MathWiki - User contributions [en]

Math 554: Foundations of Topology 2

2010-05-26T19:59:42Z

Dgw4:

== Catalog Information ==

=== Title ===
Foundations of Topology 2.

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

=== Prerequisite ===
[[Math 553]] or instructor's consent.

=== Description ===
Fundamental group, retractions and fixed points, homotopy types, separation theorems, classification of surfaces, Seifert-van Kampen Theorem,
classification of covering spaces, and applications to group theory.

== Desired Learning Outcomes ==
Students should gain a familiarity with surfaces, fundamental group, and covering spaces.

=== Minimal learning outcomes ===

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

# The Fundamenatal Group
# The topology of the plane
#* Jordan Curve Theorem
# Seifert-van Kampen Theorem
# Classification of Surfaces
# Classification of Covering Spaces
# Group Theory
#* Free groups
#* Free abelian groups
#* Presentations of groups
#* Subgroups of free groups

</div>

=== Additional topics ===

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

[[Category:Courses|554]]

Math 553: Foundations of Topology 1

2010-05-26T19:51:43Z

Dgw4: /* Description */

== Catalog Information ==

=== Title ===
Foundations of Topology 1.

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

=== Offered ===
F
=== Prerequisite ===
[[Math 451]] or instructor's consent.
=== Description ===
Naive set theory, topological spaces, product spaces, subspaces, continuous functions, connectedness, compactness, countability, separation axioms, metrization, and complete metric spaces.

== Desired Learning Outcomes ==
Students should gain a familiarity with the general topology that is used throughout mathematics.
=== Minimal learning outcomes ===
<div style="-moz-column-count:2; column-count:2;">
# Set Theory
#* Finite, countable, and uncountable sets
#* Well-ordered sets
# Topological Spaces
#* Basis for a topology
#* Product topology
#* Metric topology
# Continuous Functions
# Connectedness
# Compactness
#* Tychonoff Theorem
#Countability and Separation Axioms
#* Countable basis
#* Countable dense subsets
#* Normal spaces
#* Urysohn Lemma
#* Tietze Extension Theorem
# Metrization
#* Urysohn Metrization Theorem
# Complete Metric Spaces
</div>
=== Additional topics ===
Paracompactness, the Nagata-Smirnov Metrization Theorem, Ascoli's Theorem, Baire Spaces and dimension theory as time allows.
=== Courses for which this course is prerequisite ===
Math 554

[[Category:Courses|553]]

Math 553: Foundations of Topology 1

2010-05-26T19:48:22Z

Dgw4:

== Catalog Information ==

=== Title ===
Foundations of Topology 1.

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

=== Offered ===
F
=== Prerequisite ===
[[Math 451]] or instructor's consent.
=== Description ===
Naive set theory, topological spaces, product spaces, subspaces, continuous functions, connectedness, compactness, countability, separation axioms, metrization, complete metric spaces, function spaces, and Baire spaces.
== Desired Learning Outcomes ==
Students should gain a familiarity with the general topology that is used throughout mathematics.
=== Minimal learning outcomes ===
<div style="-moz-column-count:2; column-count:2;">
# Set Theory
#* Finite, countable, and uncountable sets
#* Well-ordered sets
# Topological Spaces
#* Basis for a topology
#* Product topology
#* Metric topology
# Continuous Functions
# Connectedness
# Compactness
#* Tychonoff Theorem
#Countability and Separation Axioms
#* Countable basis
#* Countable dense subsets
#* Normal spaces
#* Urysohn Lemma
#* Tietze Extension Theorem
# Metrization
#* Urysohn Metrization Theorem
# Complete Metric Spaces
</div>
=== Additional topics ===
Paracompactness, the Nagata-Smirnov Metrization Theorem, Ascoli's Theorem, Baire Spaces and dimension theory as time allows.
=== Courses for which this course is prerequisite ===
Math 554

[[Category:Courses|553]]

Math 553: Foundations of Topology 1

2010-05-26T19:47:13Z

Dgw4:

== Catalog Information ==

=== Title ===
Foundations of Topology 1.

