-Frobenius theorem
#* Topological entropy for subshifts of finite type
# Hyperbolic dynamical systems
#* Hartman-Grobman theorem
#* Stable manifold theorem
#* Hyperbolic sets
#* Anosov diffeomorphisms
#* Smale Horseshoe and transverse homoclinic points
#* Shadowing
#* Axiom A dynamical systems and spectral decomposition
# Low dimensional dynamical systems
#* Circle homeomorphisms, circle diffeomorphisms, and rotation number
#* Real quadratic maps
#* Expanding endomorphisms

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Ergodic theory, Markov partitions, Sharkovsky theorem, Hamiltonian dynamics, and measure theoretic entropy.

=== Recommended texts ===

Dynamical Systems, stability, symbolic dynamics, and chaos, by Clark Robinson; Introduction to Dynamical Systems, by Michael Brin and Garrett Stuck

=== Courses for which this course is prerequisite ===

[[Category:Courses|635]]

None.

Math 565: Differential Geometry

2009-02-26T23:03:30Z

Tlf2:

== Catalog Information ==

=== Title ===
Differential Geometry

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]].

=== Description ===

A rigorous treatment of the theory of differential geometry.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include differential topology, Riemannian metrics, geodesics, curvature, and integration on manifolds.

=== Prerequisites ===

Students should have taken [[Math 316]] prior to taking this course. Math 316 provides a rigorous background of analysis that is needed to understand many of the proofs in this course.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Differential topology
#* Differentiable manifolds and smooth maps
#* Tangent space, tangent bundle, derivative of a smooth map
#* Immersions, submersions, and embeddings
#* Orientation
#* Vector fields, brackets
# Riemannian metrics
#* Definition of Riemannian metrics
#* Affine connections
#* Riemannian connections
# Geodesics
#* Definition of geodesics
#* Geodesic flow
#* Minimizing properties of geodesics
#* Exponential map
#* Convex neighborhoods
# Curvature
#* Definitions of curvature, curvature tensor
#* Second fundamental form
#* Sectional and Ricci curvature
#* Jacobi fields
# Integration on manifolds
#* Tensor and vector bundles
#* Exterior algebra
#* Differential forms and exterior derivative
#* Stokes Theorem

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Hopf-Rinow theorem, spaces of constant curvature, De Rham cohomology, fixed points and intersection numbers, and Morse theory.

=== Recommended Texts ===

Riemannian geometry, by Manfredo P. Do Carmo; Differential Geometry and Topology, with a view to dynamical systems, by Keith Burns and Marian Gidea

=== Courses for which this course is prerequisite ===

None.

[[Category:Courses|565]]

Math 465: Intro to Differential Geometry

2009-02-26T23:01:23Z

Tlf2:

== Catalog Information ==

=== Title ===
Differential Geometry.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Prerequisite ===
[[Math 214]]; [[Math 315|315]] or equivalent.
Recommended: [[Math 316]] or equivalent.

=== Description ===
Geometry of smooth curves and surfaces. Topics include the first and second fundamental forms, the Gauss map, orientability of surfaces, Gaussian and mean curvature, geodesics, minimal surfaces and the Gauss-Bonnet Theorem.

== Desired Learning Outcomes ==
The main purpose of this course is to provide students with an understanding of the geometry of curves and surfaces, with the focus being on the theoretical and logical foundations of differential geometry.

=== Prerequisites ===
The prerequisite of [[Math 214]] is to ensure that students have some understanding of partial derivatives and differentiation for functions of more than one variable. [[Math 315|315]] is required so students have a rigorous understanding of the real number system and of real-valued functions. [[Math 316]] is recommended so students have a rigorous understanding of functions with more than one variable.

=== Minimal learning outcomes ===

Outlined below are topics that all successful Math 465 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.

