IN SEARCH OF EXCELLENCE
Excellence is a journey and not a destination. In science it
implies perpetual efforts to advance the frontiers of knowledge.
英文翻译中文This often leads to progressively increasing specialization and
emergence of newer disciplines. A brief summary of salient
contributions of Indian scientists in various disciplines is
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introduced in this ction.
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T
he modern period of mathematics rearch in India started with Srinivasa Ramanujan who work on analytic number theory and modular forms is
highly relevant even today
nxx. In the pre-Independence period mathematicians like S.S. Pillai,Vaidyanathaswamy, Ananda Rau and others contributed a lot.
Particular mention should be made of universities in Allahabad, Varanasi, Kolkata,Chennai and Waltair and later at Chandigarh,Hyderabad, Bangalore and Delhi (JNU). The Department of Atomic Energy came in a big way to boost mathematical rearch by starting and nurturing the Tata Institute of Fundamental Rearch (TIFR), which, under the leadership of Chandrakharan, blossomed into a great school of learning of international standard. The Indian Statistical Institute, started by P.C. Mahalanobis,made its mark in an international scene and continues to flourish. Applied mathematics community owes a great deal to the rvices of three giants Ñ N.R. Sen, B.R. Seth and P .L. Khastgir. Some of the areas in which significant contributions have been made are briefly described here.
A LGEBRA
O
ne might say that the work on modern algebra in India started with the beautiful piece of work in 1958 on the proof of SerreÕs conjecture for n =2. A particular ca of the conjecture is to imply that
a unimodular vector with polynomial entries in n vari-ables can be completed to a matrix of determinant
huazhuangone. Another important school from India was start-ed in Panjab University who work centres around Zassanhaus conjecture on groupings.
A LGEBRAIC G EOMETRY
中国国防大学分数线
T
he study of algebraic geometry began with a minal paper in 1964 on vector bundles. With further study on vector bundles that led to the mod-uli of parabolic bundles, principle bundles, algebraic differential equations (and more recently the rela-tionship with string theory and physics), TIFR has become a leading school in algebraic geometry. Of the later generation, two pieces of work need special mention: the work on characterization of affine plane purely topologically as a smooth affine surface, sim-ply connected at infinity and the work on Kodaira vanishing. There is also some work giving purely algebraic geometry description of the topologically invariants of algebraic varieties. In particular this can be ud to study the Galois Module Structure of the invariants.
L IE T HEORY
T
he inspiration of a work in Lie theory in India came from the monumental work on infinite dimensional reprentation theory by Harish Chandra, who has, in some n, brought the sub-ject from the periphery of mathematics to centre stage. In India, the initial study was on the discrete subgroups of Lie groups from number theoretic angle. The subject received an impetus after an inter-national conference in 1960 in TIFR, where the lead-ing lights on the subject, including A. Selberg partic-
M ATHEMATICAL S CIENCES
C H A P T E R V I I
ipated. Then work on rigidity questions was initiat-ed. The question is whether the lattices in arithmetic groups can have interesting deformations except for the well-known classical cas. Many important cas in this question were ttled.
D IFFERENTIAL
E QUATION
A fter the study of L-functions were found to be
uful in number theory and arithmetic geome-try, it became natural to study the L-functions arising out of the eigenvalues of discrete spectrum of the dif-ferential equations. MinakshisundaramÕs result on the corresponding result for the differential equation leading to the Epstein Zeta function and his paper with A. Pleijel on the same for the connected com-pact Riemanian manifold are works of great impor-tance. The idea of the paper (namely using the heat equation) lead to further improvement in the hands of Patodi. The results on regularity of weak solution is an important piece of work. In the later 1970s a good school on non-linear partial differential equa-tions that was t up as a joint venture between TIFR and IISc, has come up very well and an impressive lists of results to its credit.
For differential equations in applied mathematics, the result of P.L. Bhatnagar, BGK model (by Bhatnagar, Gross, Krook) in collision process in gas and an explanation of Ramdas Paradox (that the temperature minimum happens about 30 cm above the surface) will stand out as good mathematical models. Further significant contributions have been made to the area of group theoretic methods for the exact solutions of non-liner partial differential equations of physical and engineering systems.
E RGODIC T HEORY
E arliest important contribution to the Ergodic the-
ory in India came from the Indian Statistical Institute. Around 1970, there was work on spectra of unitary operators associated to non-singular trans-formation of flows and their twisted version, involv-ing a cocycle.
Two results in the subjects from 1980s and 1990s are quoted. If G is lattice in SL(2,R) and {uÐt} a unipotent one parameter subgroup of G, then all non-periodic orbits of {uÐt} on GÐ1 are uniformly distributed. If Q is non-generate in definite quadratic form in n=variables, which is not a multiple of rational form, then the number of lattice points xÐwith a< ½Q(x)½< b, ½½x½½< r, is at least comparable to the volume of the corresponding region.
