MathWiki - User contributions [en]

Math 655: Differential Topology

2022-10-14T17:06:28Z

Cpg: /* Prerequisites */

== Catalog Information ==

=== Title ===
Differential Topology

=== Credit Hours ===
3

=== Offered ===
F (even years)

=== Prerequisite ===
[[Math 342|342]] or equivalent.

=== Description ===
An introduction to manifolds and smooth manifolds and their topology.

== Desired Learning Outcomes ==

=== Prerequisites ===

Knowledge of basic point set topology will be assumed. This includes topological spaces, basis and countability, metric spaces, quotient spaces, fundamental group, and covering maps. Basic knowledge of linear algebra ([[Math 313]]) and introductory analysis ([[Math 341]] and [[Math 342|342]]) will also be assumed.

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 655 students should understand well. Students should be able to demonstrate mastery of relevant vocabulary, and use the vocabulary fluently in their work. They should know common examples and counterexamples, and be able prove that these examples and counterexamples have properties as claimed. Additionally, students should know the content (and limitations) of major theorems and the ideas of the proofs, and apply results of these theorems to solve suitable problems, or use techniques of the proofs to prove additional related results, or to make calculations and computations.

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

# Manifolds
#* Topological and smooth manifolds
#* Manifolds with boundary
#* Tangent vectors
#* Tangent bundles
#* Vector bundles and bundle maps
#* Cotangent bundles
# Submanifolds
#* Submersions, immersions, embeddings
#* Inverse and implicit function theorems
#* Transversality
#* Embedding and approximation theorems
# Differential forms and tensors
#* Wedge product
#* Exterior derivative
#* Orientations
#* Stoke's Theorem

</div>

=== Additional topics ===

At the discretion of the instructor as time allows. Topics might include Lie groups and homogeneous spaces, Morse functions, de Rham cohomology and the de Rham theorem, Jordan curve theorem, Lefschetz fixed-point theory, degree, Gauss-Bonnet theorem, etc.

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

[[Category:Courses|655]]

Math 655: Differential Topology

2022-10-14T17:05:54Z

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

== Catalog Information ==

=== Title ===
Differential Topology

=== Credit Hours ===
3

=== Offered ===
F (even years)

=== Prerequisite ===
[[Math 342|342]] or equivalent.

=== Description ===
An introduction to manifolds and smooth manifolds and their topology.

== Desired Learning Outcomes ==

=== Prerequisites ===

Knowledge of basic point set topology from [[Math 553]], [[Math 554|554]] will be assumed. This includes topological spaces, basis and countability, metric spaces, quotient spaces, fundamental group, and covering maps. Basic knowledge of linear algebra ([[Math 313]]) and introductory analysis ([[Math 341]] and [[Math 342|342]]) will also be assumed.

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 655 students should understand well. Students should be able to demonstrate mastery of relevant vocabulary, and use the vocabulary fluently in their work. They should know common examples and counterexamples, and be able prove that these examples and counterexamples have properties as claimed. Additionally, students should know the content (and limitations) of major theorems and the ideas of the proofs, and apply results of these theorems to solve suitable problems, or use techniques of the proofs to prove additional related results, or to make calculations and computations.

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

# Manifolds
#* Topological and smooth manifolds
#* Manifolds with boundary
#* Tangent vectors
#* Tangent bundles
#* Vector bundles and bundle maps
#* Cotangent bundles
# Submanifolds
#* Submersions, immersions, embeddings
#* Inverse and implicit function theorems
#* Transversality
#* Embedding and approximation theorems
# Differential forms and tensors
#* Wedge product
#* Exterior derivative
#* Orientations
#* Stoke's Theorem

</div>

=== Additional topics ===

At the discretion of the instructor as time allows. Topics might include Lie groups and homogeneous spaces, Morse functions, de Rham cohomology and the de Rham theorem, Jordan curve theorem, Lefschetz fixed-point theory, degree, Gauss-Bonnet theorem, etc.

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

[[Category:Courses|655]]

Math 751R: Advanced Special Topics in Topology

2022-10-14T17:04:11Z

Cpg:

== Catalog Information ==

=== Title ===
Advanced Special Topics in Topology.

=== Credit Hours ===
3

=== Prerequisite ===

=== Description ===
This course covers current topics of research interest.

== Desired Learning Outcomes ==
Students should become familiar with a specific area of topology undergoing current research.

=== Prerequisites ===
Graduate level differential and algebraic topology at the level of [[Math 655]] and [[Math 656|656]].

=== Minimal learning outcomes ===
Minimal learning outcomes cannot be specified for a course in which topics will vary from year to year. However, regardless of the topic, successful students will know terminology, statements and approaches to problems undergoing active research, and major results in the area and techniques used to prove them. Students will demonstrate this knowledge by working suitable problems and developing their own proofs, by presenting and writing work inside and outside of class, and participating in other activities expected of more advanced graduate students.

As an example of the level and type of material covered, the following topics were required for the course when it covered hyperbolic knot theory.