<div style="-moz-column-count:2; column-count:2;">

# Basic Properties of Curves
#* Parametrized curves
#* Regular curves
#* Arc length
# Regular surfaces
#* Regular surfaces as inverse images of regular values
#* Change of parameters
#* The tangent plane
#* The first fundamental form
#*Orientation of surfaces
# The geometry of the Gauss map
#* Definition of the Gauss map and fundamental properties
#* The Gauss map in local coordinates
#* Minimal surfaces
#Intrinsic geometry of surfaces
#* Isometries and conformal maps
#* Geodesics and parallel transport
#* The Gauss-Bonnet Theorem and applications
#* The exponential map

</div>

=== Additional topics ===

Among other topics instructors may want to cover Jacobi fields and conjugate points, covering spaces, and the Hopf-Rinow Theorem.

=== Recommended Texts ===

Differential Geometry of Curves and Surfaces by Manfredo P. Do Carmo

=== Courses for which this course is prerequisite ===

None

[[Category:Courses|465]]

Math 465: Intro to Differential Geometry

2009-02-26T22:48:21Z

Tlf2: /* Courses for which this course is prerequisite */

Math 465: Intro to Differential Geometry

2009-02-26T22:47:47Z

Tlf2:

Math 634: Theory of Ordinary Differential Equations

2008-08-28T21:33:39Z

Tlf2: /* Description */

== Catalog Information ==

=== Title ===
Theory of Ordinary Differential Equations.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]], [[Math 334]].

=== Description ===

A rigorous treatment of the theory of differential equations.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include the existence, uniqueness, and continuation of solutions to differential equations, linear differential equations, and stability of solutions for differential equations.

=== Prerequisites ===

Students should have taken [[Math 316]] and [[Math 334]] prior to taking this course. Math 316 provides a rigorous background of analysis that is needed to understand many of the proofs in this course. Math 334 provides the students an understanding of basic properties of differential equations and techniques used to solve differential equations.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>
# Solutions to ordinary differential equations
#* Existence of solutions
#* Uniqueness of solutions
#* Continuation of solutions
#* Gronwall’s inequality
#* Dependence on parameters
#* Contraction mapping principle
# Linear differential equations
#* Linear systems with constant coefficients
#* Jordan Normal Form
#* Fundamental solutions
#* Variation of constants formula
#* Floquet Theory for periodic solutions
# Stability and instability
#* Stability and asymptotic stability
#* Lyapunov functions
#* Bifurcations
# Poincare-Bendixson Theory
#* Invariant sets
#* Omega limit sets
#* Limit cycles

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Stability of nonautonomous equations, Fredholm alternative, normal forms, Hamiltonian dynamics, control theory, stable manifold theorem, and center manifold theorem.

=== Courses for which this course is prerequisite ===

[[Category:Courses|634]]

[[Math 635]]

Math 635: Dynamical Systems

2008-08-28T21:33:22Z

Tlf2: /* Description */

== Catalog Information ==

=== Title ===
Dynamical Systems.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 634]].

=== Description ===

A rigorous treatment of the theory of dynamical systems.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include topological, symbolic, and hyperbolic dynamical systems.

=== Prerequisites ===

Students are expected to have completed [[Math 634]]. This will provide the students with an understanding on the theory of ordinary differential equations.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Topological dynamical systems
#* Nonwandering set, chain recurrence
#* Topological mixing and transitivity
#* Expansive systems
#* Topological entropy
#* Topological and smooth conjugacy
# Symbolic dynamical systems
#* Subshifts of finite type
#* Perron-Frobenius theorem
#* Topological entropy for subshifts of finite type
# Hyperbolic dynamical systems
#* Hartman-Grobman theorem
#* Stable manifold theorem
#* Hyperbolic sets
#* Anosov diffeomorphisms
#* Smale Horseshoe and transverse homoclinic points
#* Shadowing
#* Axiom A dynamical systems and spectral decomposition
# Low dimensional dynamical systems
#* Circle homeomorphisms, circle diffeomorphisms, and rotation number
#* Real quadratic maps
#* Expanding endomorphisms

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Ergodic theory, Markov partitions, Sharkovsky theorem, Hamiltonian dynamics, and measure theoretic entropy.

=== Courses for which this course is prerequisite ===

[[Category:Courses|635]]

None.

