Many in the Math-community feel let down that there is no Nobel for Mathematicians. But in fact there are several prizes to encourage budding mathematicians. One such is the Abel Prize which people say is an answer to the question of ‘no Nobel for math’. The prize money is also incredibly attractive!
On May 19 2015, Louis Nirenberg and John Nash received the famed Abel prize which includes a citation, gold medal and 6 Million NOK (Norwegian Currency Norway Kroner) which approximately amount to Rs 5,00,00,000. The citation says “For striking and seminal contributions to the theory of non-linear PDE’s and it’s applications to geometric analysis”. Solutions to certain differential equations arising in nature were hard to find and then one George de Rham (a Swiss mathematician) gave the so called weak solution. The works of Louis Nirenberg and John Nash show that these solutions can be rendered ‘regular’. John Nash(of the Beautiful Mind fame) besides the work on game theory also worked on Holder estimates for the solutions of linear elliptic equations in general dimension without any regularity assumptions. This led to the solution of Hilbert’s 19th problem. Similarly Nirenberg made beautiful application of the so called maximum principles which refer to certain analytic results on the attainment of maximum bounds on some small domains in the context of non-linear elliptic partial differential equations.
Applications of these works include the solution of the prescribed curvature problem in geometry, the Navier-Stokes problems, stability of the GNS inequalities(named after Gagliardo-Nirenberg-Sobolev) and problems involving GR (General Relativity) in cosmology.
Other recipients include S. R. S. Varadhan, Yakov Sinai, and Peter Lax. The latest recipient of the Abel Prize is a leading women mathematician- Karen Uhlenbeck of the University of Texas at Austin.

In the year 1900 David Hilbert asked (among a set of 23 questions/problems) if there is a program that takes as input any given statement and decides whether it is true or not, i.e, it should output either a true or false. This will immediately bring to our mind the questions about automated reasoning,artificial intelligence pattern recognition and such other themes. However the above mentioned question came up in the context of mathematical logic and one should keep in mind the kind of computational power we had back then in the early 1900’s. In Mathematics we generally deduce statements from a set of axioms or priorly ascertained facts. So one can imagine feeding a database of mathematical facts and their interrelations and from this create a system to check whether a certain mathematical statement or conjecture is true or false. So we shall call this elusive procedure demanded by David Hilbert as ‘Hilbert’s program’. In fact a famous result in Mathematics namely the the four color problem was approached in a similar fashion.
It was K. Godel and A. Turing who showed the way toward solving Hilbert’s question and finally it was known that there were logical obstructions to get the Hilbert’s program. Godel showed that the axioms of arithmetic on which proof systems depend sometimes lead to paradoxes thus asserting that there is the so called Incompleteness in logical reasoning. Well Godel and Turing worked independently under different contexts but essentially came to the same conclusion. For more on this one can read [1].
However an Indian origin scientist Madhu Sudan developed the so called probabilistically checkable proof system which uses probability theory to check the correctness of a given proof. The crux of the matter lies in his PCP Theorem (probabilistically checkable proof theorem) for which he was awarded the Rolf Nevanlinna prize in 2002. The PCP theorem says that for some universal constant K, for every n, any mathematical proof of length n can be rewritten as a different proof of length poly(n) that is formally verifiable with 99% accuracy using a randomized algorithm. [1] H. Ramesh, V. Vinay “Who will win the toss?” Resonance Journal of Science Education, April 1998.

Observe the Squares 1², 5², 7².. these numbers viz . . 1, 25 and 49 are in an Arithmetic progression. So one wonders if there are more such examples. Consider one more such triple 7², 13² and 17²… looks like there can be infinitely many. We shall try proving such a statement .. and what if we look for 4 such squares?
let a², b², and c² be in arithmetic progression so that b²-a²=c²-b²,
or 1-(a/b)²= (c/b)²-1.
For convenience we write x=a/b, y=c/b.
This implies 1-x²= y²-1. Factorising we get
(1+x)/(y+1)=(y-1)/(1-x).........(1)
let t be the common value of (1) which gives us
1+x=t(y+1)
AND y-1=t(1-x)
OR
x-ty=t-1.................[2]
tx+y=t+1..............[3]
solving for x and y we get the following parametrization
x=(t²+2t-1)/(t²+1)
y=(-t²+2t+1)/(t²+1).
For t=1/2 we get x=1/5, y=7/5 so that we get 1, 1/25 and 49/25. Clearing denominators we get 1, 25 and 49...the triple we started with.
Similarly taking some other value for 't' we can arrive at other triples of squares that form an A.P. For instance t=2/3 leads us to 49, 169 and 289..the common difference being 120. The parametrization we have arrived at implies that we can get an infinitude of such triples!!.

Next we ask the question if there are 4 such square numbers which form an A.P. This question was asked by Pierre de Fermat and the answer is NO. One cannot get hold of 4 squares satisfying such a property. The proof for this assertion was first given by Leonhard Euler!!