<div style="-moz-column-count:2; column-count:2;">
#Link complements as 3-manifolds
#*Polyhedral decomposition associated with a link diagram
#*Alternating versus non-alternating links
#*Subdivision into tetrahedra, cusp triangulations
#Geometric structures on manifolds, particularly hyperbolic structures
#*Developing map, holonomy
#*Complete and incomplete structures
#*Mostow-Prasad rigidity
#*Gluing and completeness equations
#*Ideal hyperbolic tetrahedra, geometric triangulations
#Hyperbolic Dehn surgery
#*Hyperbolic Dehn surgery space and its shape
#*Exceptional Dehn surgeries
#*Links admitting exceptional Dehn surgeries
#Examples of hyperbolic links and their geometric properties
#*Volumes, embedded geodesic surfaces, etc.

</div>
=== Textbooks ===

Possible textbooks for this course include (but are not limited to):

*

=== Additional topics ===

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

None.

[[Category:Courses|751]]

Math 656: Algebraic Topology

2022-10-14T17:03:34Z

Cpg:

== Catalog Information ==

=== Title ===
Algebraic Topology.

=== Credit Hours ===
3

=== Offered ===
W (odd years)

=== Prerequisite ===
[[Math 371]] (or equivalent) and [[Math 553]].

=== Description ===
A rigorous treatment of the fundamentals of algebraic topology, including homotopy (fundamental group and higher homotopy groups) and homology and cohomology of spaces.

== Desired Learning Outcomes ==

=== Prerequisites ===

Topology of manifolds, tensors, and orientation is assumed from [[Math 655]].

=== Minimal learning outcomes ===

Outlined below are topics that all successful Math 656 students should understand well. Students should be able to demonstrate mastery of relevant vocabulary, and use the vocabulary fluently in their work. They should know common examples and counterexamples, and be able prove that these examples and counterexamples have properties as claimed. Additionally, students should know the content (and limitations) of major theorems and the ideas of the proofs, and apply results of these theorems to solve suitable problems, or use techniques of the proofs to prove additional related results, or to make calculations and computations.

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

# Fundamental group and homotopy
#* Constructions
#* Van Kampen Theorem
#* Covering spaces and group actions
#* Higher homotopy groups
# Homology
#* Simplicial, singular, cellular
#* Exact sequences and excision
#* Mayer-Vietoris sequences
#* Homology with coefficients
#* Homology and the fundamental group
# Cohomology
#* Universal coefficient theorem
#* Cup product
#* Poincare duality

</div>

=== Additional topics ===

At the discretion of the instructor as time allows. Possible topics include de Rham cohomology and the de Rham theorem, additional duality theorems, including Alexander and Lefschetz duality, additional homotopy theory, including Whitehead's theorem, the Hurewicz theorem, fiber bundles, fibrations, obstruction theory.

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

[[Math 751R]]

[[Category:Courses|656]]

Math 413 Advanced Linear Algebra

2019-10-24T21:33:28Z

Cpg:

== Catalog Information ==

=== Title ===
Advanced Linear Algebra

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

=== Offered ===
F and possibly W

=== Prerequisite ===
[[Math 371]] recommended, but not required.

=== Description ===
Theory and advanced topics of linear algebra.

== Desired Learning Outcomes ==

<strong>Prove:</strong> Students will be able to prove central linear-algebraic results, as well as other results with similar derivations.

<strong>Distinguish:</strong> Students will be able to distinguish between true and plausibly-sounding false propositions in the language of linear algebra.

<strong>Construct:</strong> Students will be able to construct examples and counterexamples illustrating relations between different linear-algebraic concepts.

<strong>Categorize:</strong> Students will be able to categorize linear-algebraic structures according to their properties.

<strong>Calculate:</strong> Students will be able to calculate precisely and efficiently, choosing appropriate methods.

=== Prerequisites ===
[[Math 371]] recommended, but not required.

=== Minimal learning outcomes ===
# Linear equations (row operations, matrix multiplication, and invertibility)
# Vector spaces
# Linear transformations (algebra of linear transformations, isomorphisms, linear functionals, duality)
# Polynomials and determinants (algebra of polynomials, polynomial ideals, determinant functions, permutations and uniqueness of determinants)
# Jordan canonical form and elementary canonical forms (invariant subspaces, simultaneous diagonalization and triangulation, direct-sum decompositions, rational forms, Jordan form)
# Inner product spaces (inner product spaces, linear functionals and adjoints, unitary operators, normal operators)

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

*

=== Additional topics ===

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

[[Category:Courses|413]]

Garbage 2

2015-05-04T21:04:33Z

Cpg: moved Math 320: Algorithm Design and Optimization 1 to Garbage: Wish page would go away. Delete it if you can.

Math 320: Algorithm Design and Optimization 1

2015-05-04T21:04:33Z

Cpg: moved Math 320: Algorithm Design and Optimization 1 to Garbage: Wish page would go away. Delete it if you can.

#REDIRECT [[Garbage]]

Garbage 2

2015-05-04T20:52:07Z

Cpg: Blanked the page

Garbage 2

2015-05-04T20:36:57Z

Cpg: Created page with "== Catalog Information == === Title === Algorithms Design and Optimization 1 === (Credit Hours:Lecture Hours:Lab Hours) === (3:3:0) === Offered === F === Prerequisite === [[M..."

== Catalog Information ==

=== Title ===
Algorithms Design and Optimization 1

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

=== Offered ===
F

=== Prerequisite ===
[[Math 290]], [[Math 313]], [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]

=== Description ===
A treatment of algorithms used to solve these problems. Specific topics include Complexity and Data, Approximation Theory, Recursive Algorithms, Linear Optimization, Unconstrained Optimization, Constrained Optimization, Global Optimization.