Math 565: Differential Geometry

2008-08-28T21:32:17Z

Tlf2: /* Description */

== Catalog Information ==

=== Title ===
Differential Geometry

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]].

=== Description ===

A rigorous treatment of the theory of differential geometry.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include differential topology, Riemannian metrics, geodesics, curvature, and integration on manifolds.

=== Prerequisites ===

Students should have taken [[Math 316]] prior to taking this course. Math 316 provides a rigorous background of analysis that is needed to understand many of the proofs in this course.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Differential topology
#* Differentiable manifolds and smooth maps
#* Tangent space, tangent bundle, derivative of a smooth map
#* Immersions, submersions, and embeddings
#* Orientation
#* Vector fields, brackets
# Riemannian metrics
#* Definition of Riemannian metrics
#* Affine connections
#* Riemannian connections
# Geodesics
#* Definition of geodesics
#* Geodesic flow
#* Minimizing properties of geodesics
#* Exponential map
#* Convex neighborhoods
# Curvature
#* Definitions of curvature, curvature tensor
#* Second fundamental form
#* Sectional and Ricci curvature
#* Jacobi fields
# Integration on manifolds
#* Tensor and vector bundles
#* Exterior algebra
#* Differential forms and exterior derivative
#* Stokes Theorem

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Hopf-Rinow theorem, spaces of constant curvature, De Rham cohomology, fixed points and intersection numbers, and Morse theory.

=== Courses for which this course is prerequisite ===

None.

[[Category:Courses|565]]

Math 565: Differential Geometry

2008-08-20T22:53:02Z

Tlf2: /* Desired Learning Outcomes */

== Catalog Information ==

=== Title ===
Differential Geometry

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of differential geometry.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include differential topology, Riemannian metrics, geodesics, curvature, and integration on manifolds.

=== Prerequisites ===

Students should have taken [[Math 316]] prior to taking this course. Math 316 provides a rigorous background of analysis that is needed to understand many of the proofs in this course.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Differential topology
#* Differentiable manifolds and smooth maps
#* Tangent space, tangent bundle, derivative of a smooth map
#* Immersions, submersions, and embeddings
#* Orientation
#* Vector fields, brackets
# Riemannian metrics
#* Definition of Riemannian metrics
#* Affine connections
#* Riemannian connections
# Geodesics
#* Definition of geodesics
#* Geodesic flow
#* Minimizing properties of geodesics
#* Exponential map
#* Convex neighborhoods
# Curvature
#* Definitions of curvature, curvature tensor
#* Second fundamental form
#* Sectional and Ricci curvature
#* Jacobi fields
# Integration on manifolds
#* Tensor and vector bundles
#* Exterior algebra
#* Differential forms and exterior derivative
#* Stokes Theorem

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Hopf-Rinow theorem, spaces of constant curvature, De Rham cohomology, fixed points and intersection numbers, and Morse theory.

=== Courses for which this course is prerequisite ===

None.

[[Category:Courses|565]]

Math 565: Differential Geometry

2008-08-20T22:52:37Z

Tlf2: /* Description */

== Catalog Information ==

=== Title ===
Differential Geometry

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of differential geometry.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include the differential topology, Riemannian metrics, geodesics, curvature, and integration on manifolds.

=== Prerequisites ===

Students should have taken [[Math 316]] prior to taking this course. Math 316 provides a rigorous background of analysis that is needed to understand many of the proofs in this course.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Differential topology
#* Differentiable manifolds and smooth maps
#* Tangent space, tangent bundle, derivative of a smooth map
#* Immersions, submersions, and embeddings
#* Orientation
#* Vector fields, brackets
# Riemannian metrics
#* Definition of Riemannian metrics
#* Affine connections
#* Riemannian connections
# Geodesics
#* Definition of geodesics
#* Geodesic flow
#* Minimizing properties of geodesics
#* Exponential map
#* Convex neighborhoods
# Curvature
#* Definitions of curvature, curvature tensor
#* Second fundamental form
#* Sectional and Ricci curvature
#* Jacobi fields
# Integration on manifolds
#* Tensor and vector bundles
#* Exterior algebra
#* Differential forms and exterior derivative
#* Stokes Theorem

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Hopf-Rinow theorem, spaces of constant curvature, De Rham cohomology, fixed points and intersection numbers, and Morse theory.