ADAPTED FROM THE BOOK-Numbers and the Beginning of Algebra By
Shailesh Shirali [Little Mathematical Treasures]

This title is 'lifted' from a very famous book by Hilbert and Cohn-Vossen and here I am reposting a write-up that I had written several years ago in the context of a workshop meant for Math-Teachers. The monumental work of Euclid “Elements” definitely laid a kind of a foundation for the study of geometry (and Mathematics in general) But Euclid would not have imagined the plane E^2 or the space E^3 as a set of ordered pairs and ordered triples. The notion of Cartesian (rectangular) coordinates was put forth only in the 17th century by Descartes (and independently by Fermat).

We shall see the contributions of Euclid to Mathematics and then discuss some contemperory issues. Euclid and his disciples came up with an axiomatic set-up to describe basic geometric attributes like lines, planes, triangles, points etc. Thus certain basic definitions were made and a few axioms were made that were intuitively acceptable. Theorems and propositions were proved starting from the defined terms and axioms. For instance high school geometry teaches us that the sum of angles of a triangle(drawn on a plane) equals to two right angles(180 degrees).

I take a digression here to emphasize on a certain development of Mathematical thinking: Euclid has shown us the fertile world of Imagination both geometrically as well as logically. Here I shall dwell upon the logical part. The property of triangles alluded to earlier is “deduced” by certain properties like congruences observed when a line is drawn transversely to a pair of parallel straight lines. This aspect of deduction is all pervading through mathematics. Let us see some general examples of inference. Suppose Aristotle observes on several occasions that birds are generally hatched through eggs. The scientific reasoning of the philosopher aristotle would make him believe that all birds come up by way of eggs. But this statement is rendered false the moment he observes some kind of a bird that is born in another way. (I do not know if such a bird exists,..this is just for argument sake). This kind of inference from a frequent observation is called Inductive logic and the reasoning established in this way is only probabilistically measured. The deductive logic on the other hand is stronger in the sense that as long as the basic axioms are true the deduced property/phenomena has to be true. For example If Virendra Sehwag is batting in the last two balls of the last over and the score reads 275.... and if we know that the final score is 283, no sixer was hit and no extras were given(including overthrow runs), then the only possibility we deduce is that the two balls went for two boundaries!!!. [THE BIRD EXAMPLE IS FROM The Math Explorer-by J.H.Weaver Universities Press].

Euclid's work did not involve coordinates as he did not need them. However the technological advances from 15th century onwards especially Newtonian Mechanics required precise measurement for instantaneous calculations. As far as Euclid's geometry (Note that what Mathematicians call “Euclidean geometry” is different) is concerned, mundane calculations like area of closed curves, surfaces and volumes of certain surfaces like pyramids etc all of them could be done easily. However the thoughts of the renaissance era required study of dynamics involving time, like the motion of stars, predicting the time of eclipses etc. Hence the gift of Cartesian geometry became indispensable. At this point I quote Peter Doyle (of the University of Illinois, Urbana Champaign) “The spirit of Mathematics is not captured by spending 3 hours of solving 20 look alike homework problems. Mathematics is thinking, comparing, analyzing, inventing and understanding. The main point is not quantity or speed, the main point is quality of thought....reach a more complete and a better understanding...”

Marriage between Algebra and geometry: The introduction of Cartesian coordinates lead to a sort of marriage between algebra and geometry. Geometry on one hand uses the spatial sense of the brain say while thinking about navigation, painting symmetrical designs , thinking aesthetically for interior designs and architecture etc. On the other hand proving theorems in geometry requires the verbal function of the brain where one analyzes logically the properties that ought to be satisfied by the shapes in consideration. Remarkably geometers invented a way to look at geometry without ever drawing any figure. For instance we may imagine a person chewing gum and then assume that we can establish a relation between the amount of gum he/she chews and the linear distance traveled by him. Let us say this relation is given by the
equation y²=5x-x³. Now irrespective of the geometric shape of this function, one still analyses the relation. This is true as the geometer in this case analyses only the functions involved theire-in.
Analytic geometry is a more appropriate word since we have several coordinates other than Cartesian coordinates like the spherical ones, polar coordinates etc. In fact an attempt to understand geometry in a coordinate free form lead to generalizations like Riemannian geometry. Here Mathematicians wanted to freely (and smoothly) traverse between one coordinate system to another. To conclude this piece, we can look at other generalizations like the plane and space (R² and R³) will take us further into an n-dimensional euclidean space. One can even go on to replace the real number system by the complex numbers to obtain the space Cⁿ. Algebraic geometers use appropriate rings in place of R and C(These are also rings !) to define affine spaces on which geometry is done.

Computer Science has definitely taken the drivers seat changing in many ways our lifestyles and preferences. In this post we would be glancing through some historical milestones especially the technical ones involved. The computer and the Internet undoubtedly have revolutionized our planet and very few people know how this revolution happened. The Internet has facilitated information exchange and resulted in efficient management of e-commerce and thus a greater consumer choice. The efficient usage of search engines has been a big game changer. Also the social fabric of our society has seen a sea change with the advent of the so called ‘social media’. One must note this accelerating rate of acceptance of ICT –Information and communication Technology.