== Desired Learning Outcomes ==

=== Prerequisites ===
[[Math 290]], [[Math 313]], [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]

=== Minimal learning outcomes ===
Students will have a solid understanding of the concepts listed below. They will be able to prove many of the theorems that are central to this material. They will understand the model specifications for the optimization algorithms, and be able to recognize whether they apply to a given application or not. They will be able to perform the relevant computations on small, simple problems. They will be able to describe the optimization and approximation algorithms well enough that they could program simple versions of them, and will have a basic knowledge of the computational strengths and weaknesses of the algorithms covered.

# Complexity and Data
#* Asymptotic Analysis
#* Combinatorics
#* Graphs and Trees
#* Complexity (P, NP, NP Complete)
# Approximation Theory
#* Interpolation and Splines
#* Stone-Weierstrass Theorem
#* Bezier Curves
#* B-Splines
# Recursive Algorithms
#* Difference Calculus, including Summation by Parts
#* Simple linear recurrences
#* General linear recurrences
#* Generating functions
# Linear Optimization
#* Problem Formulation
#* Simplex Method
#* Duality
#* Applications
# Unconstrained Optimization
#* Steepest Descent
#* Newton
#* Broyden
#* Conjugate Gradient
#* Applications
# Constrained Optimization
#* Equality Constrained, Lagrange Multipliers
#* Inequality Constrained, KKT Condition
#* Applications
# Global Optimization
#* Interior Point Methods
#* Genetic Algorithms
#* Simulated Annealing

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

*

=== Additional topics ===

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

[[Category:Courses|320]]

Math 439: Modeling with Dynamics and Control 2 Lab

2012-06-06T18:29:46Z

Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."

== Catalog Information ==

=== Title ===

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

=== Offered ===

=== Prerequisite ===

=== Description ===

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

*

=== Additional topics ===

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

[[Category:Courses|439]]

Math 438: Modeling with Dynamics and Control 2

2012-06-06T18:29:23Z

Math 437: Modeling with Dynamics and Control 1 Lab

2012-06-06T18:29:08Z

Math 436: Modeling with Dynamics and Control 1

2012-06-06T18:28:55Z

Math 405: Modeling with Uncertainty and Data 2 Lab

2012-06-06T18:28:31Z

Math 404: Modeling with Uncertainty and Data 2

2012-06-06T18:28:15Z

Math 403: Modeling with Uncertainty and Data 1 Lab

2012-06-06T18:28:00Z

Math 402: Modeling with Uncertainty and Data 1

2012-06-06T18:27:45Z

Math 347: Mathematical Analysis 2 Lab

2012-06-06T18:27:20Z

Math 346: Mathematical Analysis 2

2012-06-06T18:27:06Z

Math 345: Mathematical Analysis 1 Lab

2012-06-06T18:26:50Z

Math 344: Mathematical Analysis 1

2012-06-06T18:26:35Z

Math 323: Algorithm Design and Optimization 2 Lab

2012-06-06T18:26:13Z

Math 322: Algorithm Design and Optimization 2

2012-06-06T18:25:55Z

Math 321: Algorithm Design and Optimization 1 Lab

2012-06-06T18:25:35Z

Garbage 3

2012-06-06T18:25:09Z

Math 480

2012-06-06T18:20:57Z

Cpg: Blanked the page

Math 480

2012-06-06T18:20:22Z

Cpg:

== Catalog Information ==

=== Title ===
Mathematical Models.

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

=== Offered ===
On demand

=== Prerequisite ===
[[Math 334]], [[Math 410|410]].

=== Description ===
Construction, solution, and interpretation of discrete and continuous models applied to problems in the physical, natural, and social sciences.

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===

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

</div>
=== Textbooks ===

Possible textbooks for this course include (but are not limited to):

*

=== Additional topics ===

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

Math 97

2012-05-22T19:13:01Z

Cpg:

== Catalog Information ==

=== Title ===
Intermediate Algebra

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(0:2:1)

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

=== Prerequisite ===
High school algebra.

=== Description ===
Elementary logic, real number system, equations and inequalities (linear, polynomial, rational, and radical expressions), graphing, function notation, inverse function, exponential functions, systems of equations, variations.

== Desired Learning Outcomes ==
???

=== Prerequisites ===
???

=== Minimal learning outcomes ===
???

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

*

=== Additional topics ===
???

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

[[Category:Courses|097]]

Math 97

2012-05-22T19:12:33Z

Cpg: Created page with "== Catalog Information == === Title === Intermediate Algebra === (Credit Hours:Lecture Hours:Lab Hours) === (0:2:1) === Offered === F, W, Sp, Su === Prerequisite === High sch..."

Math 431: Probability Theory

2012-05-16T15:32:59Z

Cpg:

== Catalog Information ==

=== Title ===
Probability Theory.

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

=== Offered ===
F

=== Prerequisite ===
[[Math 313]].

=== Description ===
Axiomatic probability theory, conditional probability, discrete / continuous random variables, expectation, conditional expectation, moments, functions of random variables, multivariate distributions, laws of large numbers, central limit theorem.