=== Courses for which this course is prerequisite ===

None.

[[Category:Courses|565]]

Math 565: Differential Geometry

2008-08-20T22:52:08Z

Tlf2: New page: == Catalog Information == === Title === Differential Geometry === Credit Hours === 3 === Prerequisite === Math 316. === Description === This course is designed to provide students...

== Catalog Information ==

=== Title ===
Differential Geometry

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of differential equations.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include the differential topology, Riemannian metrics, geodesics, curvature, and integration on manifolds.

=== Prerequisites ===

Students should have taken [[Math 316]] prior to taking this course. Math 316 provides a rigorous background of analysis that is needed to understand many of the proofs in this course.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Differential topology
#* Differentiable manifolds and smooth maps
#* Tangent space, tangent bundle, derivative of a smooth map
#* Immersions, submersions, and embeddings
#* Orientation
#* Vector fields, brackets
# Riemannian metrics
#* Definition of Riemannian metrics
#* Affine connections
#* Riemannian connections
# Geodesics
#* Definition of geodesics
#* Geodesic flow
#* Minimizing properties of geodesics
#* Exponential map
#* Convex neighborhoods
# Curvature
#* Definitions of curvature, curvature tensor
#* Second fundamental form
#* Sectional and Ricci curvature
#* Jacobi fields
# Integration on manifolds
#* Tensor and vector bundles
#* Exterior algebra
#* Differential forms and exterior derivative
#* Stokes Theorem

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Hopf-Rinow theorem, spaces of constant curvature, De Rham cohomology, fixed points and intersection numbers, and Morse theory.

=== Courses for which this course is prerequisite ===

None.

[[Category:Courses|565]]

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:26:23Z

Tlf2: /* Prerequisite */

== Catalog Information ==

=== Title ===
Theory of Ordinary Differential Equations.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]], [[Math 334]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of differential equations.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include the existence, uniqueness, and continuation of solutions to differential equations, linear differential equations, and stability of solutions for differential equations.

=== Prerequisites ===

Students should have taken [[Math 316]] and [[Math 334]] prior to taking this course. Math 316 provides a rigorous background of analysis that is needed to understand many of the proofs in this course. Math 334 provides the students an understanding of basic properties of differential equations and techniques used to solve differential equations.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>
# Solutions to ordinary differential equations
#* Existence of solutions
#* Uniqueness of solutions
#* Continuation of solutions
#* Gronwall’s inequality
#* Dependence on parameters
#* Contraction mapping principle
# Linear differential equations
#* Linear systems with constant coefficients
#* Jordan Normal Form
#* Fundamental solutions
#* Variation of constants formula
#* Floquet Theory for periodic solutions
# Stability and instability
#* Stability and asymptotic stability
#* Lyapunov functions
#* Bifurcations
# Poincare-Bendixson Theory
#* Invariant sets
#* Omega limit sets
#* Limit cycles

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Stability of nonautonomous equations, Fredholm alternative, normal forms, Hamiltonian dynamics, control theory, stable manifold theorem, and center manifold theorem.

=== Courses for which this course is prerequisite ===

[[Category:Courses|634]]

[[Math 635]]

Math 635: Dynamical Systems

2008-08-12T22:22:53Z

Tlf2: /* Minimal learning outcomes */

== Catalog Information ==

=== Title ===
Dynamical Systems.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 634]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of dynamical systems.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include topological, symbolic, and hyperbolic dynamical systems.

