Week 1
4, 6, 8 Aug 08
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Combinatorics: Basic counting, sets, multisets,
Suggested reading: pages 1--20 of [S].
Set Theory: Basic axioms. Suggested reading
pages 1--11 of [H]
Algebra: Matrices, row-echelon form.
Suggested reading: pages 1-35 of [A]; try the
exercises.
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Week 2
11, 13 Aug 08
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Combinatorics: Permutations, cycle structure, standard
form, the Stirling numbers of the first kind.
Homework (due 25 Aug 08): [PDF].
Set Theory: Unions, intersections, ordered pairs.
Suggested reading: pages 12-25 of [H].
Homework (due 3 Sep 08):
attempt the exercises on page 10,
page 16, page 25 and the
`non-trivial' exercise on
page 23 of [H].
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Week 3
18, 20, 22 Aug 08
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Combinatorics: Cycle structure, inversions.
Set Theory: Cartesian products, relations, functions.
Suggested reading: pages 16--33 of [H].
Algebra: Determinants. Homework (due 12 Sep 08):
[PDF].
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Week 4
25, 27, 29 Aug 08
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Combinatorics: permutations and trees,
counting functions.
Set Theory: families, inverses, compositions, natural numbers.
Suggested reading: pages 34--45 of [H].
Algebra: Permutation matrices, Cramer's rule, Groups.
Suggested reading Chapter 2 of [A].
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Week 5
1, 5 Sep 08
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Combinatorics: Inclusion-exclusion.
Set Theory: Holiday
Suggested reading: pages 34--45 of [H].
Algebra: Subgroups, homomorphisms, isomorphism,
kernel, image.
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Week 6
8, 10, 12 Sep 08
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Combinatorics: Inclusion-exclusion formula,
graph reconstruction, derangements.
Homework (due 25 Sep 08):
[PDF].
Set Theory: Natural Numbers, Peano's axioms.
Suggested reading: pages 46--49 of [H].
Algebra: Modular arithmetic,
cosets, quotient groups, normal subgroups.
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Week 7
17, 19 Sep 08
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Combinatorics: Mobius inversion (Amitava Bhattacharya
was the guest lecturer while Jaikumar was away).
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Week 8
22, 24, 26 Sep 08
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Set theory: Recursion theorem. Suggested reading:
Homework (due 22 Oct 08): Bottom of page 49,
bottom of 53 of [H]. Suggested reading:
pages 51--53 of [H].
Algebra: Products of groups,
Quotient groups, first isomorphism theorem.
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Week 9
29 Sep 08 1, 3 Oct 08
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Combinatorics: Hall's theorem.
Homework (due 3 November 08):
[PDF]
Set Theory: Arithmetic, order in natural
numbers. Suggested reading: 54--57 of
[H].
Algebra: Chinese remainder theorem. Homework
(assigned 1 Nov 08, due 14 Nov 08):
problem 11 on page 70, problems 12 and 14 on
page 71, problem 8 on page 75, problem 6 page
76, problem 4 on page 77 (all from [A]).
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Week 10
6, 8, 10 Oct 08
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Combinatorics: Homework, Linear codes.
Set Theory: Axiom of choice.
Suggested reading: 59--61 of [H].
Algebra: Vector Spaces. Suggested
reading: pages 78--103 of [A].
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Week 11
13 Sep 08
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Combinatorics: Group actions, orbits, the orbit
counting lemma.
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Week 12
20, 22 Oct 08
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Combinatorics: Counting objects under symmetry, cycle
index polynomial.
Set theory: Zorn's lemma. Suggested reading pages
62--65 of [H].
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Week 13
3, 5, 7 Nov 08
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Combinatorics:
Set theory:
Algebra
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