== Desired Learning Outcomes ==

This course is a calculus-based first course in probability. It is cross-listed with EC En 370.

=== Prerequisites ===

The current prerequisite is linear algebra. Because of the need to work with joint distributions of continuous random variables in Math 431, the department should consider adding multivariable calculus as a prerequisite.

=== Minimal learning outcomes ===

Primarily, students should be able to do basic computation of probabilistic quantities, including those involving applications. Students should be able to recall the most common types of discrete and continuous random variables and describe and compute their properties. Students should understand the theory of probability <i>in an elementary context</i>.

<div style="column-count:2;-moz-column-count:2;-webkit-column-count:2">
# Basic principles of counting
#* Product sets
#* Disjoint unions
#* Combinations
#* Permutations
# Axiomatic probability
#* Outcomes
#* Events
#* Probability measures
#** Additivity
#** Continuity
# Discrete random variables
#* Probability mass function
#* Cumulative distribution function
#* Moments
#** Expectation
#*** Of a function of a random variable
#** Variance
#* Common types
#** Bernoulli
#** Binomial
#** Poisson <br><br><br><br>
# Continuous random variables
#* Probability density function
#* Cumulative distribution function
#* Moments
#** Expectation
#*** Of a function of a random variable
#** Variance
#* Common types
#** Uniform
#** Exponential
#** Normal
# Conditional probability
#* As a probability
#* Bayes' Formula
#* Independence
#** Events
#** Random variables
# Joint distributions
#* Covariance
#* Conditional distributions
# Conditional expectation
# Limit theorems
#* Weak Law of Large Numbers
#* Strong Law of Large Numbers
#* Central Limit Theorem
</div>

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Sheldon Ross, ''A First Course in Probability (8th edition)'', Prentice Hall, 2009.

=== Additional topics ===

If time permits, geometric, negative binomial, hypergeometric, gamma, Weibull, Cauchy, and/or beta random variables might be studied.

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

Currently, Math 431 is only a prerequisite for [[Math 435]]. Consideration should perhaps be given to making it a prerequisite for [[Math 543]].

[[Category:Courses|431]]

Math 540: Linear Analysis

2011-11-14T18:34:25Z

Cpg: /* Textbooks */

== Catalog Information ==

=== Title ===
Linear Analysis.
=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Recommended ===
[[Math 342]] or equivalent.

=== Description ===
Normed vector spaces and linear maps between them.

== Desired Learning Outcomes ==
The course is designed to cover elementary abstract linear functional analysis. "Elementary" means that methods dependent on complex analysis or measure-theoretic integration are not core topics. "Abstract" means that applications to specific function spaces are not core topics.

=== Prerequisites ===
The official prerequisite is [[Math 342]]. What's important is that incoming students be familiar with linear algebra and metric spaces and be mathematically mature.

=== Minimal learning outcomes ===
Students should obtain a thorough understanding of the topics listed below. In particular they should be able to define and use relevant terminology, compare and contrast closely-related concepts, and state (and, where feasible, prove) major theorems.

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

# Normed spaces
#* Basics
#** Banach spaces
#** Special linear operators
#*** Continuous/bounded
#*** Compact
#*** Finite rank
#* Duality
#** Dual spaces
#*** Their completeness
#** Adjoints of bounded linear operators
#** Second duals
#*** Reflexivity
#** Weak and weak-star topologies
#*** Banach-Alaoglu theorem
#* Structure
#** Hamel and Schauder bases
#** Biorthogonal systems
#** Separability
#** Direct sums
#** Quotient spaces
#* Finite-dimensional spaces
#** Equivalence of all norms
#** Completeness
#** Continuity of all linear operators
#** Characterization: unit ball is compact
#* Fundamental theorems
#** Baire category theorem
#** Hahn-Banach extension theorem
#** Banach-Steinhaus theorem
#** Open mapping theorem
#** Closed graph theorem
#** Bounded inverse theorem
# Inner product spaces
#* Basics
#** Hilbert spaces
#** Special linear operators
#*** Self-adjoint
#*** Unitary
#*** Normal
#*** Orthogonal projections
#*** Hilbert-Schmidt operators
#* Structure
#** Orthogonality
#*** Complements and direct sums
#*** Bases
#** Representation theorems
#*** Riesz-Frechet theorem
#*** Lax-Milgram theorem
#** Abstract Fourier theory
#*** Riesz-Fischer theorem
#*** Bessel’s inequality
#*** Parseval’s identities
# Spectral theory
#* Banach algebras
#* Bounded operators on Banach spaces
#** Gelfand’s spectral-radius formula
#* Compact operators on Banach spaces
#** Riesz-Schauder theory including Fredholm Alternative
#* Compact normal operators on Hilbert spaces
#* Compact self-adjoint operators on Hilbert spaces<br><br><br><br><br><br>

</div>
=== Textbooks ===

Possible textbooks for this course include (but are not limited to):

* David Promislow, <i>A First Course in Functional Analysis</i>, Wiley, 2008.

=== Additional topics ===
While the focus of the course is on abstract theory, this theory should probably be motivated and illustrated with appropriate concrete examples.

=== Courses for which this course is prerequisite ===
This course is recommended for [[Math 640]]. Indirectly (through the [[Math 640]]), this course will possibly become be a prerequisite for [[Math 647]].