=== Prerequisites ===

Students are expected to have completed [[Math 634]]. This will provide the students with an understanding on the theory of ordinary differential equations.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Topological dynamical systems
#* Nonwandering set, chain recurrence
#* Topological mixing and transitivity
#* Expansive systems
#* Topological entropy
#* Topological and smooth conjugacy
# Symbolic dynamical systems
#* Subshifts of finite type
#* Perron-Frobenius theorem
#* Topological entropy for subshifts of finite type
# Hyperbolic dynamical systems
#* Hartman-Grobman theorem
#* Stable manifold theorem
#* Hyperbolic sets
#* Anosov diffeomorphisms
#* Smale Horseshoe and transverse homoclinic points
#* Shadowing
#* Axiom A dynamical systems and spectral decomposition
# Low dimensional dynamical systems
#* Circle homeomorphisms, circle diffeomorphisms, and rotation number
#* Real quadratic maps
#* Expanding endomorphisms

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Ergodic theory, Markov partitions, Sharkovsky theorem, Hamiltonian dynamics, and measure theoretic entropy.

=== Courses for which this course is prerequisite ===

[[Category:Courses|635]]

None.

Math 635: Dynamical Systems

2008-08-12T22:22:13Z

Tlf2: /* Desired Learning Outcomes */

== Catalog Information ==

=== Title ===
Dynamical Systems.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 634]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of dynamical systems.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include topological, symbolic, and hyperbolic dynamical systems.

=== Prerequisites ===

Students are expected to have completed [[Math 634]]. This will provide the students with an understanding on the theory of ordinary differential equations.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Topological dynamical systems
#* Nonwandering set, chain recurrence
#* Topological mixing and transitivity
#* Expansive systems
#* Topological entropy
#* Topological and smooth conjugacy
# Symbolic dynamical systems
#* Subshifts of finite type
#* Perron-Frobenius Theorem
#* Topological entropy for subshifts of finite type
# Hyperbolic dynamical systems
#* Hartman-Grobman theorem
#* Stable manifold theorem
#* Hyperbolic sets
#* Anosov diffeomorphisms
#* Smale Horseshoe and transverse homoclinic points
#* Shadowing
#* Axiom A dynamical systems and spectral decomposition
# Low dimensional dynamical systems
#* Circle homeomorphisms, circle diffeomorphisms, and rotation number
#* Real quadratic maps
#* Expanding endomorphisms

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Ergodic theory, Markov partitions, Sharkovsky theorem, Hamiltonian dynamics, and measure theoretic entropy.

=== Courses for which this course is prerequisite ===

[[Category:Courses|635]]

None.

Math 635: Dynamical Systems

2008-08-12T22:20:49Z

Tlf2: /* Minimal learning outcomes */

== Catalog Information ==

=== Title ===
Dynamical Systems.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 634]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of dynamical systems.

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Topological dynamical systems
#* Nonwandering set, chain recurrence
#* Topological mixing and transitivity
#* Expansive systems
#* Topological entropy
#* Topological and smooth conjugacy
# Symbolic dynamical systems
#* Subshifts of finite type
#* Perron-Frobenius Theorem
#* Topological entropy for subshifts of finite type
# Hyperbolic dynamical systems
#* Hartman-Grobman theorem
#* Stable manifold theorem
#* Hyperbolic sets
#* Anosov diffeomorphisms
#* Smale Horseshoe and transverse homoclinic points
#* Shadowing
#* Axiom A dynamical systems and spectral decomposition
# Low dimensional dynamical systems
#* Circle homeomorphisms, circle diffeomorphisms, and rotation number
#* Real quadratic maps
#* Expanding endomorphisms

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Ergodic theory, Markov partitions, Sharkovsky theorem, Hamiltonian dynamics, and measure theoretic entropy.

=== Courses for which this course is prerequisite ===

[[Category:Courses|635]]

None.

Math 635: Dynamical Systems

2008-08-12T22:20:25Z

Tlf2: /* Minimal learning outcomes */

== Catalog Information ==

=== Title ===
Dynamical Systems.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 634]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of dynamical systems.

