Fall 2014 Section 0978 and Spring 2015 Section 1239
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Current Announcements
- Wed Jan 28: We continue with Bringmann and Ono’s paper. Later we will relate it to Zagier’s paper Ramanujan’s mock theta functions and their applications [d’après Zwegers and Bringmann-Ono].
- Fri Jan 16: We continue the Seminar by first working through
Bringmann and Ono’s paper Dyson’s ranks and Maass forms
Annals of Mathematics, 171 (2010), pp. 419-449
- Fri Oct 10: Proved Mittag-Leffler Theorem. Continued with the proof of Zwegers’s Lemma 3.3.
- Fri Oct 3: Showed equivalence of Watson and Zwegers version of Theorem. Proof identity at the end of Watson’s paper. Did Lemma from Atkin and Swinnerton-Dyer’s paper to prepare for proof of Watson’s quintuple product identity which is connected with Zwegers’s functions g0,g1 and g2.
- Wed Oct 1: Finished proof of Watson’s transformation formula for f(q). Started work on Zwegers’ paper on Real Analytic Modular Forms.
- Mon Sep 22: We proved another identity for the 3rd order functions phi(-q) and psi(-q) again using Chan’s identity.
- Fri Sep 19: We used a case of S.H. Chan’s identity to prove an identity for the 3rd functions phi(q) and f(q).
- Fri Sep 12: We discussed some of Homework 2 problems including an overview of the Delta function as a modular form of weight 12. We showed that the proof of the even k case will follow from an identity of Zudilin. This involves Watson’s q-Whipple transformation which we started.
- Wed Sep 10: Bilateral form of the rank function. Another identity for the crank function using Bailey’s 6phi5.
- Mon Sep 8: Combinatorics of the crank function and introduction to basic hypergeometric series. Stated Bailey’s 2psi2 transformation and started proof of an identity of Ramanujan. Collected Homework 1.
- Fri Sep 5: Started work on Zudilin’s proof of the Folsom-Ono-Rhoades result for f(q). Discussed the rank and crank functions.
- Fri Aug 29: Applied Poisson Summation Formula to prove Ramanujan’s result (A) for the partition function. Homework 1 handed out. Defined the Dedekind eta-function.
- Wed Aug 27: We proved Jacobi’s Triple Product Identity and derived the Poisson Summation Formula.
- Mon Aug 25: First day of classes. Syllabus and
Ramanujan’s Last Letter handed out. We discussed Ramanujan’s functions (A) and (B). We proved Euler’s Theorem for the generating function of the partition function. We introduced concept of Durfee square to explain that (B)
is the generating function of partitions into parts where difference between parts is at least 2. See Q-SERIES NOTES (below) for more detail on some of these topics.
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LINKS:
PERSONALITIES:
RAMANUJAN
NOTES FROM MY Q-SERIES COURSE
- Chapter 1 – Partitions and Infinite Products
- Chapter 2 – Infinite Series Generating Functions and Hypergeometric Series
- Chapter 3 – Ramanujan’s Partition Congruences
- Chapter 4 – Restricted Partitions and Permutations
- Chapter 5 – Identities of the Rogers-Ramanujan Type
- Chapter 6 – Sums of Squares
Old News
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Last update made October 1, 2014, 8:45pm. |