[[Category:Courses|540]]

Math 640: Nonlinear Analysis

2011-08-03T21:48:35Z

Cpg:

== Catalog Information ==

=== Title ===
Nonlinear Analysis.
=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Recommended(?) ===
[[Math 540]].

=== Description ===
Differential calculus in normed spaces, fixed point theory, and abstract critical point theory.

== Desired Learning Outcomes ==
This course is intended as a natural nonlinear sequel to [[Math 540]]. Like its prequel, the focus would be on operators on abstract Banach spaces.

=== Prerequisites ===
Students need to have a good understanding of basic linear analysis, whether this comes from taking the [[Math 540]] or some other way.

=== Minimal learning outcomes ===
Students should obtain a thorough understanding of the topics listed below. In particular they should be able to define and use relevant terminology, compare and contrast closely-related concepts, and state (and, where feasible, prove) major theorems.

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

# Differential calculus on normed spaces
#* Fréchet derivatives
#* Gâteaux derivatives
#* Inverse Function theorem
#* Implicit Function theorem
#* Lyapunov-Schmidt reduction
# Fixed point theory
#* Metric spaces
#** Banach’s contraction mapping principle
#** Parametrized contraction mapping principle
#* Finite-dimensional spaces
#** Brouwer fixed point theorem
#* Normed spaces
#** Schauder fixed point theorem
#** Leray-Schauder alternative
#* Ordered Banach spaces
#** Monotone iterative method
#* Monotone operators
#** Browder-Minty theorem
# Abstract critical point theory
#* Functional properties
#** Convexity
#** Coercivity
#** Lower semi-continuity
#* Existence of global minimizers
#* Existence of constrained minimizers
#* Minimax results
#** Ambrosetti-Rabinowitz mountain pass theorem
</div>
=== Textbooks ===

Possible textbooks for this course include (but are not limited to):

*

=== Additional topics ===
In addition to the minimal learning outcomes above, instructors should give serious consideration to covering the following specific topics:

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

# Differential calculus on normed spaces
#* Nash-Moser theorem
# Fixed point theory
#* Metric spaces
#** Caristi fixed point theorem
#* Hilbert spaces
#** Browder-Göhde-Kirk theorem
#* Ordered Banach spaces
#** Krasnoselski’s fixed point theorem
#** Krein-Rutman theorem
#* Monotone operators
#** Hartman-Stampacchia theorem
# Abstract critical point theory
#* Minimax results
#** Ky Fan’s minimax inequality
#** Ekeland’s variational principle
#** Schechter’s bounded mountain pass theorem
#** Rabinowitz saddle point theorem
#** Rabinowitz linking theorem

</div>

Furthermore, it is anticipated that instructors will want to motivate the abstract theory by considering appropriate concrete examples.

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

It is proposed that this course be a prerequisite for [[Math 647]].

[[Category:Courses|640]]

Math 648: Theory of Partial Differential Equations 2

2011-05-31T19:12:25Z

Cpg: /* Desired Learning Outcomes */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 2.

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

=== Offered ===
F

=== Prerequisite ===
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.

=== Description ===
Advanced theory of partial differential equations. Functional-analytic techniques.

== Desired Learning Outcomes ==

=== Prerequisites ===
Students need a thorough understanding of real analysis.

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 648 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;">
# Second-order elliptic equations
#* Classification
#* Weak solutions
#** Lax-Milgram theorem
#** Energy estimates
#** Fredholm alternative
#* Regularity
#** Interior
#** Boundary
#* Maximum principles
#** Weak
#** Strong
#** Harnack's inequality
#* Eigenpairs of elliptic operators
#** Symmetric
#** Nonsymmetric
# Linear Evolution Equations
#* Second-order parabolic equations
#** Weak solutions
#** Regularity
#** Maximum principles
#* Second-order hyperbolic equations
#** Weak solutions
#** Regularity
# Calculus of Variations
#* Euler-Lagrange equation
#* Coercivity
#* Convexity
#* Semicontinuity
#* Weak Solutions
#* Regularity
#* Constraints
#* Critical points
#** Mountain pass theorem
# Hamilton-Jacobi equations
#* Viscosity solutions
</div>

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===
If time permits, topics that could be discussed include hyperbolic systems, semigroup theory, systems of convservation laws, and nonvariational techniques for nonlinear equations.

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

[[Category:Courses|648]]
None

Math 648: Theory of Partial Differential Equations 2

2011-05-31T19:12:08Z

Cpg: /* Prerequisites */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 2.

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

=== Offered ===
F

=== Prerequisite ===
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.

=== Description ===
Advanced theory of partial differential equations. Functional-analytic techniques.

== Desired Learning Outcomes ==
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.

=== Prerequisites ===
Students need a thorough understanding of real analysis.