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

: Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Topological dynamical systems
#* Nonwandering set, chain recurrence
#* Topological mixing and transitivity
#* Expansive systems
#* Topological entropy
#* Topological and smooth conjugacy
# Symbolic dynamical systems
#* Subshifts of finite type
#* Perron-Frobenius Theorem
#* Topological entropy for subshifts of finite type
# Hyperbolic dynamical systems
#* Hartman-Grobman theorem
#* Stable manifold theorem
#* Hyperbolic sets
#* Anosov diffeomorphisms
#* Smale Horseshoe and transverse homoclinic points
#* Shadowing
#* Axiom A dynamical systems and spectral decomposition
# Low dimensional dynamical systems
#* Circle homeomorphisms, circle diffeomorphisms, and rotation number
#* Real quadratic maps
#* Expanding endomorphisms

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Ergodic theory, Markov partitions, Sharkovsky theorem, Hamiltonian dynamics, and measure theoretic entropy.

=== Courses for which this course is prerequisite ===

[[Category:Courses|635]]

None.

Math 635: Dynamical Systems

2008-08-12T22:19:09Z

Tlf2: /* Additional topics */

Math 635: Dynamical Systems

2008-08-12T22:18:56Z

Tlf2: /* Additional topics */

Math 635: Dynamical Systems

2008-08-12T22:18:30Z

Tlf2: /* Courses for which this course is prerequisite */

Math 635: Dynamical Systems

2008-08-12T22:18:13Z

Tlf2: /* Description */

Math 635: Dynamical Systems

2008-08-12T22:17:56Z

Tlf2: /* Prerequisite */

== Catalog Information ==

=== Title ===
Dynamical Systems.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 634]].

=== Description ===

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

=== Additional topics ===

=== Courses for which this course is prerequisite ===

[[Category:Courses|635]]

Math 635: Dynamical Systems

2008-08-12T22:17:38Z

Tlf2: /* Title */

== Catalog Information ==

=== Title ===
Dynamical Systems.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 315]], [[Math 334|334]].

=== Description ===

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

=== Additional topics ===

=== Courses for which this course is prerequisite ===

[[Category:Courses|635]]

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:17:01Z

Tlf2: /* Title */

== Catalog Information ==

=== Title ===
Theory of Ordinary Differential Equations.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]], [[Math 334|334]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of differential equations.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include the existence, uniqueness, and continuation of solutions to differential equations, linear differential equations, and stability of solutions for differential equations.

=== Prerequisites ===

Students should have taken [[Math 316]] and [[Math 334]] prior to taking this course. Math 316 provides a rigorous background of analysis that is needed to understand many of the proofs in this course. Math 334 provides the students an understanding of basic properties of differential equations and techniques used to solve differential equations.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>
# Solutions to ordinary differential equations
#* Existence of solutions
#* Uniqueness of solutions
#* Continuation of solutions
#* Gronwall’s inequality
#* Dependence on parameters
#* Contraction mapping principle
# Linear differential equations
#* Linear systems with constant coefficients
#* Jordan Normal Form
#* Fundamental solutions
#* Variation of constants formula
#* Floquet Theory for periodic solutions
# Stability and instability
#* Stability and asymptotic stability
#* Lyapunov functions
#* Bifurcations
# Poincare-Bendixson Theory
#* Invariant sets
#* Omega limit sets
#* Limit cycles

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Stability of nonautonomous equations, Fredholm alternative, normal forms, Hamiltonian dynamics, control theory, stable manifold theorem, and center manifold theorem.

=== Courses for which this course is prerequisite ===

[[Category:Courses|634]]

[[Math 635]]

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:16:30Z

Tlf2: /* Title */

== Catalog Information ==

=== Title ===
Theory of ordinary differential equations.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]], [[Math 334|334]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of differential equations.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include the existence, uniqueness, and continuation of solutions to differential equations, linear differential equations, and stability of solutions for differential equations.

=== Prerequisites ===

Students should have taken [[Math 316]] and [[Math 334]] prior to taking this course. Math 316 provides a rigorous background of analysis that is needed to understand many of the proofs in this course. Math 334 provides the students an understanding of basic properties of differential equations and techniques used to solve differential equations.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>
# Solutions to ordinary differential equations
#* Existence of solutions
#* Uniqueness of solutions
#* Continuation of solutions
#* Gronwall’s inequality
#* Dependence on parameters
#* Contraction mapping principle
# Linear differential equations
#* Linear systems with constant coefficients
#* Jordan Normal Form
#* Fundamental solutions
#* Variation of constants formula
#* Floquet Theory for periodic solutions
# Stability and instability
#* Stability and asymptotic stability
#* Lyapunov functions
#* Bifurcations
# Poincare-Bendixson Theory
#* Invariant sets
#* Omega limit sets
#* Limit cycles

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Stability of nonautonomous equations, Fredholm alternative, normal forms, Hamiltonian dynamics, control theory, stable manifold theorem, and center manifold theorem.

