Connected at Infinity II

The problems relating to the existence and construction of orthogonal Latin squares have fascinated researchers for several centuries now. Though many important discoveries have been made, some problems still remain unresolved. Latin squares and orthogonal Latin squares have a beautiful underlying structure and are related to other combinatorial objects. These have applications in different areas, including statistical design of experiments and cryptology. Comprehensive accounts of the theory and applications of Latin squares are available in the books by J. Dénes and A. D. Keedwell (1974, 1991) and C. F. Laywine and G. L. Mullen (1998).

This article focuses on an important piece of work of the world renowned Indian statistician, Calyampudi Radhakrishna Rao. In 1945, C. R. Rao (25 years old then) published a pathbreaking paper [43], which had a profound impact on subsequent statistical research. Roughly speaking, Rao obtained a lower bound to the variance of an estimator. The importance of this work can be gauged, for instance, by the fact that it has been reprinted in the volume Breakthroughs in Statistics: Foundations and Basic Theory [32]. There have been two major impacts of this work:

Frobenius splittings were introduced by V. B. Mehta and A. Ramanathan in [6] and refined further by S. Ramanan and Ramanathan in [9]. Frobenius splittings have proven to be a amazingly effective when they apply. Proofs involving Frobenius splittings tend to be very efficient. Other methods usually require a much more detailed knowledge of the object under study. For instance, while showing that the intersection of one union of Schubert varieties with another union of Schubert varieties is reduced, one does not need to know where that intersection is situated, let alone what it looks like exactly.

In 1968 Roddam Narasimha (RN) published a paper in Journal of Sound and Vibration (JSV) deriving the equation for the transverse displacement v(x, t) of a vibrating string of length l, where v = (v, w) is a two-dimensional vector in the yz plane orthogonal to the x axis, \(v_x^2 = v_x^2 + w_x^2\) is the squared x derivative of v, R is a damping coefficient, Γ′ a nonlinearity parameter that involves a characteristic amplitude of the string motion and the material properties of the string (see equation 4.25 below) and f₀ is an external force acting on the string. Note that for R ≡ 0 and Γ′ ≡ 0 this is the standard linear wave equation that is the canonical example for a second order hyperbolic equation in one space dimension which ignores the coupling (usually nonlinear) between the transverse and longitudinal displacements of the string. In the usual text book derivation of the wave equation for the vibrating string it is assumed that the motion of the string is entirely transversal. That this is not strictly true was realized by Kirchhoff [1876] and Lord Rayleigh in the nineteenth century: while the former went on to derive an equation similar to (1.1), Rayleigh [1883] (with no reference to Kirchhoff) restricted himself to the classical van der Pol oscillator to model the vibrating string. More such models were proposed by Osgood in the 1920’s, Carrier and Coulson in the 1940’s, followed by Oplinger, Murthy and Ramakrishna, Narasimha and Anand in the 1960’s.

Lie groups and Lie algebras occupy a prominent and central place in mathematics, connecting differential geometry, representation theory, algebraic geometry, number theory, and theoretical physics. In some sense, the heart of (classical) representation theory is in the study of the semisimple Lie groups. Their study is simultaneously simple in its beauty, as well as complex in its richness. From Killing, Cartan, and Weyl, to Dynkin, Harish-Chandra, Bruhat, Kostant, and Serre, many mathematicians in the twentieth century have worked on building up the theory of semisimple Lie algebras and their universal enveloping algebras. Books by Borel, Bourbaki, Bump, Chevalley, Humphreys, Jacobson, Varadarajan, Vogan, and others form the texts for (introductory) graduate courses on the subject.

Symmetries have always been a natural source of inspiration and a strong guiding principle for both scientists and artists, as it is remarkably discussed in a book by Hermann Weyl [53]. In particular, the notions of group of symmetries and group actions lie at the heart of deep concepts both in physics and in mathematics. In this context, Dirac operators provide elegant and powerful bridges between quantum physics, geometry and representation theory. In particular, using representation theoretic Dirac operators, Parthasarathy provided deep and elegant answers to each of the following important representation theoretic questions.

The title of this paper is chosen to echo the title of the paper [44] by Daya-Nand Verma. Our aim is to explain the main results and conjectures there and to describe the impact they have had in representation theory in the years since then.

The work of Norbert Wiener and Pesi Masani started at Indian Statistical Institute in Kolkata during the visit of Wiener in 1955–56. Wiener [18] had done substantial work on the prediction problem for univariate (weakly) stationary (henceforth, stationary) processes and had partial results for multivariate stationary processes. A.N. Kolmogorov [7] had studied the univariate case in detail, with emphasis on a fundamental theorem of H. Wold. In addition, V. Zasuhin [20] had announced partial results. Because of the connection between prediction theory (of interest to Wiener) and factorization of matrix-valued functions (of interest to Wiener [18] p.150 and Masani [12]), their collaboration produced results and techniques which had a lasting influence on prediction theory, analysis and operator theory (Nagy and Foias, see [13]).