Frank Garvan
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On Dyson's Crank Conjecture and the Uniform Asymptotic
ON DYSON’S CRANK CONJECTURE AND THE UNIFORM ASYMPTOTIC BEHAVIOR OF CERTAIN INVERSE THETA FUNCTIONS KATHRIN BRINGMANN AND JEHANNE DOUSSE Abstract. In this paper we prove a longstanding conjecture by Freeman Dyson concerning the limiting shape of the crank generating function. We fit this function in a more general family of inverse theta functions which play a key role in physics. 1. Introduction and statement of results Dyson’s crank was introduced to explain Ramanujan’s famous partition congru- ences with modulus 5, 7, and 11. Denoting for n 2 N by p(n) the number of integer partitions of n, Ramanujan [22] proved that for n ≥ 0 p (5n + 4) ≡ 0 (mod 5); p (7n + 5) ≡ 0 (mod 7); p (11n + 6) ≡ 0 (mod 11): A key ingredient of his proof is the modularity of the partition generating function 1 1 X 1 q 24 P (q) := p(n)qn = = ; (q; q) η(τ) n=0 1 Qj−1 ` 2πiτ where for j 2 N0 [ f1g we set (a)j = (a; q)j := `=0(1 − aq ), q := e , and 1 1 1 24 Q n η(τ) := q n=1(1 − q ) is Dedekind’s η-function, a modular form of weight 2 . Ramanujan’s proof however gives little combinatorial insight into why the above congruences hold. In order to provide such an explanation, Dyson [8] famously intro- duced the rank of a partition, which is defined as its largest part minus the number of its parts. He conjectured that the partitions of 5n + 4 (resp. 7n + 5) form 5 (resp. -
Nominations for President
Nominations for President combinatorics which was then starting to become fashion- Nomination of able; and some number theory, which has never gone out George Andrews of fashion although the parts of it George dealt with were not fashionable then. George spent a year on leave at MIT. Richard Askey At Rota’s suggestion, he edited the “Collected Papers” of P. A. MacMahon. He also wrote a book, “The Theory of Who is George Andrews? According to Freeman Dyson, Partitions”, for the Encyclopedia of Mathematics and Its George Andrews is the chief gardener in Ramanujan’s Gar- Applications series which Rota edited. den. This is true, but is only part of who George Andrews George came to the University of Wisconsin-Madison for is. He is a number theorist with an honorary doctorate in a year, 1976-77. In the spring he went to Europe for two physics. He is a long-time user of computers in his own meetings in France. Since he was not teaching and airfare research, who has written about the harm technology can abroad was less expensive if you spent 21 to 45 days, he do in mathematics education. He is primarily a problem also went to Cambridge to look for old work in the Wren solver, yet one paper with Rodney Baxter and Peter For- Library of Trinity College. This changed his life. rester has had a major impact in mathematical physics What George found was a bit over 100 pages of math- with currently 490 citations in the Web of Science. ematical claims in the distinctive handwriting of Srinivasa George Andrews received his Ph.D. -
Freeman Dyson, the Mathematician∗
GENERAL ARTICLE Freeman Dyson, the Mathematician∗ B Sury A mathematician of the highest rank with ideas ever-brimming with a full tank. He designed special lenses to ‘see’ Ramanujan’s congruences partitioned through the yet-invisible crank! The one person who comes closest to the legacy of Hermann B Sury is with the Indian Weyl was Freeman Dyson, who contributed enormously to Statistical Institute in Bengaluru. He is the National both Physics and Mathematics. His books and talks are tes- Co-ordinator for the taments to his prolificity in writing widely on the world at Mathematics Olympiad large and on science and mathematics in particular. To name Programme in India, and also a few of his books, he has (talked and) written on ‘Bombs and the President of the Indian Mathematical Society. Poetry’, ‘Imagined Worlds’, ‘Origins of Life’ and ‘Birds and Frogs’. Remarkably, Dyson contributed handsomely to what is termed ‘pure mathematics’. One would expect a physicist- mathematician to interest himself mainly in problems of an ‘applied’ nature. Here, we take a necessarily brief peek into some of his ‘purely mathematical’ work. The suggested read- ing at the end can be referred to for more details. 