N UMBER T HEORY
T he tradition on number theory started with Ramanujan. His work on the cusp form for the full modular group was a breakthrough in the study of modular form. His conjectures on the coefficient of this cusp form (called RamanujanÕs tau function) and the connection of the conjectures with conjectures of A. Weil in algebraic geometry opened new rearch areas in mathematics. RamanujanÕs work (with Hardy) on an asymptotic formula for the parti-tion of n, led a new approach
路倒(in the hands of Hardy-Littlewood) to attack such problems called circle method. This idea was further refined and S.S. Pillai ttled WaringÕs Conjecture for the 6th power by this method. Later the only remaining ca namely 4th powers was ttled in mid-1980s. After Independence, the major work in number theory was in analytic number theory, by the school in TIFR and in geometry of numbers by the school in Panjab University. The work on elliptic units and the con-struction of ray class fields over imaginary quadratic fields of elliptic units are some of the important achievements of Indian number theory school. Pioneering work in BakerÕs Theory of linear forms in logarithms and work on geometry of numbers and in particular the MinkowskiÕs theorem for n = 5 are worth mentioning.
P ROBABILITY T HEORY
S ome of the landmarks in rearch in probability theory at the Indian Statistical Institute are the following:
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q A comprehensive study of the topology of weak convergence in the space of probability measures on topological spaces, particularly, metric spaces. This includes central limit theorems in locally com
pact abelian groups and Milhert spaces, arithmetic of probability distributions under convolution in topological groups, Levy-khichini reprentations for characteristic functions of probability distributions on group and vector spaces.
q Characterization problems of mathematical statistics with emphasis on the derivation of probability laws under natural constraints on statistics evaluated from independent obrvations.
q Development of quantum stochastic calculus bad on a quantum version of ItoÕs formula for non-commutative stochastic process in Fock spaces. This includes the study of quantum stochastic integrals and differential equations leading to the construction of operator Markov process describing the evolution of irreversible quantum process.
q Martingale methods in the study of diffusion process in infinite dimensional spaces.
q Stochastic process in financial mathematics.
C OMBINATORICSdownton abbey
risky
T hough the work in combinatorics had been ini-tiated in India purely through the efforts of R.C.Bo at the Indian Statistical Institute in late thirties, it reached its peak in late fifties at the University of No
rth Carolina, USA, where he was joined by his former student S.S.Shrikhande. They provided the first counter-example to the celebrat-ed conjecture of Euler (1782) and jointly with Parker further improved it. The last result is regarded a classic.
In the abnce of the giants there was practically no rearch activity in this area in India. However, with the return of Shrikhande to India in 1960 activities in the area flourished and many notable results in the areas of embedding of residual designs in symmetric designs, A-design conjecture and t-designs and codes were reported.
T HEORY OF R ELATIVITY
I n a strict n the subject falls well within the purview of physics but due to the overwhelming respon by workers with strong foundation in applied mathematics the activity could blossom in some of the departments of mathematics of certain universities/institutes. Groups in BHU, Gujarat University, Ahmedabad, Calcutta University, and IIT, Kharagpur, have contributed generously to the area of exact solutions of Einstein equations of gen-eral relativity, unified field theory and others. However, one exact solution which has come to be known as Vaidya metric and ems to have wide application in high-energy astrophysics derves a special mention.
N UMERICAL A NALYSIS
T he work in this area commenced with an attempt to solve non-linear partial differential equations governing many a physical and engineering system with special reference to the study of Navier-Stabes equations and cross-viscous forces in non-Newtonian fluids. The work on N-S equation has turned out to be a basic paper in the n that it reappeared in the volume, Selected Papers on Numerical Solution of Equations of Fluid Dynamics, Applied Mathematics, through the Physical Society of Japan. The work on non-Newtonian fluid has found a place in the most prestigious volume on Principles of Classical Mechanics & Field Theory by Truesdell and Toupin. The other works which derve mention are the development of extremal point collocation method and stiffy stable method.
A PPLIED M ATHEMATICS
T ill 1950, except for a group of rearch enthusi-asts working under the guidance of N.R.Sen at Calcutta University there was practically no output in applied mathematics. However, with directives from the centre to emphasize on rearch in basic
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and applied sciences and liberal central fundings through central and state sponsored laboratories, the activity did receive an impetus. The department of mathematics at IIT, Kharagpur, established at the very inception of the institute of national importance in 1951, under the dynamic leadership of B.R.Seth took the lead role in developing a group of excellence in certain areas of mathematical sciences. In fact, the rearch carried out there in various disciplines of applied mathematics such as elasticity-plasticity, non-linear mechanics, rheological fluid mechanics, hydroelasticity, thermoelasticity, numerical analysis, theory of relativity, cosmology, magneto hydrody-namics and high-temperature gasdynamics turned out to be a trend tting one for other IITs, RECs, other Technical Institutes and Universities that were in the formative stages. B.R. SethÕs own rearches on the study of Saint-VenamtÕs problem and transi-tion theory to unify elastic-plastic behaviour of mate-rials earned him the prestigious EulerÕs bronze medal of the Soviet Academy of Sciences in 1957. The other areas in which applied mathematicians con-tributed generously are biomechanics, CFD, chaotic dynamics, theory of turbulence, bifurcation analysis, porous media, magnetics fluids and mathematical
physiology.
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