Srinivasa Ramanujan: in celebration of the centenary of his election as FRS

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.

09:40-09:55 Discussion

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.

10:25-10:40 Discussion

10:40-11:05 Coffee

11:05-11:35 Ramanujan and quadratic forms

Professor Manjul Bhargava, Princeton University, USA

11:35-11:50 Discussion

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.

12:20-12:35 Discussion

13:30-14:00 Srinivasa Ramanujan and signal-processing problems

Professor P P Vaidyanathan, California Institute of Technology, USA

Abstract

In 1918, Srinivasa Ramanujan introduced a summation, known today as the Ramanujan-sum. He used this to express several arithmetic functions in the form of infinite series. For many years this sum was used by other mathematicians to prove important results in number theory. In recent years, engineers and physicists have found applications of this sum in digital signal processing, especially in identifying periodic components of signals with integer periods. This has also led to generalization of the Ramanujan-sum decomposition in several directions, opening up new theory as well as applications in the representation and identification of periodic signals. Key to these developments is the so-called Ramanujan subspace which is a space containing a specific class of integer-periodic functions. These subspaces turn out to be fundamental in the representation of arbitrary periodic sequences. The new developments include Ramanujan dictionaries for sparse representation of periodic signals, Farey dictionaries for the same, and Ramanujan filter banks for tracking periodicity as it evolves and changes in time. The well known subspace algorithm for parameter identification called the multiple signal classification (MUSIC) algorithm has also been generalized to obtain the so-called iMUSIC algorithm for integer periodicities. The results have applications in the study of periodic signals with inherently integer periods, such as segments of DNA and protein sequences among others. This talk gives an overview of the theory and outlines some of these applications.

14:00-14:15 Discussion

14:15-14:45 From Ramanujan graphs to Ramanujan complexes

Professor Alex Lubotzky, Hebrew University, Israel

Abstract

Ramanujan graphs are k-regular graphs with all non trivial eigenvalues are bounded (in absolute value) by 2SR(k-1). They are optimal expanders (from spectral point of view). Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL(2) over a local field F, by the action of suitable congruence subgroups of arithmetic groups.

The spectral bound was proved using works of Hecke, Deligne and Drinfeld on the 'Ramanujan conjecture' in the theory of automorphic forms.

The work of Lafforgue, extending Drinfeld  from GL(2) to GL(n), opened the door for the construction of Ramanujan complexes as quotients of the Bruhat-Tits buildings associated with GL(n) over F. This way one gets finite simplical complxes which on one hand are 'random like' and at the same time have strong symmetries. These seemingly contradicting properties make them very useful for constructions of various external objects.

Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties. The last application is probably the most suprising one, showing that there exist d-dimensional BOUNDED DEGREE simplical complexes X with the following remarkable property: for every continuous map from X to the d-dimensional Euclidean space, there is a point which is covered by a fixed fraction of the d-dimensional faces of X.

14:45-15:00 Discussion

15:00-15:30 Coffee

15:30-16:00 Optimal point configurations and Fourier interpolation formulas

Professor Maryna Viazovska, Ecole Polytechnique Federale de Lausanne, Switzerland

Abstract

The sphere packing problem asks for the densest configuration of non-intersecting open unit balls at the Euclidean space. This classical geometric problem is solved only in dimensions 1, 2, 3, 8, and 24.  In this talk, Professor Viazovska will present a solution of the sphere packing problem in dimensions 8 and 24. It seems that each dimension has its own features and requires a different approach. One method of estimating the density of a sphere packing from above was suggested by H Cohn and N Elkies in 2003. Their approach is based on Fourier optimization. Namely, they showed that the existence of a function satisfying certain inequalities for the function itself and for its Fourier transform leads to an upper bound of the density of a sphere packing. Using this method Cohn and Elkies were able to prove almost sharp bounds in dimensions 8 and 24. Professor Viazovska will show that functions providing exact bounds can be constructed explicitly. Moreover, Professor Viazovska will present a new type of Fourier interpolation formula.

16:00-16:15 Discussion

16:15-16:45 Ramanujan conjecture, Ramanujan graphs and Ramanujan complexes

Professor Wen-Ching Winnie Li, Pennsylvania State University, USA

Abstract

The purpose of this talk is to give a panorama of the three topics in the title carrying Ramanujan's name.

In 1916 Ramanujan predicted that the pth Fourier coeffcient of the weight 12 discriminant function is bounded by 2p^11/2 for all primes p. This conjecture is now proved for classical holomorphic cuspidal Hecke eigenforms by the work of Eichler-Shimura, Deligne, and Deligne-Serre. After the connection between the classical modular forms and representations of GL(2) over Q was understood, the Ramanujan conjecture is expected to hold in great generality, revealing deep arithmetic properties of automorphic forms. In particular, it is established for automorphic cuspidal representations of GL(n) over global function fields by Drinfeld and L La orgue.

The Bruhat-Tits building attached to PGL(n + 1) over a p-adic field F is an n-dimensional simplicial complex on which PGL(n+1, F) acts. The orbit space of a discrete cocompact subgroup Γ of PGL(n+1, F) is a finite n-dimensional simplicial complex X_Γ. The validity of the Ramanujan conjecture is used to select infinitely many such Γ so that the spaces X_Γ are spectrally optimal, called Ramanujan graphs (for n = 1) and Ramanujan complexes (for n ≥ 2). There are zeta functions counting closed geodesics in X_Γ of given dimension and type. The Ramanujan graphs/complexes are those whose zeta functions satisfy the Riemann Hypothesis. The prime geodesic theorems for X_Γ follow from the analytic behaviour of the associated zeta functions, and for Ramanujan graphs/complexes, one also obtains a good error estimate, similar to what happens for prime numbers.