1. Statistical Mechanics – Dyson’s Conjecture In an enormously influential paper, ‘Statistical theory of energy Keywords levels of complex systems’, I. J. Math. Phys., 3:140–156, 1962, Congruences for partitions, rank and crank, random matrix the- Dyson proposed a new type of ensemble transforming the whole ory, Riemann zeta function, Dyson subject into purely group-theoretic language. In the paper, Dyson transform, quasi-crystals. -
What Is the Crank of a Partition? Daniel Glasscock, July 2014
What is the crank of a partition? Daniel Glasscock, July 2014 These notes complement a talk given for the What is ... ? seminar at the Ohio State University. Introduction The crank of an integer partition is an integer statistic which yields a combinatorial explanation of Ramanujan's three famous congruences for the partition function. The existence of such a statistic was conjectured by Freeman Dyson in 1944 and realized 44 years later by George Andrews and Frank Gravan. Since then, groundbreaking work by Ken Ono, Karl Mahlburg, and others tells us that not only are there infinitely many congruences for the partition function, but infinitely many of them are explained by the crank. In this note, we will introduce the partition function, define the crank of a partition, and show how the crank underlies congruences of the partition function. Along the way, we will recount the fascinating history behind the crank and its older brother, the rank. This exposition ends with the recent work of Karl Mahlburg on crank function congruences. Andrews and Ono [3] have a good quick survey from which to begin. The partition function The partition function p(n) counts the number of distinct partitions of a positive integer n, where a partition of n is a way of writing n as a sum of positive integers. For example, p(4) = 5 since 4 may be written as the sum of positive integers in 5 essentially different ways: 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1. The numbers comprising a partition are called parts of the partition, thus 2 + 1 + 1 has three parts, two of which are odd, and is not composed of distinct parts (since the 1 repeats). -
Ramanujan, His Lost Notebook, Its Importance
RAMANUJAN, HIS LOST NOTEBOOK, ITS IMPORTANCE BRUCE C. BERNDT 1. Brief Biography of Ramanujan Throughout history most famous mathematicians were educated at renowned centers of learning and were taught by inspiring teachers, if not by distinguished research mathematicians. The one exception to this rule is Srinivasa Ramanujan, born on December 22, 1887 in Erode in the southern Indian State of Tamil Nadu. He lived most of his life in Kumbakonam, located to the east of Erode and about 250 kilometers south, southwest of Madras, or Chennai as it is presently named. At an early age, he won prizes for his mathematical ability, not mathematics books as one might surmise, but books of English poetry, reflecting British colonial rule at that time. At the age of about fifteen, he borrowed a copy of G. S. Carr's Synopsis of Pure and Applied Mathematics, which was the most influential book in Ramanujan's development. Carr was a tutor and compiled this compendium of approximately 4000{5000 results (with very few proofs) to facilitate his tutoring at Cambridge and London. One or two years later, Ramanujan entered the Government College of Kumbakonam, often called \the Cambridge of South India," because of its high academic standards. By this time, Ramanujan was consumed by mathematics and would not seriously study any other subject. Consequently he failed his examinations at the end of the first year and lost his scholarship. Because his family was poor, Ramanujan was forced to terminate his formal education. At about the time Ramanujan entered college, he began to record his mathematical discoveries in notebooks. -