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 648 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;">
# Second-order elliptic equations
#* Classification
#* Weak solutions
#** Lax-Milgram theorem
#** Energy estimates
#** Fredholm alternative
#* Regularity
#** Interior
#** Boundary
#* Maximum principles
#** Weak
#** Strong
#** Harnack's inequality
#* Eigenpairs of elliptic operators
#** Symmetric
#** Nonsymmetric
# Linear Evolution Equations
#* Second-order parabolic equations
#** Weak solutions
#** Regularity
#** Maximum principles
#* Second-order hyperbolic equations
#** Weak solutions
#** Regularity
# Calculus of Variations
#* Euler-Lagrange equation
#* Coercivity
#* Convexity
#* Semicontinuity
#* Weak Solutions
#* Regularity
#* Constraints
#* Critical points
#** Mountain pass theorem
# Hamilton-Jacobi equations
#* Viscosity solutions
</div>

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===
If time permits, topics that could be discussed include hyperbolic systems, semigroup theory, systems of convservation laws, and nonvariational techniques for nonlinear equations.

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

[[Category:Courses|648]]
None

Math 648: Theory of Partial Differential Equations 2

2011-05-31T19:11:28Z

Cpg: /* Additional topics */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 2.

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

=== Offered ===
F

=== Prerequisite ===
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.

=== Description ===
Advanced theory of partial differential equations. Functional-analytic techniques.

== Desired Learning Outcomes ==
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.

=== Prerequisites ===
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 648 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;">
# Second-order elliptic equations
#* Classification
#* Weak solutions
#** Lax-Milgram theorem
#** Energy estimates
#** Fredholm alternative
#* Regularity
#** Interior
#** Boundary
#* Maximum principles
#** Weak
#** Strong
#** Harnack's inequality
#* Eigenpairs of elliptic operators
#** Symmetric
#** Nonsymmetric
# Linear Evolution Equations
#* Second-order parabolic equations
#** Weak solutions
#** Regularity
#** Maximum principles
#* Second-order hyperbolic equations
#** Weak solutions
#** Regularity
# Calculus of Variations
#* Euler-Lagrange equation
#* Coercivity
#* Convexity
#* Semicontinuity
#* Weak Solutions
#* Regularity
#* Constraints
#* Critical points
#** Mountain pass theorem
# Hamilton-Jacobi equations
#* Viscosity solutions
</div>

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===
If time permits, topics that could be discussed include hyperbolic systems, semigroup theory, systems of convservation laws, and nonvariational techniques for nonlinear equations.

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

[[Category:Courses|648]]
None

Math 648: Theory of Partial Differential Equations 2

2011-05-31T19:09:17Z

Cpg: /* Minimal learning outcomes */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 2.

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

=== Offered ===
F

=== Prerequisite ===
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.

=== Description ===
Advanced theory of partial differential equations. Functional-analytic techniques.

== Desired Learning Outcomes ==
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.

=== Prerequisites ===
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 648 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;">
# Second-order elliptic equations
#* Classification
#* Weak solutions
#** Lax-Milgram theorem
#** Energy estimates
#** Fredholm alternative
#* Regularity
#** Interior
#** Boundary
#* Maximum principles
#** Weak
#** Strong
#** Harnack's inequality
#* Eigenpairs of elliptic operators
#** Symmetric
#** Nonsymmetric
# Linear Evolution Equations
#* Second-order parabolic equations
#** Weak solutions
#** Regularity
#** Maximum principles
#* Second-order hyperbolic equations
#** Weak solutions
#** Regularity
# Calculus of Variations
#* Euler-Lagrange equation
#* Coercivity
#* Convexity
#* Semicontinuity
#* Weak Solutions
#* Regularity
#* Constraints
#* Critical points
#** Mountain pass theorem
# Hamilton-Jacobi equations
#* Viscosity solutions
</div>

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===

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

[[Category:Courses|648]]
None

Math 648: Theory of Partial Differential Equations 2

2011-05-31T19:02:33Z

Cpg: /* Additional topics */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 2.

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

=== Offered ===
F

=== Prerequisite ===
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.

=== Description ===
Advanced theory of partial differential equations. Functional-analytic techniques.

== Desired Learning Outcomes ==
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.

=== Prerequisites ===
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.

=== Minimal learning outcomes ===
#The Bochner Integral
#*The Pettis theorem
#*The spaces ''L''<sup>''p''</sup>(Ω; ''X'')
#*Vector measures and Radon Nikodym property in Banach space
#*Riesz representation theorem for the duals of ''L''<sup>''p''</sup>(Ω;''X'')
#*Embedding results of Lions and Simon
#Surjectivity of nonlinear set valued operators.
#Lion's method of elliptic regularization and evolution equations of mixed type.
#Weak Derivatives
#*Morrey's inequality and Rademacher's theorem
#*Area formula
#*Integration on manifolds
#Sobolev spaces
#*Embedding theorems for ''W''<sup>''m'', ''p''</sup>('''R'''<sup>''n''</sup>)
#*Extension theorems for Lipschitz domains
#*General embedding theorems
#Korn's Inequality on bounded Lipschitz domains
#Elliptic regularity and Nirenberg differences
#The trace spaces of Lions
#*Traces of Sobolev spaces and fractional order spaces
#*The half space
#*A right inverse for the trace for a half space
#*Intrinsic norms
#*Fractional order Sobolev spaces
#*Reflexivity of fractional order Sobolev spaces
#Sobolev spaces on manifolds
#*Basic definitions
#*The trace on the boundary of an open set

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

</div>
=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===

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

[[Category:Courses|648]]
None

Math 648: Theory of Partial Differential Equations 2

2011-05-31T19:01:53Z

Cpg: /* Description */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 2.

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

=== Offered ===
F

=== Prerequisite ===
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.