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

=== Offered ===
F
=== Prerequisite ===
[[Math 451]] or instructor's consent.

=== Description ===
Naive set theory, topological spaces, product spaces, subspaces, continuous functions, connectedness, compactness, countability, separation axioms, metrization, complete metric spaces, function spaces, and Baire spaces.

== Desired Learning Outcomes ==

Students should gain a familiarity with the general topology that is used throughout mathematics.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">
# Set Theory
#* Finite, countable, and uncountable sets
#* Well-ordered sets
# Topological Spaces
#* Basis for a topology
#* Product topology
#* Metric topology
# Continuous Functions
# Connectedness
# Compactness
#* Tychonoff Theorem
#Countability and Separation Axioms
#* Countable basis
#* Countable dense subsets
#* Normal spaces
#* Urysohn Lemma
#* Tietze Extension Theorem
# Metrization
#* Urysohn Metrization Theorem
# Complete Metric Spaces

</div>

=== Additional topics ===
Paracompactness, the Nagata-Smirnov Metrization Theorem, Ascoli's Theorem, Baire Spaces and dimension theory as time allows.
=== Courses for which this course is prerequisite ===
Math 554

[[Category:Courses|553]]

Math 553: Foundations of Topology 1

2010-05-26T19:46:09Z

Dgw4:

== Catalog Information ==

=== Title ===
Foundations of Topology 1.

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

=== Offered ===
F

=== Prerequisite ===
[[Math 451]] or instructor's consent.

=== Description ===
Naive set theory, topological spaces, product spaces, subspaces, continuous functions, connectedness, compactness, countability, separation axioms, metrization, complete metric spaces, function spaces, and Baire spaces.

== Desired Learning Outcomes ==

Students should gain a familiarity with the general topology that is used throughout mathematics.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">
# Set Theory
#* Finite, countable, and uncountable sets
#* Well-ordered sets
# Topological Spaces
#* Basis for a topology
#* Product topology
#* Metric topology
# Continuous Functions
# Connectedness
# Compactness
#* Tychonoff Theorem
#Countability and Separation Axioms
#* Countable basis
#* Countable dense subsets
#* Normal spaces
#* Urysohn Lemma
#* Tietze Extension Theorem
# Metrization
#* Urysohn Metrization Theorem
# Complete Metric Spaces

</div>

=== Additional topics ===
Paracompactness, the Nagata-Smirnov Metrization Theorem, Ascoli's Theorem, Baire Spaces and dimension theory as time allows.
=== Courses for which this course is prerequisite ===
Math 554

[[Category:Courses|553]]

Math 553: Foundations of Topology 1

2010-05-26T19:45:38Z

Dgw4:

== Catalog Information ==

=== Title ===
Foundations of Topology 1.

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

=== Offered ===
F

=== Prerequisite ===
[[Math 451]] or instructor's consent.

=== Description ===
Naive set theory, topological spaces, product spaces, subspaces, continuous functions, connectedness, compactness, countability, separation axioms, metrization, complete metric spaces, function spaces, and Baire spaces.

== Desired Learning Outcomes ==

Students should gain a familiarity with the general topology that is used throughout mathematics.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">
# Set Theory
#* Finite, countable, and uncountable sets
#* Well-ordered sets
# Topological Spaces
#* Basis for a topology
#* Product topology
#* Metric topology
# Continuous Functions
# Connectedness
# Compactness
#* Tychonoff Theorem
#Countability and Separation Axioms
#* Countable basis
#* Countable dense subsets
#* Normal spaces
#* Urysohn Lemma
#* Tietze Extension Theorem
# Metrization
#* Urysohn Metrization Theorem
# Complete Matric Spaces

</div>

=== Additional topics ===
Paracompactness, the Nagata-Smirnov Metrization Theorem, Ascoli's Theorem, Baire Spaces and dimension theory as time allows.
=== Courses for which this course is prerequisite ===
Math 554

[[Category:Courses|553]]

Math 111: Trigonometry

2010-05-20T22:02:25Z

Dgw4:

== Catalog Information ==

=== Title ===
Trigonometry.