Dyson's Ranks and Maass Forms
ANNALS OF MATHEMATICS Dyson’s ranks and Maass forms By Kathrin Bringmann and Ken Ono SECOND SERIES, VOL. 171, NO. 1 January, 2010 anmaah Annals of Mathematics, 171 (2010), 419–449 Dyson’s ranks and Maass forms By KATHRIN BRINGMANN and KEN ONO For Jean-Pierre Serre in celebration of his 80th birthday. Abstract Motivated by work of Ramanujan, Freeman Dyson defined the rank of an inte- ger partition to be its largest part minus its number of parts. If N.m; n/ denotes the number of partitions of n with rank m, then it turns out that n2 X1 X1 X1 q R.w q/ 1 N.m; n/wmqn 1 : I WD C D C Qn .1 .w w 1/qj q2j / n 1 m n 1 j 1 D D1 D D C C We show that if 1 is a root of unity, then R. q/ is essentially the holomorphic ¤ I part of a weight 1=2 weak Maass form on a subgroup of SL2.Z/. For integers 0 Ä r < t, we use this result to determine the modularity of the generating function for N.r; t n/, the number of partitions of n whose rank is congruent to r .mod t/. We I extend the modularity above to construct an infinite family of vector valued weight 1=2 forms for the full modular group SL2.Z/, a result which is of independent interest. 1. Introduction and statement of results The mock theta-functions give us tantalizing hints of a grand synthe- sis still to be discovered. -
Polyhedral Geometry, Supercranks, and Combinatorial Witnesses of Congruences for Partitions Into Three Parts
Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts Felix Breuer∗ Research Institute for Symbolic Computation (RISC) Johannes Kepler University, A-4040 Linz, Austria [email protected] Dennis Eichhorn Department of Mathematics University of California, Irvine Irvine, CA 92697-3875 [email protected] Brandt Kronholm∗ Research Institute for Symbolic Computation (RISC) Johannes Kepler University, A-4040 Linz, Austria [email protected] August 8, 2018 Abstract In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn allows us to define statistics called supercranks that combinatorially witness every instance of divisibility of p(n; 3) by any prime m ≡ −1 (mod 6), where p(n; 3) is the number of partitions of n into three parts. A rearrangement of lattice points allows us to demonstrate with explicit bijections how to divide these sets of partitions into m equinumerous classes. The behavior for primes m0 ≡ 1 (mod 6) is also discussed. 1 Introduction arXiv:1508.00397v1 [math.CO] 3 Aug 2015 Ramanujan [18] observed and proved the following congruences for the ordinary partition function: p(5n + 4) ≡ 0 (mod 5) p(7n + 5) ≡ 0 (mod 7) p(11n + 6) ≡ 0 (mod 11): In 1944, Freeman Dyson [9] called for direct proofs of Ramanujan's congruences that would give concrete demonstrations of how the sets of partitions of 5n+4, 7n+5, and 11n + 6 could be split into 5; 7; and 11 equinumerous classes, respectively. -
In Memoriam: Freeman Dyson (1923–2020) George E
In Memoriam: Freeman Dyson (1923–2020) George E. Andrews, Jürg Fröhlich, and Andrew V. Sills 1. A Brief Biography 1.1. Personal background. Freeman John Dyson was born in Crowthorne, Berkshire, in the United Kingdom, on December 15, 1923. His father was the musician and composer Sir George Dyson; his mother, Mildred Lucy, n´eeAtkey, was a lawyer who later became a social worker. Freeman had an older sister, Alice, who said that, as a boy, he was constantly calculating and was always surrounded by encyclopedias. According to his own testimony, Free- man became interested in mathematics and astronomy around age six. At the age of twelve, he won the first place in a scholar- ship examination to Winchester College, where his father was the Director of Music; one of the early manifestations of Freeman’s extraordinary talent. Dyson described his ed- ucation at Winchester as follows: the official curriculum at the College was more or less limited to imparting basic skills in languages and in mathematics; everything else was in the responsibility of the students. In the company of