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21-235/236: Math Studies

Course Description


Instructor and TA's

Course Schedule and Homework

Grading Information

Other Course Policies

Frequently Asked Questions

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Schedule and Homework

For each week there will be a link to a page with a reading assignment and a homework assignment.

This schedule is tentative. It will get more accurate as the semester progresses. No week's topics should be taken as final until the homework is linked.

Week #1: 
Aug 30 - Sep. 3
Metric Spaces
Compact Sets
Finite, countable infinite and uncountable sets.
NxN and Q are countable, binary operations.
Groups, Examples, identity and inverse elements.
Week #2: 
Sep. 6 - 10
Compact Sets
Connected Sets
Rings, Examples
Fields, Examples, the Real numbers
Administrative Note: Monday 6 September is Labor Day. Class will not meet.
Week #3: 
Sep 13 - 17
Problem Session 1
Convergent Sequences
Cauchy Sequences
Homomorphisms of groups, normal subgroups.
Week #4: 
Sep. 20 - 24
Problem Session 2
Series of Nonnegative terms
The Root and Ratio Tests
Power Series
Summation by Parts
Quotient groups, the First Isomorphism Theorem.
Week #5: 
Sep. 27 - Oct. 1
Problem Session 3
Absolute Convergence
Addition and Multiplication of Series
Limits of Functions
Continuous Functions
Week #6: 
Oct. 4 - 8
Problem Session 4
Continuity and Compactness
Continuity and Connectedness
Monotonic Functions
Infinite Limits and Linits at Infinity
Connected, Path Connected and Simply Connected.
Homotopies and the Fundamental Group.
Week #7: 
Oct 11 - 15
Problem Session 5
The Derivative of a Real Function
Mean Value Theorems
The Continuity of Derivatives
L'Hospital's Rule
Derivatives of Higher Order
Week #8: 
Oct 18 - 22
Problem Session 6
Taylor's Theorem
Differentiation of Vector-valued Functions
Sequences and Series of Functions: Discussion of Main Problem
Administrative note: Friday 22 October is the Mid-Semester Break. Class will not meet.
Week #9: 
Oct 25 - 29
Uniform Convergence
Uniform Convergence and Contuinity
Uniform Convergence and Differentiation
Week #10: 
Nov 1 - 5
Problem Session 8
Equicontinuous Families of Functions
Definition and Existence of the (Riemann) Integral
Properties of the Integral
Week #11: 
Nov 8 - 12
Integration and Differentiation
Integration of Vector Valued Functions
Rectifiable Curves
Week #12: 
Nov 15 - 19
Problem Session 10
Set Functions
Construction of the Lebesgue Measure
Week #13: 
Nov 22 - 26
Measure Spaces
Week #14: 
Nov 29 - Dec 3
Measurable Functions
Simple Functions
Week #15: 
Dec 6 - 10
Comparison with the Riemann Integral
Integration of Complex Functions
Functions of Class L2

Administrative note: Friday 10 December is the last day of class.
Final Exam
Dec 10 - 21
The Final Exam will be a take some exam.