Special Topics in MATH: Integer partitions and q-series. (MATH 58003)

2022 Fall
Faculty of Engineering and Natural Sciences
Mathematics(MATH)
3
10
Kağan Kurşungöz kursungoz@sabanciuniv.edu,
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English
Doctoral, Master
--
Formal lecture,Interactive lecture
Communicative
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CONTENT

Integer partitions; q-series, elementary identities (q-binomial theorem, Heine's transformation, Jacobi's triple product identity, Ramanujan's 1-psi-1 transformation) and corollaries; q-series as partition generating functions; Ramanujan's congruences for the partition function, Rogers- Ramanujan generalizations.

OBJECTIVE

* to understand the place of integer partitions among counting problems
* to understand various methods of proving a partition identity, and to be able to imitate those proofs to find similar identities
* to be able to construct and manipulate q-series using well-known transformation formulas
* to be able to recognize q-series as partition generating functions
* to be able to manipulate q-series in modular arithmetic, and prove partition congruences.
* to be able to use symbolic computation software and do large scale computational experiments for new discoveries.

ASSESSMENT METHODS and CRITERIA

  Percentage (%)
Presentation 40
Homework 60

RECOMENDED or REQUIRED READINGS

Textbook

Andrews, G.E., 1998. The theory of partitions (No. 2). Cambridge university press.

Berndt, B.C., 2006. Number theory in the spirit of Ramanujan (Vol. 34). American Mathematical Soc..

Readings

Andrews, G.E. and Eriksson, K., 2004. Integer partitions. Cambridge University Press.

Gasper, G., Rahman, M. and George, G., 2004. Basic hypergeometric series (Vol. 96). Cambridge university press.

Hirschhorn, M.D., 2017. The Power of q. In Developments in Mathematics (Vol. 49). Springer.