some of his fellow students, he thus tried to absorb what- ever he found interesting, wherever he could find it. That included, for example, basic Russian that he needed in or- der to be able to understand Vinogradov’s Introduction to Figure 1. the Theory of Numbers. George E. Andrews is Evan Pugh Professor of Mathematics at the Pennsylvania In 1941, Dyson won a scholarship to Trinity College in State University. His email address is [email protected]. Cambridge. He studied physics with Paul Dirac and Sir Jürg Fröhlich is a professor emeritus of theoretical physics at the Institute for The- Arthur Eddington and mathematics with G. -
Frank Garvan
Curriculum Vitae FRANK GARVAN Office Address: University of Florida Office Phone: +1{352{294{2305 Mathematics Department Email Address: fgarvan@ufl.edu PO Box 118105 Homepage: http://people.clas.ufl.edu/fgarvan Gainesville, FL 32611-8105 Birthdate: March 9, 1955 Last modified: January 5, 2021 EDUCATION/EMPLOYMENT 2000 { Professor of Mathematics, University of Florida 1994 { 1999 Associate Professor of Mathematics, University of Florida 1990 { 1993 Assistant Professor of Mathematics, University of Florida 1991 NSERC International fellow, Dalhousie University, Halifax 1988 { 1990 Postdoctoral Research Fellow, Macquarie University, Sydney, Australia 1987 { 1988 Postdoctoral Fellow, I.M.A., University of Minnesota, Minneapolis 1986 { 1987 Visiting Assistant Professor of Mathematics, University of Wisconsin, Madison 1985 { 1986 Mathematics Instructor, Pennsylvania State University, York Campus 1986 Ph.D. Pennsylvania State University, Mathematics (advisor: George Andrews) 1982 M.Sc. University of New South Wales, Kensington, Australia, Mathematics (advisor: Mike Hirschhorn) 1978 { 1980 High School Teacher, New South Wales, Australia 1977 Dip.Ed. University of New South Wales, Kensington, Australia, Education 1976 B.Sc. University of New South Wales, Kensington, Australia, Mathematics (with honours) RESEARCH INTERESTS Number Theory, Combinatorics, Special Functions, Symbolic Computation RESEARCH GRANTS 2014 { 2019 Simons Foundation Collaboration Grant (ref. 318714) ($35,000), Analytic, Combinatorial and Computational Study of Partitions 2009 { 2012 CoPI for NSA Grant (ref. H98230-09-1-0051) ($131,450), Some Problems in the Theory of Partitions 2006 { 2009 CoPI for NSA Grant (ref. H98230-07-1-0011) ($123,380), Some Problems in the Theory of Partitions 1998 { 2001 NSF grant (ref. DMS-9870052) ($77,895) Combinatorial, Differential & Modular Partition Problems 1992 { 1995 NSF grant (ref. -
The Pennsylvania State University the Graduate School STUDIES of PARTITION FUNCTIONS with CONDITIONS on PARTS and PARITY a Disse
The Pennsylvania State University The Graduate School STUDIES OF PARTITION FUNCTIONS WITH CONDITIONS ON PARTS AND PARITY A Dissertation in Mathematics by Donny Passary © 2019 Donny Passary Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2019 The dissertation of Donny Passary was reviewed and approved∗ by the following: George Andrews Evan Pugh Professor of Mathematics Dissertation Co-Advisor, Co-Chair of Committee Ae Ja Yee Associate Professor of Mathematics Dissertation Co-advisor, Co-Chair of Committee Wen-Ching W. Li Distinguished Professor of Mathematics James Sellers Associate Professor of Mathematics Donald Richards Professor of Statistics Alexei Novikov Professor of Mathematics Chair of Graduate Program ∗Signatures are on file in the Graduate School. ii Abstract This dissertation explores four topics in partition theory, with main themes on parts and parities conditions. The first topic studies many properties of the EO- partitions by Andrews, especially the results analogous to the work of Atkin and Swinnerton-Dyer. The second topic deals with the concept of Andrews’ separable integer partition classes, which yields alternate proofs for many partition identities including little Göllnitz identities. The third topic gives infinite families of congruences for partition functions arising from Ramanujan’s mock theta functions. Some other related identities analogue to Euler’s pentagonal number theorem are also proved. Finally, the fourth topic presents a combinatorial proof for an overpartition identity, derived from the truncated version of a certain theta series identity. iii Table of Contents List of Figures vi Acknowledgments vii Chapter 1 Introduction 1 1.1 EO−partitions . 1 1.2 Separable Integer Partition Classes . -