=== Courses for which this course is prerequisite ===

[[Category:Courses|634]]

[[Math 635]]

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:16:09Z

Tlf2: /* Prerequisites */

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:13:15Z

Tlf2: /* Courses for which this course is prerequisite */

== Catalog Information ==

=== Title ===
Theory of Ordinary Differential Equations.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]], [[Math 334|334]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of differential equations.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include the existence, uniqueness, and continuation of solutions to differential equations, linear differential equations, and stability of solutions for differential equations.

=== Prerequisites ===

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>
# Solutions to ordinary differential equations
#* Existence of solutions
#* Uniqueness of solutions
#* Continuation of solutions
#* Gronwall’s inequality
#* Dependence on parameters
#* Contraction mapping principle
# Linear differential equations
#* Linear systems with constant coefficients
#* Jordan Normal Form
#* Fundamental solutions
#* Variation of constants formula
#* Floquet Theory for periodic solutions
# Stability and instability
#* Stability and asymptotic stability
#* Lyapunov functions
#* Bifurcations
# Poincare-Bendixson Theory
#* Invariant sets
#* Omega limit sets
#* Limit cycles

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Stability of nonautonomous equations, Fredholm alternative, normal forms, Hamiltonian dynamics, control theory, stable manifold theorem, and center manifold theorem.

=== Courses for which this course is prerequisite ===

[[Category:Courses|634]]

[[Math 635]]

Math 334: Ordinary Differential Equations

2008-08-12T22:10:16Z

Tlf2: /* Courses for which this course is prerequisite */

== Catalog Information ==

=== Title ===
Ordinary Differential Equations.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Offered ===
F, W, Sp, Su

=== Prerequisite ===
[[Math 113]], [[Math 343|343]].

=== Description ===
Methods and theory of ordinary differential equations.

== Desired Learning Outcomes ==
This course is aimed at students majoring in mathematical and physical sciences and mathematical education. The main purpose of the course is to introduce students to the theory and methods of ordinary differential equations. The course content contributes to all the expected learning outcomes of the Mathematics BS (see [http://learningoutcomes.byu.edu]).

=== Prerequisites ===
Students are expected to have completed [[Math 113]], and [[Math 343]] or be concurrently enrolled in [[Math 343]].

=== Minimal learning outcomes ===
Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.

<div style="-moz-column-count:2; column-count:2;">
# First order equations
#* Linear, separable, and exact equations
#* Existence and uniqueness of solutions
#* Linear versus nonlinear equations
#* Autonomous equations
#* Models and Applications
# Higher order equations
#* Theory of linear equations
#* Linear independence and the Wronskian
#* Homogeneous linear equations with constant coefficients
#* Nonhomogeneous linear equations, method of undetermined coefficients and variation of parameters
#* Mechanical and electrical vibrations
#* Power series solutions
#* The Laplace transform – definitions and applications
# Systems of equations
#* General theory
#* Eigenvalue-eigenvector method for systems with constant coefficients
#* Homogeneous linear systems with constant coefficients
#* Fundamental matrices
#* Nonhomogeneous linear systems, method of undetermined coefficients and variation of parameters
#* Stability, instability, asymptotic stability, and phase plane analysis
#* Models and applications

</div>

=== Additional topics ===
These are at the instructor's discretion as time allows; applications to physical problems are particularly helpful.

=== Courses for which this course is prerequisite ===
This course is required for [[Math 347]], [[Math 480]], [[Math 521]], [[Math 534]], [[Math 547]], and [[Math 634]].