To begin with the Internet in its primitive avataar was built by a partnership among three groups: the military, universities and private corporation. What made the process even more fascinating was that this was not merely a loose-knot consortium with each group pursuing its own local goals. They foresaw the gigantic role this tool would play in the future. From the academic stand point one should look at the collaboration of people playing with symbolic logic, people playing with electronic devices and people involved in pure Math. First Boolean algebra was conceptualized, which converts numerical 1 and 0 to true and false, and with mathematical logic one could build a system that could process logic, instructions, and mathematics for calculations and decisions. The core concepts of the electronic computer – Central Processing Unit, a set of circuits that can perform Boolean logic and mathematics, a rapid memory, data storage, and the idea of a program converting data, all was developed by Von Neumann during the Manhattan project.

Along with Bell , Morse , Babbage and Ada Byron, one of the most important names in the history of computing is arguably John von Neumann, a Hungarian-American polymath and Manhattan Project veteran. Von Neumann joined Princeton’s Institute for Advanced Study (IAS) in 1933, the same year as his mentor, Albert Einstein. Like many of those initially hired by IAS, Von Neumann was a mathematician by training.

On any list of its inventions, the most notable is probably the transistor, invented in the 1940’s, which is now the building block of all digital products and contemporary life. These tiny devices can accomplish a multitude of tasks. The most basic is the amplification of an electric signal. But with small bursts of electricity, transistors can be switched on and off, and effectively be made to represent a “bit” of information, which is digitally expressed as a 1 or 0. Billions of transistors now reside on the chips that power our phones and computers.

One of the earliest digital computers was brought into usage on February 14^th, 1946, when the University of Pennsylvania announced the “Electronic Numerical Integrator and Computer”: ENIAC. Constructed at the Moore School of Electrical Engineering, ENIAC was built for the purpose of calculating artillery-firing tables, which provided information to help artillerymen aim their weapons. ENIAC weighed more than 60,000 pounds, covered 1800 square feet of area, consumed 150 kilowatts of power, and cost $500,000 to build (about $6,000,000 in today’s dollars) Roughly after 20 years of invention of the digital computer the Internet was first invented for military purposes, and then expanded to the purpose of communication among scientists.

Two organisations played a pivotal role. NSF-the National Science Foundation of the United States and DARPA (Defense Advanced Research Projects Agency). Researchers began to realize that an interconnected network of computers could provide services that transcended the capabilities of a single system. At this time, computers were becoming increasingly powerful, and a number of scientists were beginning to consider applications that went far beyond simple numerical calculation. Perhaps the most important e arly description of these opportunities was presented by J.C.R. Licklider (1960), who predicted that, within a few years, computers would become sufficiently powerful to cooperate with humans in solving scientific and technical problems. Licklider, a psychologist at the Massachusetts Institute of Technology (MIT), would begin realizing his vision when he became director of the Information Processing Techniques Office (IPTO) at the Advanced Research Projects Agency (ARPA) in 1962. Licklider remained at ARPA until 1964 (and returned for a second tour in 1974-1975), and he convinced his successors, Ivan Sutherland and Robert Taylor, of the importance of attacking difficult, long-term problems.

Most of this early networking research concentrated on packet switching, a technique of breaking up a written script(text matter) into small, independent units, each of which carries the address of its destination and is routed through the network independently. Specialized computers at the branching points in the network can vary the route taken by packets on a moment-to-moment basis in response to network congestion or link failure. One of the earliest pioneers of packet switching was Paul Baran of the RAND Corporation, who was interested in methods of organizing networks to withstand nuclear attack. Baran proposed a richly interconnected set of network nodes, with no centralized control system—both properties of today’s Internet. Similar work was under way in the United Kingdom, where Donald Davies and Roger Scantlebury of the National Physical Laboratory (NPL) coined the term “packet.” The United States already had an extensive communications network, the public switched telephone network (PSTN), in which digital switches and transmission lines were deployed as early as 1962. At the first Association for Computing Machinery (ACM) Symposium on Operating System Principles in 1967, Lawrence Roberts, then an IPTO program manager, presented an initial design for the packet-switched network that was to become the ARPANET. There was the concept of a hybrid network involving analog and digital devices.


The first innovation of data communication came with the file transfer protocol developed at MIT by Abhay Bhushan. His protocol enabled a user on one system to connect to another system for the purpose of either sending or retrieving a particular file. With FTP, users could now move files to their own machines and work with them as local files. This capability led to several new areas of activity, including distributed client-server computing and network-connected file systems. The first e-mail program was developed in 1972 by Ray Tomlinson of BBN. Telnet, FTP, and e-mail were examples of the leverage that research typically provided in early network development. As each new capability was added, the efficiency and speed with which knowledge could be disseminated dramatically improved. E-mail and FTP made it possible for geographically distributed researchers to collaborate and share results much more effectively…and the rest as they say is history!

References
Innovation and the Bell-Labs Miracle, NY Times Funding a Revolution: Government support for Computing Research, Ch-7. National Academic Press