© 2010 Byung Chan Kim ARITHMETIC of PARTITION FUNCTIONS and Q-COMBINATORICS
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Illinois Digital Environment for Access to Learning and Scholarship Repository © 2010 Byung Chan Kim ARITHMETIC OF PARTITION FUNCTIONS AND q-COMBINATORICS BY BYUNG CHAN KIM DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 2010 Urbana, Illinois Doctoral Committee: Professor Alexandru Zaharescu, Chair Professor Bruce Berndt, Director of Research Associate Professor Scott Ahlgren, Director of Research Assistant Professor Alexander Yong Abstract Integer partitions play important roles in diverse areas of mathematics such as q-series, the theory of modular forms, representation theory, symmetric functions and mathematical physics. Among these, we study the arithmetic of partition functions and q-combinatorics via bijective methods, q-series and modular forms. In particular, regarding arithmetic properties of partition functions, we examine partition congruences of the overpartition function and cubic partition function and inequalities involving t-core partitions. Concerning q-combinatorics, we establish various combinatorial proofs for q-series identities appearing in Ramanujan's lost notebook and give combinatorial interpretations for third and sixth order mock theta functions. ii To my parents iii Acknowledgments First of all, I am grateful to my thesis advisers, Scott Ahlgren and Bruce Berndt. It was an algebraic number theory course given by Ahlgren which led me to number theory. In the 2006 Fall semester, a modular forms course also by Ahlgren and a reading course with Berndt led me to the theory of partitions. -
Curriculum Vitae FRANK GARVAN
Curriculum Vitae FRANK GARVAN Office Address: University of Florida Office Phone: +1{352{294{2305 Mathematics Department Email Address: fgarvan@ufl.edu PO Box 118105 Homepage: http://people.clas.ufl.edu/fgarvan Gainesville, FL 32611-8105 Birthdate: March 9, 1955 Last modified: May 11, 2021 EDUCATION/EMPLOYMENT 2000 { Professor of Mathematics, University of Florida 1994 { 1999 Associate Professor of Mathematics, University of Florida 1990 { 1993 Assistant Professor of Mathematics, University of Florida 1991 NSERC International fellow, Dalhousie University, Halifax 1988 { 1990 Postdoctoral Research Fellow, Macquarie University, Sydney, Australia 1987 { 1988 Postdoctoral Fellow, I.M.A., University of Minnesota, Minneapolis 1986 { 1987 Visiting Assistant Professor of Mathematics, University of Wisconsin, Madison 1985 { 1986 Mathematics Instructor, Pennsylvania State University, York Campus 1986 Ph.D. Pennsylvania State University, Mathematics (advisor: George Andrews) 1982 M.Sc. University of New South Wales, Kensington, Australia, Mathematics (advisor: Mike Hirschhorn) 1978 { 1980 High School Teacher, New South Wales, Australia 1977 Dip.Ed. University of New South Wales, Kensington, Australia, Education 1976 B.Sc. University of New South Wales, Kensington, Australia, Mathematics (with honours) RESEARCH INTERESTS Number Theory, Combinatorics, Special Functions, Symbolic Computation RESEARCH GRANTS 2014 { 2019 Simons Foundation Collaboration Grant (ref. 318714) ($35,000), Analytic, Combinatorial and Computational Study of Partitions 2009 { 2012 CoPI for NSA Grant (ref. H98230-09-1-0051) ($131,450), Some Problems in the Theory of Partitions 2006 { 2009 CoPI for NSA Grant (ref. H98230-07-1-0011) ($123,380), Some Problems in the Theory of Partitions 1998 { 2001 NSF grant (ref. DMS-9870052) ($77,895) Combinatorial, Differential & Modular Partition Problems 1992 { 1995 NSF grant (ref.