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

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

=== Prerequisite ===
[[Math 110]] or equivalent.

=== Description ===
Circular functions, triangle relationships, identities, inverse trig functions, trigonometric equations, complex numbers, DeMoivre's theorem.

== Desired Learning Outcomes ==
Students should gain familiarity and proficiency with the basic theorems of trigonometry.
=== Prerequisites ===

=== Minimal learning outcomes ===

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

</div>
====1. Trigonometric Functions====

Include angles and their measure, the six trigonometric functions via the unit circle, properties of trigonometric functions (including domain, range, period, fundamental identities, etc.), and graphs of trigonometric functions.

====2. Analytic Trigonometry====

Include inverse trigonometric functions, trigonometric identities (including sum and difference formulas,, double-angle and half-angle formulas), and solving trigonometric equations.

====3. Applications of Trigonometric Functions====

Include the Law of Sines, the Law of Cosines, and finding the area of a triangle (including Heron's Formula).

====4. Polar Coordinates====

Include polar coordinates, graphs in polar coordinates, the complex plane, and De Moivre's Theorem.

=== Additional topics ===

Vectors.

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

[[Math 112]]
[[Category:Courses|111]]

Math 111: Trigonometry

2010-05-20T22:01:36Z

Dgw4: This is a major edit.

Math 111: Trigonometry

2010-05-20T22:00:24Z

Dgw4:

Math 362: Survey of Geometry

2010-05-20T21:42:03Z

Dgw4:

== Catalog Information ==

=== Title ===
Survey of Geometry.

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

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

=== Prerequisite ===
[[Math 290]].

=== Description ===
This course studies the foundations of geometry going back more than two thousand years to Euclid and the ancient Greeks. This course is especially aimed at understanding the importance of Euclid’s parallel postulate and the alternative non-Euclidean geometries that arise from alternative axioms. This course places an emphasis on logical thinking and clear mathematical writing. Geometry software such as Geometer’s Sketchpad should be used throughout the course when appropriate.

== Desired Learning Outcomes ==
Students should gain familiarity with axioms of geometry, both Euclidean and non-Euclidean. Students should be able to prove the major theorems of geometry based on the axioms.
=== Prerequisites ===
A knowledge of calculus and a maturity developed in mathematical communication.

=== 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 many of the theorems.

====1. Euclid’s Elements====

a. Understand the historical importance of Euclid’s Elements.

b. Understand and interpret Euclid’s definitions, axioms, and common notions.

c. Identify the logical gaps in Euclid’s proofs.

====2. Axiomatic Systems and Incidence Geometry====

a. Give examples of axiomatic systems and understand the axioms for Incidence Geometry.

b. Give examples of systems that satisfy the axioms of Incidence Geometry.

c. State theorems about Incidence Geometry.

d. Write direct and indirect proofs of theorems of Incidence Geometry.

====3. Set Theory and Real Numbers====

a. Understand basic facts from set theory including sets, elements, intersections, unions, and distinguish between “element of” and “subset of.”

b. Write correct mathematical statements using the vocabulary of set theory.

c. Understand the properties of the real numbers including trichotomy, density of rational numbers, the Archimedean property, and the least upper bound property.

====4. Axioms for Plane Geometry====

a. Identify undefined terms for neutral geometry.

b. Understand existence and incidence postulates and vocabulary for neutral geometry in terms of basic facts from set theory.

c. Give examples of different metrics to show an understanding of the Ruler Postulate and coordinate functions on lines.

d. Understand the need for and use the Plane Separation Postulate, and use it to define the interior of polygon.

e. Define angles, and angle measure using the Protractor Postulate.

f. Define and prove theorems about betweenness for points and rays.

g. Prove theorems about vertical angles, linear pairs, and perpendicular bisectors.

h. Define congruence of triangles and the need for the side-angle-side Postulate.

i. Understand the Euclidean, elliptic, and hyperbolic parallel postulates and examples of geometries that satisfy each of the parallel postulates.

