Srinivasa Ramanujan and signal-processing problems
Professor P P Vaidyanathan, California Institute of Technology, USA
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.
From Ramanujan graphs to Ramanujan complexes
Professor Alex Lubotzky, Hebrew University, Israel
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.
Optimal point configurations and Fourier interpolation formulas
Professor Maryna Viazovska, Ecole Polytechnique Federale de Lausanne, Switzerland
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.
Ramanujan conjecture, Ramanujan graphs and Ramanujan complexes
Professor Wen-Ching Winnie Li, Pennsylvania State University, USA
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 2p11/2 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 XΓ. The validity of the Ramanujan conjecture is used to select infinitely many such Γ so that the spaces XΓ are spectrally optimal, called Ramanujan graphs (for n = 1) and Ramanujan complexes (for n ≥ 2). There are zeta functions counting closed geodesics in XΓ of given dimension and type. The Ramanujan graphs/complexes are those whose zeta functions satisfy the Riemann Hypothesis. The prime geodesic theorems for XΓ 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.