[[Category:Courses|334]]

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:09:06Z

Tlf2: /* Desired Learning Outcomes */

== Catalog Information ==

=== Title ===
Theory of Ordinary Differential Equations.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]], [[Math 334|334]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of differential equations.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include the existence, uniqueness, and continuation of solutions to differential equations, linear differential equations, and stability of solutions for differential equations.

=== Prerequisites ===

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>
# Solutions to ordinary differential equations
#* Existence of solutions
#* Uniqueness of solutions
#* Continuation of solutions
#* Gronwall’s inequality
#* Dependence on parameters
#* Contraction mapping principle
# Linear differential equations
#* Linear systems with constant coefficients
#* Jordan Normal Form
#* Fundamental solutions
#* Variation of constants formula
#* Floquet Theory for periodic solutions
# Stability and instability
#* Stability and asymptotic stability
#* Lyapunov functions
#* Bifurcations
# Poincare-Bendixson Theory
#* Invariant sets
#* Omega limit sets
#* Limit cycles

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Stability of nonautonomous equations, Fredholm alternative, normal forms, Hamiltonian dynamics, control theory, stable manifold theorem, and center manifold theorem.

=== Courses for which this course is prerequisite ===

[[Category:Courses|634]]

Math 635

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:08:20Z

Tlf2: /* Description */

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:07:27Z

Tlf2: /* Minimal learning outcomes */

== Catalog Information ==

=== Title ===
Theory of Ordinary Differential Equations.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]], [[Math 334|334]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of differential equations. The topics include the existence, uniqueness, and continuation of solutions to differential equations, linear differential equations, and stability of solutions for differential equations.

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>
# Solutions to ordinary differential equations
#* Existence of solutions
#* Uniqueness of solutions
#* Continuation of solutions
#* Gronwall’s inequality
#* Dependence on parameters
#* Contraction mapping principle
# Linear differential equations
#* Linear systems with constant coefficients
#* Jordan Normal Form
#* Fundamental solutions
#* Variation of constants formula
#* Floquet Theory for periodic solutions
# Stability and instability
#* Stability and asymptotic stability
#* Lyapunov functions
#* Bifurcations
# Poincare-Bendixson Theory
#* Invariant sets
#* Omega limit sets
#* Limit cycles

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Stability of nonautonomous equations, Fredholm alternative, normal forms, Hamiltonian dynamics, control theory, stable manifold theorem, and center manifold theorem.

=== Courses for which this course is prerequisite ===

[[Category:Courses|634]]

Math 635

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:04:45Z

Tlf2: /* Additional topics */

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:04:28Z

Tlf2: /* Courses for which this course is prerequisite */

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:03:54Z

Tlf2: /* Description */

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:03:27Z

Tlf2: /* Description */

Math 634: Theory of Ordinary Differential Equations

2008-08-12T22:02:26Z

Tlf2: /* Prerequisite */

== Catalog Information ==

=== Title ===
Theory of Ordinary Differential Equations.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]], [[Math 334|334]].

=== Description ===

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

=== Additional topics ===

=== Courses for which this course is prerequisite ===

[[Category:Courses|634]]

Curriculum Proposals

2008-08-11T16:38:11Z

Tlf2:

# Make all course titles unique. We currently have 11 2-course sequences in which the 2 courses have identical titles. Distinguish these 2 titles by appending a "1" to the title of the first course and a "2" to the title of the second course.--[[User:Cpg|cpg]]
# Delete [[Math 480]], which has not been taught in over 10 years (and apparently was only taught by Don Snow prior to that).--[[User:Cpg|cpg]]
# Remove [[Math 343]] as a prerequisite for [[Math 315]], and add it as a prerequisite for [[Math 316]]. The status quo is an artifact of the time when [[Math 315]] had single and multiple variable differentiation; it doesn't anymore.--[[User:Cpg|cpg]]
# Is there a reason for Math 460R? It seems odd to have one 400 level topics class and to have this topic be geometry.
# Math 499R is a senior thesis class, but we don't have a departmental senior thesis option at present.
# Math 561 and 562 have Math 671 as a prerequisite.