====5. Theorems in Neutral Geometry====

a. Prove theorems about isosceles triangles and perpendicular lines.

b. Prove and understand the Exterior Angle Theorem.

c. Prove and understand congruence theorems for triangles.

d. Prove and understand triangle inequality theorems relating sides and angles (Scalene Inequality, Triangle Inequality, Hinge Theorem).

e. Prove and understand theorems about parallel lines cut by a transversal.

f. Prove and understand the Saccheri-Legendre Theorem about the sum of the angles of a triangle.

g. Prove and understand theorems about quadrilaterals including Saccheri and Lambert quadrilaterals.

h. Be able list and prove that statements are equivalent to the parallel postulate.

====6. Basic Theorems of Euclidean Geometry====

a. Prove and understand the Euclidean theorems about parallel lines cut by a transversal.

b. Prove and understand the angle sum theorem for triangles.

c. Prove and understand theorems about quadrilaterals including squares, rectangles, parallelograms, and trapezoids.

d. Prove and understand theorems about the ratio of lengths of segments of transversals to three parallel lines.

e. Prove and understand theorems about similar triangles.

f. Prove and understand the Pythagorean Theorem.

g. Work with triangles including altitudes, medians, angle bisectors and perpendicular bisectors. Prove and understand the concurrency theorems and the Euler Line Theorem.

====7. Hyperbolic Geometry====

a. Prove and understand the angle sum theorem for triangles and define the angle defect of a triangle.

b. Prove and understand theorems about quadrilaterals including Saccheri and Lambert quadrilaterals.

c. Prove and understand theorems about parallel lines and transversals.

d. Prove and understand that triangles with congruent angles are congruent.

e. Describe and understand limiting parallel rays and asymptotically parallel lines.

f. Model the hyperbolic plane with the Poincaré disk or the upper half plane.

====8. Transformations====

a. Contrast the transformational perspective with the Euclidean perspective of Euclid.

b. Define an isometry and prove that isometries for a geometry from a group.

c. Show that translations, rotations, and reflections are isometries.

d. Prove that the group of isometries for the plane is generated by reflections about a line.

e. Develop plane geometry by replacing the side-angle-side postulate by the reflection postulate and defining congruence in terms of isometries.

====9. Van Hiele Levels====

a. Be familiar with the van Hiele Model of the development of geometric thought.

=== Additional topics ===
These are at the instructor's discretion as time allows; paper models of the hyperbolic plane and area in Euclidean and hyperbolic geometry would be useful.

=== Courses for which this course is prerequisite ===
[[Mathed 562]]

[[Category:Courses|362]]

Math 362: Survey of Geometry

2010-05-20T21:35:53Z

Dgw4: This is a major edit.

== Catalog Information ==

=== Title ===
Survey of Geometry.

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

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

=== Prerequisite ===
[[Math 290]].

=== Description ===
This course studies the foundations of geometry going back more than two thousand years to Euclid and the ancient Greeks. This course is especially aimed at understanding the importance of Euclid’s parallel postulate and the alternative non-Euclidean geometries that arise from alternative axioms. This course places an emphasis on logical thinking and clear mathematical writing. Geometry software such as Geometer’s Sketchpad should be used throughout the course when appropriate.

== Desired Learning Outcomes ==
Students should gain familiarity with axioms of geometry, both Euclidean and non-Euclidean. Students should be able to prove the major theorems of geometry based on the axioms.
=== Prerequisites ===
A knowledge of calculus and a maturity developed in mathematical communication.

=== 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 many of the theorems.

1. Euclid’s Elements

a. Understand the historical importance of Euclid’s Elements.

b. Understand and interpret Euclid’s definitions, axioms, and common notions.

c. Identify the logical gaps in Euclid’s proofs.

