Japan Society for Promotion of Science (JSPS) Core-to-Core Program (2006 April-)

"New Developments of Arithmetic Geometry, Motive, Galois Theory, and Their Practical Applications"

Purpose of this program (explanation for non-specialists)

• Develop the advanced modern mathematics
• Search classical/contemporary mathematics for practical applications, such as cryptography and randomnumber generations, and implement them in computer programs for public use
• Feedback the requirements from practical applications.

Idea

Mathematics has two aspects:
1. Most beatiful research area, studied purely theoretical interests
2. Most powerful tool for practical applications.

Issac Newton stated in the Introduction of the famous "Principia" (1687), where he resolved the law of gravity, mechanics, and differentiation/integration: "From these laws proved in this book, we see the following. If, we throw a thing with a higher speed, then the reaching point is farther. And then, beyond a certain speed, the thing will not land again, and circulating along an elliptic orbit. Moreover, beyond another certain speed, the thing is going farther and farther away from the earth, and moves along an orbit that never returns to the original point."

As a clear answer to the strange behaviors of the planets, Newton stated that if the laws of mechanics and gravity hold, then elliptic orbit of the planets, comets, satelites of Jupiter, tide of the sea, all these can be explained in a unified manner. The proof is to solve the differential equation (using theorems of plain geometry), that is a pure mathematics.

This is a well-cited example showing the power of mathemtaics. However, it is not well-cited that he found the principle of artificial satellites in the introduction.

Just a few decades after the heliocentric theory accepted, Newton explicitly stated that if one can throw a thing with a sufficient speed, then the thing becomes an artificial satellite, and if one can do with a certain higer speed, then the thing will go infinitely farther away, with a mathematical proof.

This latter fact can not be certificated by any experiments nor observations, in the era of Newton. At that time, the human beings could never observe heavenly bodies that go further and further from the earth.

At that time, this statement of Newton should be "meaningless" from the pragmatic point of view. We need to wait until 20 century when the technology realized the artificial satellites, which are forcasted by Newton's mechanics. Now, the artificial satellites are of pragmatic use, such as navigation systems of automobiles or portable phones.

Thus, the power of Mathematics (over any other research areas) lies in the fact that one can PROVE a fact which can never certificated by experiments nor observations.

Moreover, with Mathematics, we can study an invisible world, the world where common senses can never reach.

If a pupil writes "1+1=0" in a school, then what the teachers will consider. They must re-educate the pupil, to correct the mistake.

However, Galois (1811-1832) studied how the matematics would be if we assumed 1+1=0. (Actually, Galois considers more general "finite fields", and these are different from the famous "Galois theory, though.)

The theory of finite fields that Galois introduces is now widely used, after more than hundred years had passed. In the digital computers, any information is expressed in terms of sequences of 1 and 0. To communicate exact information through noizy channels (Coding Theory), to protect information passing through leeky channels from adversary (cryptography), and to generate random numbers to simulate probabilistic events in computer simulations (randomnumber generation), the mathematics based on 1+1=0 is very efficiently used.

The world where 1+1=0 holds is invisible in the ordinal life. In the era of Galois, such mathematics should be "purely theoretic, non-pragmatic imaginary objects." However, nowadays, every codes and ciphers are based on Galois fields. Your CD player and portable phones are relying on the mathematics of 1+1=0.

On the other hand, the proof of Fermat's Last Theorem (open for 350 years, and in 1995 Wiles solved this using most advanced mathematics in the world) utilizes theory of unusual numbers, such as finite fields, p-adic fields, Galois groups, and do on. The advace of mathematics defeated a number of most difficult open problems, in 20 century.

Thus, we conclude:

1. Mathematics can study other worlds of truth, where the common sense can not reach.
2. Among such studies, some show unexpected effectiveness in the pragmatic applications. Such results can not substituted by any other studies.
3. Mathematics are developping rapidly, powerfully, with beatiful structures (as if they constitute the rich ecosystem), from ancient era to now.
4. Mathematics are sometimes developped self-objectively (Galois), and sometimes by the requirement from another research area (Newton).
Through these observations, the purpose of this program is:
• To develop the pure mathematics
• To mine Mathematics for practical applications
• To feedback the requirements from practical applications to the pure mathematicians.

Method

The idea above is too broad: it goes beyond the whole area of mathematics, and the result may be after decades.

Because of the limited time (basically two years from 2006 April) and budget, the program concentrates on:

• Algebraic Geometry and Number Theory, which are ones of most abstract and advanced mathematics presently. These areas and related areas are mainly studied, the researchers in these areas are exchanged internationally.
• Mining classical and advanced mathematics for practical applications, such as randomnumber generations and cryptography, and make them into practical computer prgrams.
• In the workshops, ask for lectures by researchers on practical sides, and feed back their requirements to pure mathematicians.
Through these, the program aims at
• purely theoretical results
• practical results
• establish continuous reseach-exchanges and developping core institutions
• grow researchers who see through from pure theory to pragmatic applications.

On the other hand, it is not preferable to mix everything: some pure mathematicians should study pure mathematics from pure interest (as deep as possible, neglecting practical applications), and some applied scientists should study sciences from purely applicational purpose (as pragmatic as possible, neglecting the beauty of the theory). Then, a real interesting result is obtained, when they exchange their researches.

Thus, this program puts a strong stress on the development of pure mathematics as well. Among others, this program's theme is arithmetic geometry. This area uses "geometric approaches in studying algebra and number theory," and achieved a great success such as proving Weil's Conjecture, Shimura-Taniyama Conjecture, and Fermat's Last Theorem, in 20th century. In 21st Century, this area is rapidly developping (such as theory of motives and cateogorical arithmetic geometry), with its relations with other branches of mathematics, such as homotopy theory. The title of this program refers to some keywords in this area. From practical viewpoints, this area has a great success. The (sequence of) codes which goes beyond the Gilbert-Varshamov bound is obtained from Ihara's algebraic curve. The most efficient public-key cipher is based on the group of rational points of elliptic curves over finite fields. The fastest integer-factorization algorithm, the number field sieve method, is based on the norm map from the algebraic extension field. A randomnumber generation algorithm with a period length of 2^19937-1, Mersenne Twister, is optimized by using the geometry of lattices over Laurant series field over finite fields.

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