=== Description ===
Advanced theory of partial differential equations. Functional-analytic techniques.

== Desired Learning Outcomes ==
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.

=== Prerequisites ===
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.

=== Minimal learning outcomes ===
#The Bochner Integral
#*The Pettis theorem
#*The spaces ''L''<sup>''p''</sup>(Ω; ''X'')
#*Vector measures and Radon Nikodym property in Banach space
#*Riesz representation theorem for the duals of ''L''<sup>''p''</sup>(Ω;''X'')
#*Embedding results of Lions and Simon
#Surjectivity of nonlinear set valued operators.
#Lion's method of elliptic regularization and evolution equations of mixed type.
#Weak Derivatives
#*Morrey's inequality and Rademacher's theorem
#*Area formula
#*Integration on manifolds
#Sobolev spaces
#*Embedding theorems for ''W''<sup>''m'', ''p''</sup>('''R'''<sup>''n''</sup>)
#*Extension theorems for Lipschitz domains
#*General embedding theorems
#Korn's Inequality on bounded Lipschitz domains
#Elliptic regularity and Nirenberg differences
#The trace spaces of Lions
#*Traces of Sobolev spaces and fractional order spaces
#*The half space
#*A right inverse for the trace for a half space
#*Intrinsic norms
#*Fractional order Sobolev spaces
#*Reflexivity of fractional order Sobolev spaces
#Sobolev spaces on manifolds
#*Basic definitions
#*The trace on the boundary of an open set

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

</div>
=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===
The above might be all there is time for, but if there is time for more, it would be nice to consider Mihlin's theorem and the $L^{p}$ theory of elliptic regularity. Other topics could include methods of interpolation in Banach space.

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

[[Category:Courses|648]]
None

Math 648: Theory of Partial Differential Equations 2

2011-05-31T19:01:03Z

Cpg: /* Prerequisite */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 2.

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

=== Offered ===
F

=== Prerequisite ===
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.

=== Description ===
This course develops abstract methods for studying partial differential equations and inclusions.

== Desired Learning Outcomes ==
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.

=== Prerequisites ===
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.

=== Minimal learning outcomes ===
#The Bochner Integral
#*The Pettis theorem
#*The spaces ''L''<sup>''p''</sup>(Ω; ''X'')
#*Vector measures and Radon Nikodym property in Banach space
#*Riesz representation theorem for the duals of ''L''<sup>''p''</sup>(Ω;''X'')
#*Embedding results of Lions and Simon
#Surjectivity of nonlinear set valued operators.
#Lion's method of elliptic regularization and evolution equations of mixed type.
#Weak Derivatives
#*Morrey's inequality and Rademacher's theorem
#*Area formula
#*Integration on manifolds
#Sobolev spaces
#*Embedding theorems for ''W''<sup>''m'', ''p''</sup>('''R'''<sup>''n''</sup>)
#*Extension theorems for Lipschitz domains
#*General embedding theorems
#Korn's Inequality on bounded Lipschitz domains
#Elliptic regularity and Nirenberg differences
#The trace spaces of Lions
#*Traces of Sobolev spaces and fractional order spaces
#*The half space
#*A right inverse for the trace for a half space
#*Intrinsic norms
#*Fractional order Sobolev spaces
#*Reflexivity of fractional order Sobolev spaces
#Sobolev spaces on manifolds
#*Basic definitions
#*The trace on the boundary of an open set

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

</div>
=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===
The above might be all there is time for, but if there is time for more, it would be nice to consider Mihlin's theorem and the $L^{p}$ theory of elliptic regularity. Other topics could include methods of interpolation in Banach space.

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

[[Category:Courses|648]]
None

Math 647: Theory of Partial Differential Equations 1

2011-05-31T18:59:39Z

Cpg: /* Minimal learning outcomes */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 1.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.

=== Description ===
Proposed: Classical theory of canonical linear PDEs. Introduction to Sobolev spaces.

== Desired Learning Outcomes ==

=== Prerequisites ===
Students should understand analysis at the first-year graduate level.

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 647 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;">
# Classical theory for canonical linear PDEs
#* Transport equation
#* Laplace's equation
#** Fundamental solution
#** Mean-value and maximum principles
#** Energy methods
#* Heat equation
#** Fundamental solution
#** Mean-value and maximum principles
#** Energy methods
#* Wave equation
#** Spherical means
#** Energy methods
# Method of characteristics
# Sobolev spaces
#* Traces
#* Sobolev inequalities
#* Compactness
</div>

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===

If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.

=== Courses for which this course is prerequisite ===
[[Math 648]]

[[Category:Courses|647]]

Math 647: Theory of Partial Differential Equations 1

2011-05-31T18:59:10Z

Cpg: /* Description */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 1.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.

=== Description ===
Proposed: Classical theory of canonical linear PDEs. Introduction to Sobolev spaces.

== Desired Learning Outcomes ==

=== Prerequisites ===
Students should understand analysis at the first-year graduate level.

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 647 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;">
# Foundational theory for canonical linear PDEs
#* Transport equation
#* Laplace's equation
#** Fundamental solution
#** Mean-value and maximum principles
#** Energy methods
#* Heat equation
#** Fundamental solution
#** Mean-value and maximum principles
#** Energy methods
#* Wave equation
#** Spherical means
#** Energy methods
# Method of characteristics
# Sobolev spaces
#* Traces
#* Sobolev inequalities
#* Compactness
</div>

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===

If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.

