Date
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Lecture Topics / Reading / Handouts
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Thurs., Jan. 12
|
Definition
of partitions (Andrews 1.1); Motivating examples: finitely
generated abelian groups, and conjugacy classes in S_n
|
Tues., Jan. 17
|
Partition
function p(n); Euler's Odd-Distinct Theorem (Andrews 1.2;
page 10 of H. Wilf's lecture
notes); Ferrers Diagrams (Andrews 1.3)
|
Thurs., Jan. 19
|
Partition
statistics, including largest part and length; Conjugate of
a partition (Andrews 1.3); Sylvester's proof of Euler's
Odd-Distinct Theorem (p. 12 of I. Pak's lecture)
|
Tues., Jan. 24
|
Sylvester's
proof of Euler's Theorem (cont.); Introduction to generating
functions; Van der Monde's Identity (Wikipedia);
Partition generating functions; Generating function proof of
Euler's Theorem (Andrews 1.2)
|
Thurs., Jan. 26
|
Congruences
modulo 2 for partition generating functions; Two-parameter
generating functions; Pentagonal Number Theorem and
Franklin's proof via "almost"-involution (Andrews 1.3)
|
Tues., Jan. 31
|
Consequences
of Pentagonal Number Theorem, including computing p(n)
recursively; Introduction to q-factorials (Andrews 2.1)
|
Thurs., Feb. 2
|
Properties
of q-factorials; Partitions with at most m parts; Cauchy's
Theorem (Andrews 2.1), proof by series expansion and
q-difference equations; Binomial theorem as corollary
|
Tues., Feb. 7
|
Euler's
identities, as corollaries of Cauchy's Theorem (Andrews
2.1), and as combinatorial partition identities; Jacobi
Triple Product (Andrews 2.2), and Pentagonal Number Theorem
as corollary
|
Thurs., Feb. 9
|
Proof of
Jacobi Triple Product (Andrews 2.2); Heine's transformation
and summation formulas; Basic hypergeometric series (Digital
Library of Mathematical Functions at NIST); Examples of
theta functions
|
Tues, Feb. 14
|
Partition
identities from basic hypergeometric series (Andrews 2.3);
Combinatorial proofs, Durfee squares, counting largest part
and number of parts simultaneously
|
Thurs., Feb. 16
|
Classical
hypergeometric functions (DLMF
at NIST); Examples and Identities, Euler's transformation;
Basic hypergeometric series
|
Tues., Feb. 21
|
Finite/polynomial
q-series; Motivating example: inversions in permutations;
q-binomial coefficients
|
Thurs., Feb. 23
|
Gaussian
polynomials and restricted partitions (Andrews 3.2 and 3.3);
q-Binomial Theorem, analytic and combinatorial proofs
|
Thurs., Mar. 2
|
q-Chu-Vandermonde
identity, analytic proof (Andrews 3.3); q-analog of Hockey
Stick Theorem (Wikipedia);
Permutations of multisets, the inversion statistic, and
q-multinomial coefficients (Andrews 3.4)
|
Tues., Mar. 7
|
Proof of
generating function for inversion statistic (Andrews 3.4);
Greater index statistic; Reciprocal and unimodal polynomials
(Andrews 3.5, R. Stanley's survey)
|
Thurs., Mar. 9
|
Basic
results on reciprocal and unimodal polynomials (D.
Zeilberger's paper);
Combinatorics of inversions, including combinatorial proof
of Andrews' Theorem 3.11
|
Tues., Mar. 14
|
Discussion
of O'Hara's q-series proof of unimodality for Gaussian
polynomials (K. O'Hara's paper);
Szekeres' proof of unimodality of restricted partitions (G.
Szekeres' paper);
Relations between restricted partitions and compositions
(Andrews 4.2)
|
Thurs., Mar. 16
|
Proof of
Erdos-Lehner Theorem (Andrews 4.2); Introduction to
Ramanujan congruences (Andrews Chapter 10, S. Ramanujan's paper)
|
Tues., Mar. 21
|
Proof of
Ramanujan congruence modulo 5 (Andrews Examples 10.7 -
10.13); Discussion of partition congruences, including work
of Atkin, Newman, Ono (K. Ono's survey);
Ramanujan's congruences modulo powers of 5 (Andrews 10.3);
Introduction to product expansions of formal power series
|
Thurs., Mar. 23
|
Product
expansions of formal power series; Sum-Product Identities
and partitions with gap conditions, including Euler,
Rogers-Ramanujan, and Schur (Andrews Chapter 7; I. Pak's survey)
|
Tues., Mar. 28
|
Proof of
Schur's second partition Theorem using recurrences and
q-difference equations (Andrews Examples 7.1 - 7.4; Andrews'
paper)
|
Thurs., Mar. 30
|
Appell's
Comparison Theorem; Heine's transformation and Schur's
theorem; Proof of Lehmer/Alder's result on nonexistence of
gap-product identities, d=3 case (Lehmer's paper);
Discussion of Alder-Andrews Conjecture
|
Tues., Apr. 4
|
k-modular
partition diagrams (Andrews Examples 1.6 - 1.7); Conjugation
of 2-modular diagrams; Bressoud's bijective proof of Schur's
Theorem (D. Bressoud's paper, I. Pak's survey);
Double sum representation for Schur partitions
(Andrews-Bringmann-Mahlburg's paper)
|
Thurs., Apr. 6
|
Ehrenpreis
problem (K. Kadell's paper);
Andrews and Baxter's "Motivated proof" of Rogers-Ramanujan
(Andrews-Baxter's paper)
|
Tues., Apr. 18
|
Proof of
Rogers-Ramanujan identities using q-difference equations
(Rogers and Ramanujan's paper in archived journal
volume); Ramanujan congruences and Dyson's rank
statistic (A.O.L. Atkin and P. Swinnerton-Dyer's paper)
|
Thurs., Apr. 20
|
Guest
Speaker: Ken Ono, Partitions with restricted smallest parts
and arithmetic densities (K. Ono, R. Schneider, and I.
Wagner's preprint);
Garvan and Andrews-Garvan crank statistic and congruences
(K. Mahlburg's paper)
|
Tues., Apr. 22
|
Circle
Method and Hardy-Ramanujan's asymptotic formula for the
partition function (G. Hardy and S. Ramanujan's paper);
Hardy-Ramanujan Tauberian Theorem; Asymptotic behavior of
log(P(q))
|
Thurs., Apr. 27
|
Wright's
Circle Method (K. Bringmann and K. Mahlburg's paper), definition of Major
and Minor Arc; Modular inversion formula for P(q); Proof of
asymptotic main term for p(n), and error term
|