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

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.

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.

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.

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.

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.

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.

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.

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.