=== Courses for which this course is prerequisite ===
[[Math 648]]

[[Category:Courses|647]]

Math 647: Theory of Partial Differential Equations 1

2011-05-31T18:57:57Z

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

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 1.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.

=== Description ===

== Desired Learning Outcomes ==

=== Prerequisites ===
Students should understand analysis at the first-year graduate level.

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 647 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;">
# Foundational theory for canonical linear PDEs
#* Transport equation
#* Laplace's equation
#** Fundamental solution
#** Mean-value and maximum principles
#** Energy methods
#* Heat equation
#** Fundamental solution
#** Mean-value and maximum principles
#** Energy methods
#* Wave equation
#** Spherical means
#** Energy methods
# Method of characteristics
# Sobolev spaces
#* Traces
#* Sobolev inequalities
#* Compactness
</div>

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===

If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.

=== Courses for which this course is prerequisite ===
[[Math 648]]

[[Category:Courses|647]]

Math 647: Theory of Partial Differential Equations 1

2011-05-31T18:57:41Z

Cpg: /* Prerequisites */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 1.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.

=== Description ===

== Desired Learning Outcomes ==

=== Prerequisites ===
Students should understand analysis at the first-year graduate level.

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 647 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;">
# Foundational theory for canonical linear PDEs
#* Transport equation
#* Laplace's equation
#** Fundamental solution
#** Mean-value and maximum principles
#** Energy methods
#* Heat equation
#** Fundamental solution
#** Mean-value and maximum principles
#** Energy methods
#* Wave equation
#** Spherical means
#** Energy methods
# Method of characteristics
# Sobolev spaces
#* Traces
#* Sobolev inequalities
#* Compactness
</div>

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===

If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.

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

[[Category:Courses|647]]

Math 647: Theory of Partial Differential Equations 1

2011-05-31T18:56:58Z

Cpg: /* Minimal learning outcomes */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 1.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.

=== Description ===

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 647 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;">
# Foundational theory for canonical linear PDEs
#* Transport equation
#* Laplace's equation
#** Fundamental solution
#** Mean-value and maximum principles
#** Energy methods
#* Heat equation
#** Fundamental solution
#** Mean-value and maximum principles
#** Energy methods
#* Wave equation
#** Spherical means
#** Energy methods
# Method of characteristics
# Sobolev spaces
#* Traces
#* Sobolev inequalities
#* Compactness
</div>

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===

If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.

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

[[Category:Courses|647]]

Math 647: Theory of Partial Differential Equations 1

2011-05-31T18:56:02Z

Cpg: /* Additional topics */

Math 647: Theory of Partial Differential Equations 1

2011-05-31T18:54:59Z

Cpg: /* Minimal learning outcomes */

Math 647: Theory of Partial Differential Equations 1

2011-05-31T18:52:34Z

Cpg: /* Minimal learning outcomes */

Math 647: Theory of Partial Differential Equations 1

2011-05-31T18:49:08Z

Cpg: /* Minimal learning outcomes */

Math 648: Theory of Partial Differential Equations 2

2011-05-31T18:24:53Z

Cpg: /* Textbooks */

== Catalog Information ==

=== Title ===
Theory of Partial Differential Equations 2.

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

=== Offered ===
F

=== Prerequisite ===
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]

=== Description ===
This course develops abstract methods for studying partial differential equations and inclusions.

== Desired Learning Outcomes ==
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.

=== Prerequisites ===
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.

=== Minimal learning outcomes ===
#The Bochner Integral
#*The Pettis theorem
#*The spaces ''L''<sup>''p''</sup>(Ω; ''X'')
#*Vector measures and Radon Nikodym property in Banach space
#*Riesz representation theorem for the duals of ''L''<sup>''p''</sup>(Ω;''X'')
#*Embedding results of Lions and Simon
#Surjectivity of nonlinear set valued operators.
#Lion's method of elliptic regularization and evolution equations of mixed type.
#Weak Derivatives
#*Morrey's inequality and Rademacher's theorem
#*Area formula
#*Integration on manifolds
#Sobolev spaces
#*Embedding theorems for ''W''<sup>''m'', ''p''</sup>('''R'''<sup>''n''</sup>)
#*Extension theorems for Lipschitz domains
#*General embedding theorems
#Korn's Inequality on bounded Lipschitz domains
#Elliptic regularity and Nirenberg differences
#The trace spaces of Lions
#*Traces of Sobolev spaces and fractional order spaces
#*The half space
#*A right inverse for the trace for a half space
#*Intrinsic norms
#*Fractional order Sobolev spaces
#*Reflexivity of fractional order Sobolev spaces
#Sobolev spaces on manifolds
#*Basic definitions
#*The trace on the boundary of an open set

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

</div>
=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.

=== Additional topics ===
The above might be all there is time for, but if there is time for more, it would be nice to consider Mihlin's theorem and the $L^{p}$ theory of elliptic regularity. Other topics could include methods of interpolation in Banach space.

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

[[Category:Courses|648]]
None

Math 647: Theory of Partial Differential Equations 1

2011-05-31T18:24:35Z

Cpg: /* Textbooks */