Chairs
Professor Ken Ono, Emory University, USA
Professor Ken Ono, Emory University, USA
Ono is the Asa Griggs Candler Professor of Mathematics at Emory University and Vice President of the American Mathematical Society. His contributions include several monographs and more than 160 research and popular articles in number theory, combinatorics and algebra. He earned his Ph.D. from UCLA and has received many awards for his research, including a Guggenheim Fellowship, a Packard Fellowship and a Sloan Research Fellowship. He was awarded a Presidential Early Career Award for Science and Engineering (PECASE) by Bill Clinton in 2000 and was named a Distinguished Teaching Scholar by the National Science Foundation in 2005. He is also a member of the US National Committee for Mathematics and the National Academy of Sciences. He was also an Associate Producer of the film “The Man Who Knew Infinity”, the Hollywood biopic about Srinivasa Ramanujan which starred Jeremy Irons and Dev Patel.
09:00-09:05
Welcome by the Royal Society
09:10-09:40
Living with Ramanujan for forty years
Professor Bruce Berndt, University of Illinois, USA
Abstract
Beginning in May 1977, the speaker began to devote all of his research efforts to proving the approximately 3000 claims made by Ramanujan without proofs in his notebooks. While completing this task a little over 20 years later, with the help, principally, of his former and then current graduate students, he began to work with George Andrews on proving Ramanujan's claims from his "lost notebook." After another 20 years, with the help of several mathematicians, we think we have found proofs of all the claims in the lost notebook. But one entry, connected with the famous Dirichlet Divisor Problem, remained painfully difficult to prove. In analogy with G.N. Watson's retiring address to the London Mathematical Society in November 1935 on the "final problem," arising from Ramanujan's last letter to Hardy, we have called this entry the "final problem," because it was the last entry from the lost notebook to be proved. Early this summer, a proof was finally given by Junxian Li, who just completed her doctorate at the University of Illinois; Alexandru Zaharescu (her advisor); and myself. We will discuss the identity comprising the "final problem" as well as other highlights from our 40 year investigation of the (earlier) notebooks and lost notebook.
Show speakers
Professor Bruce Berndt, University of Illinois, USA
Professor Bruce Berndt, University of Illinois, USA
Bruce Berndt has been on the mathematics faculty at the University of Illinois at Urbana-Champaign since 1967. While on sabbatical leave at the Institute for Advanced Study in Princeton in 1974, he became keenly interested in the work of Ramanujan, primarily through two papers by Emil Grosswald. In 1977, Berndt began to edit the earlier notebooks of Srinivasa Ramanujan. Earlier notes by G.N. Watson and B.M. Wilson from their attempts in the late 1920's and early 1930's to edit the notebooks were very helpful. By "editing," we mean providing proofs for the over 3300 formulas and claims in these notebooks, which arise from approximately the period 1904--1914. This task was completed in 1998 with the fifth volume published by Springer. Beginning in the late 1990's, Berndt and George Andrews began "editing" Ramanujan's lost notebook. Their fifth and final volume on the lost notebook was published by Springer earlier this year. Berndt and R.A. Rankin have also published two volumes with the American Mathematical Society on Ramanujan's correspondence and essays about his work.
09:55-10:25
Ramanujan's Lost Notebook in Five Volumes: Future Directions
Professor George E Andrews, Pennsylvania State University, USA
Abstract
Ramanujan composed two Notebooks of his discoveries prior to coming to Cambridge in 1914. Upon his return to India in 1919, he filled another 100 plus pages with formulas discovered during this last year of his life. The latter is referred to as his Lost Notebook; it lay unexamined until 1976. Bruce Berndt devoted 5 volumes (published by Springer) to the mathematics contained in the original two Notebooks. Four volumes have already been published on the Lost Notebook explicating the formulas contained in the Lost Notebook. The fifth and final volume is in press. Ramanujan left no proofs of the thousands of results in these notebooks.
It is important to note a couple things concerning the Notebooks. First, through the efforts of many currently active researchers, every formula in the Notebooks has now been proved (or in a minimally few cases, disproved). However, there are many results that have great importance currently (e.g. assertions about the mock theta functions) where it is almost certain that the modern proofs are radically different from Ramanujan's understanding of the results. To put it another way, there are many results in the Lost Notebook (especially those dealing with the mock theta functions) which seem impossible to discover (even by Ramanujan) without some overarching theory. Furthermore the modern proofs contain intermediate results which, owing to their elegance and simplicity, Ramanujan certainly would have included in his Lost Notebook had he known them. All this leads to the very natural conclusion that Ramanujan knew many things and had many methods that are currently unknown to us. The object of this talk will be to draw attention to aspects of the Lost Notebook where Ramanujan's discoveries have left mysteries that are well worth exploring. It is hoped that this will point to and encourage further investigation.