16:45-17:00 Discussion

09:00-09:30 The Ramanujan Conjectures

Professor Peter Sarnak, Princeton University, USA

Abstract

One of the most influential insights by Ramanujan was his conjecture concerning the size of Fourier coefficients of modular forms. These lead naturally to some of the fundamental conjectures in the modern theory of automorphic forms, and the special cases that have been proven (including Ramanujan's original by Deligne) constitute high points of this theory. Professor Sarnak will review these briefly and as well as some number theoretic applications.

09:30-09:45 Discussion

09:45-10:15 Selmer groups and Galois representations

Professor Sujatha Ramdorai, University of British Columbia, Canada

Abstract

Selmer groups are important objects that are studied in the Iwasawa theory of Galois representations. Considered as modules over the corresponding Iwasawa algebras, they have certain invariants associated to them which provide information on their exact structure. This talk will provide an overview of Selmer groups that arise in the study of Galois representations associated to elliptic curves and modular forms, revisiting some of the classical results as well as exposing new results on their algebraic structure.

10:15-10:30 Discussion

10:30-11:00 Coffee

11:00-11:30 Mock modular and quantum modular forms

Professor Amanda Folsom, Amherst College, USA

Abstract

Ramanujan’s last letter to Hardy surrounds his mock theta functions, certain curious q-hypergeometric series. In the decades following his death in 1920, while the mock theta functions were of great importance and interest, it was unclear how exactly they fit into the theory of modular forms – Dyson called this “a challenge for the future” at the 1987 Ramanujan Centenary Conference. Zwegers finally recognised that Ramanujan had discovered glimpses of special families of nonholomorphic modular forms, which we now know to be harmonic Maass forms, as defined by Bruinier and Funke in 2004. Studying Ramanujan’s mock theta functions within the modern context of harmonic Maass forms, and their holomorphic parts called mock modular forms, has resulted in their deeper understanding, and a great deal of research activity and applications; however, as of a few years ago, a fundamental question remained. In his last letter, Ramanujan claimed that as q approaches any even-ordered root of unity radially from within the unit disk, either the sum or difference between his mock theta function f(q) and a modular form b(q) is bounded.  The author, together with Ono and Rhoades, revisited this claim, which led to a fascinating connection between mock theta functions and quantum modular forms, which were not defined until 2010 by Zagier, 90 years after Ramanujan’s death. In this talk, Professor Folsom will bring together past and present, and study the relationship between mock modular forms and quantum modular forms, with Ramanujan’s mock theta functions as motivation. In particular, she will discuss related work of Bringmann-Rolen, Choi-Lim-Rhoades, Griffin-Ono-Rolen, and others.

11:30-11:45 Discussion

11:45-12:15 Ramanujan and the anatomy of integers

Professor K Soundararajan, Stanford University, USA

12:15-12:30 Discussion

13:30-14:00 Ramanujan’s influence on string theory, black holes and moonshine

Professor Jeff Harvey, University of Chicago, USA

Abstract

Many problems in string theory involve counting certain kinds of quantum states of strings or black holes. These counting problems can be recast in terms of the coefficients of modular forms that were central to the work of Ramanujan. The Hardy-Ramanujan asymptotic estimates for the growth of these coefficients are turned in physical problems into statements about a limiting temperature in string theory or the entropy of black holes. More recently the mock modular forms studied by Ramanujan in the last year of his life have also appeared in string theory through the study of K3 surfaces. In particular this has led to the new 'moonshine' phenomenon that link Ramanujan’s mock modular forms to the representation theory of certain finite groups, in particular the sporadic Mathieu group M24. Professor Harvey will survey these connections between Ramanujan’s work and aspects of string theory with a particular emphasis on the new moonshine phenomena linking K3 and M24 and its generalization to umbral moonshine.

14:00-14:15 Discussion

14:15-14:45 Ramanujan and recent work on the Riemann hypothesis

Professor Ken Ono, Emory University, USA

Abstract

Ramanujan and Hardy famously developed the "circle method" to approximate the values of the partition function. This work has been generalized and fully developed with a wide array of applications in the world of modular forms (e.g. string theory, moonshine, topological invariants, to name a few). The speaker will describe how the implementation of the circle method to a conjecture on partitions led to a general theorem about the hyperbolicity of real polynomials. This general theorem includes results on the Polya-Jensen criterion for the Riemann Hypothesis. This lecture represents joint work with Michael Griffin, Larry Rolen, and Don Zagier.

14:45-15:00 Discussion

15:00-15:30 Coffee

15:30-16:00 Mock modular forms are everywhere

Professor C. N. Cheng, University of Amsterdam, Netherlands

Abstract

One of the most important legacies of Ramanujan is the introduction of mock theta functions. Nearly a century later, we have understood that they have the following three key properties: 'nice' Fourier coefficients, radial limits, and shadows.

First, a conspicuous property of the q-series Ramanujan wrote down is that they have integral coefficients. Second, the original characterisation of the mock theta functions, given by Ramanujan himself, concerns their behaviour near roots of unity. Third, after a long quest, a modern definition of mock modular forms was found in the first decade of the millenium. This definition states that, upon adding a non-holomorphic modular correction dictated by the 'shadow' function, the mock modular form becomes a harmonic Maass form transforming just like an ordinary modular form.

Not even two decades after the modern definition was discovered, mock modular forms have already found a wide range of applications in various fields in mathematics and theoretical physics. In this talk, Professor Cheng will mention moonshine, stringy black holes, and three-dimensional topology. Quite evidently, the Fourier coefficients and to a lesser degree the shadows of the mock modular forms play an important role in the context of moonshine and black holes. Professor Cheng will explain how all three properties of mock modular forms, in particular their connection to quantum modular forms, are crucial for applications in three-dimensional topology.

16:00-16:15 Discussion