2. Axiomatic Systems and Incidence Geometry

a. Give examples of axiomatic systems and understand the axioms for Incidence Geometry.

b. Give examples of systems that satisfy the axioms of Incidence Geometry.

c. State theorems about Incidence Geometry.

d. Write direct and indirect proofs of theorems of Incidence Geometry.

3. Set Theory and Real Numbers

a. Understand basic facts from set theory including sets, elements, intersections, unions, and distinguish between “element of” and “subset of.”

b. Write correct mathematical statements using the vocabulary of set theory.

c. Understand the properties of the real numbers including trichotomy, density of rational numbers, the Archimedean property, and the least upper bound property.

4. Axioms for Plane Geometry

a. Identify undefined terms for neutral geometry.

b. Understand existence and incidence postulates and vocabulary for neutral geometry in terms of basic facts from set theory.

c. Give examples of different metrics to show an understanding of the Ruler Postulate and coordinate functions on lines.

d. Understand the need for and use the Plane Separation Postulate, and use it to define the interior of polygon.

e. Define angles, and angle measure using the Protractor Postulate.

f. Define and prove theorems about betweenness for points and rays.

g. Prove theorems about vertical angles, linear pairs, and perpendicular bisectors.

h. Define congruence of triangles and the need for the side-angle-side Postulate.

i. Understand the Euclidean, elliptic, and hyperbolic parallel postulates and examples of geometries that satisfy each of the parallel postulates.

5. Theorems in Neutral Geometry

a. Prove theorems about isosceles triangles and perpendicular lines.

b. Prove and understand the Exterior Angle Theorem.

c. Prove and understand congruence theorems for triangles.

d. Prove and understand triangle inequality theorems relating sides and angles (Scalene Inequality, Triangle Inequality, Hinge Theorem).

e. Prove and understand theorems about parallel lines cut by a transversal.

f. Prove and understand the Saccheri-Legendre Theorem about the sum of the angles of a triangle.

g. Prove and understand theorems about quadrilaterals including Saccheri and Lambert quadrilaterals.

h. Be able list and prove that statements are equivalent to the parallel postulate.

6. Basic Theorems of Euclidean Geometry

a. Prove and understand the Euclidean theorems about parallel lines cut by a transversal.

b. Prove and understand the angle sum theorem for triangles.

c. Prove and understand theorems about quadrilaterals including squares, rectangles, parallelograms, and trapezoids.

d. Prove and understand theorems about the ratio of lengths of segments of transversals to three parallel lines.

e. Prove and understand theorems about similar triangles.

f. Prove and understand the Pythagorean Theorem.

g. Work with triangles including altitudes, medians, angle bisectors and perpendicular bisectors. Prove and understand the concurrency theorems and the Euler Line Theorem.

7. Hyperbolic Geometry

a. Prove and understand the angle sum theorem for triangles and define the angle defect of a triangle.

b. Prove and understand theorems about quadrilaterals including Saccheri and Lambert quadrilaterals.

c. Prove and understand theorems about parallel lines and transversals.

d. Prove and understand that triangles with congruent angles are congruent.

e. Describe and understand limiting parallel rays and asymptotically parallel lines.

f. Model the hyperbolic plane with the Poincaré disk or the upper half plane.

8. Transformations

a. Contrast the transformational perspective with the Euclidean perspective of Euclid.

b. Define an isometry and prove that isometries for a geometry from a group.

c. Show that translations, rotations, and reflections are isometries.

d. Prove that the group of isometries for the plane is generated by reflections about a line.

e. Develop plane geometry by replacing the side-angle-side postulate by the reflection postulate and defining congruence in terms of isometries.

9. Van Hiele Levels

a. Be familiar with the van Hiele Model of the development of geometric thought.

=== Additional topics ===
These are at the instructor's discretion as time allows; paper models of the hyperbolic plane and area in Euclidean and hyperbolic geometry would be useful.

=== Courses for which this course is prerequisite ===
[[Mathed 562]]

[[Category:Courses|362]]