Show speakers
Professor George E Andrews, Pennsylvania State University, USA
Professor George E Andrews, Pennsylvania State University, USA
George E. Andrews was born in Salem, Oregon. He received his B.S. and M.A. degrees from Oregon State University, and his Ph.D. from the University of Pennsylvania. Andrews is Evan Pugh University Professor in Mathematics at the Pennsylvania State University. An expert on q-series, he has written and published more than 300 papers and has just completed (jointly with Bruce Berndt) the fifth and final volume explicating Ramanujan's Lost Notebook. Andrews was elected to the American Academy of Arts and Sciences in 1997, and to the National Academy of Sciences (USA) in 2003. He was awarded an honorary professorship at Nankai University in 2008. In 2009 he became a SIAM Fellow. He holds honorary degrees from Parma, Florida, Waterloo, Illinois and SASTRA University (India). Andrews served as President of the American Mathematical Society from February 1, 2009 to January 31, 2011, and became a Fellow of the AMS in 2012
11:05-11:35
Ramanujan's Legacy: The Work of the SASTRA Prize Winners
Professor Krishnaswami Alladi, University of Florida, USA
Abstract
The SASTRA Ramanujan Prize, launched in 2005, is a $10,000 annual award given to mathematicians not exceeding the age of 32, for path-breaking contributions in areas infuenced by Srinivasa Ramanujan. The age limit has been set at 32 because Ramanujan lived only for 32 years and in that brief life span made revolutionary contributions; so the challenge for the prize candidates is to show what they have achieved in that same time frame! The prize is given each year at SASTRA University in Kumbakonam (Ramanujan's hometown) in South India around December 22 (Ramanujan's birthday). The prize has been unusually effective in recognizing extremely gifted mathematicians at an early stage of their careers, and so is now considered to be one of the most prestigious and coveted mathematics awards in the world. I will describe briefly the spectacular work for which the awardees were recognised and focus on some aspect aspects of their research that relate to Ramanujan.
Show speakers
Professor Krishnaswami Alladi, University of Florida, USA
Professor Krishnaswami Alladi, University of Florida, USA
Krishnaswami Alladi is professor of mathematics at the University of Florida, where he was department chairman from 1998-2008. He received his PhD from UCLA in 1978. His area of research is number theory - especially analytic number theory and the theory of partitions and q-hypergeometric series. He is the Founder and Editor-in-Chief of The Ramanujan Journal, devoted to all areas of mathematics influenced by Ramanujan, and published by Springer. He helped create the SASTRA Ramanujan Prize given to very young mathematicians for outstanding contributions to areas influenced by Ramanujan, and has chaired the prize committee since its inception in 2005.
11:50-12:20
Circle method
Professor Trevor Wooley FRS, University of Bristol, UK
Abstract
Hardy and Ramanujan’s introduction of the circle method in 1916 as a means of analysing the behaviour of the partition function led very rapidly to pivotal work of Hardy and Littlewood, and later, of Vinogradov, concerning Waring’s problem and the Goldbach problem. Now, a century later, applications of the circle method are legion across analytic number theory, quantitative arithmetic geometry, the theory of Diophantine approximation, discrete harmonic analysis, and beyond. With the exception of work concerning problems that might be characterised as dominated by linear behaviour, conclusions have usually remained far from the sharpest conjectured to hold.
The speaker will describe recent progress on non-linear problems that attains the sharpest bounds conjectured to hold, focusing on the resolution of the main conjecture in Vinogradov’s mean value theorem. The latter and its relatives provide key input into the sharpest estimates available in Waring’s problem concerning sums of powers, the zero-free region for the Riemann zeta function, the existence of rational points on varieties of large dimension over number fields, and so on. Progress on these problems, and its absence, will be described so as to highlight recent success, and the formidable challenges that remain.
Show speakers
Professor Trevor Wooley FRS, University of Bristol, UK
Professor Trevor Wooley FRS, University of Bristol, UK
Trevor D.Wooley gained his PhD from Imperial College in 1990, He was faculty member of the University of Michigan, Ann Arbor, 1991 - 2007 (serving as Chair 2002-2005), and has been based at the University of Bristol from 2007 - present. His research is centred on the Hardy-Littlewood (circle) method, a method based on the use of Fourier series that delivers asymptotic formulae for counting functions associated with arithmetic problems. In the 21st Century, this method has become immersed in a turbulent mix of ideas on the interface of Diophantine equations and inequalities, arithmetic geometry, harmonic analysis and ergodic theory, and arithmetic combinatorics. Wooley has been awarded the Salem Prize (1998), Frohlich Prize (2012), elected FRS (2007) and FAMS (2012). He was a 45-minute speaker at ICM in Beijing (2002) and Seoul (2014).