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Jugendtraum Focus Week Institute for the Physics and Mathematics of the Universe World Premier International Research Center Initiative 数物連携宇宙研究機構 世界トップレベル研究拠点プログラム The University of Tokyo 東京大学 Feature Jugendtraum Interview with George F. Smoot Focus Week: NEWS Condensed Matter Physics Meets High Energy Physics No. 9 March 2010 IPMU NEWS CONTENTS English Japanese 3 Director’s Corner Hitoshi Murayama 23 Director’s Corner 村山 斉 Science of the Universe 宇宙を科学する 4 Feature Kyoji Saito 24 Feature 斎藤 恭司 Jugendtraum of a mathematician 一数学者の青春の夢 10 Our Team Won Sang Cho 30 Our Team 趙 元相 Johanna Knapp ヨハンナ・クナップ Takahiro Nishimichi 西道 啓博 Masaomi Tanaka 田中 雅臣 Masahito Yamazaki 山﨑 雅人 12 IPMU Interview with George F. Smoot 32 IPMU Interview ジョージ・スムート教授に聞く 16 Workshop Report 36 Workshop Report Focus Week: Condensed Matter Physics Meets フォーカスウィーク：物性と素粒子の対話 High Energy Physics 18 News 38 News 22 Temperature anisotropies in cosmic 40 宇宙マイクロ波背景放射の温度揺らぎ microwave background radiation 杉山 直 Naoshi Sugiyama Kyoji Saito is IPMU Professor and a principal investigator. He is a world-leading mathematician in complex analytic geometry. His theories on primitive forms and innite dimensional Lie algebras appear in recent theoretical physics, in particular, in topological string theory as the Landau-Ginzburg theory. He graduated from the University of Tokyo in 1967 and received a Doctorate from Göttingen University in 1971. He became an associate professor at the University of Tokyo in 1976. In 1979 he moved to RIMS (Research Institute for Mathematical Sciences), Kyoto University. At RIMS he became a professor in 1987 and served as Director in 1996 - 1998. Since 2008 he has been IPMU Professor. 斎藤恭司：東京大学IPMUの特任教授で主任研究員を兼ねる。専門は数学で、複素解析幾 何学の世界的権威。その創始した原始形式の理論や無限次元リー環の理論は近年の理論 物理学、特にトポロジカル超弦理論にランダウ－ギンズブルク理論として現れる。1967年に 東京大学理学部数学科卒業、1971年にドイツのゲッティンゲン大学で博士の学位を取得。 1976年に東京大学助教授、1979年に京都大学数理解析研究所助教授、1987年に同教授、 2008年からIPMU特任教授。この間、1996年から1998年まで数理解析研究所長を務める。 Director’s Corner Director of IPMU Science of the Universe Hitoshi Murayama We had a pleasure of having a very distinguished there and he could observe them in the CMB. These visitor at IPMU as one of our members in February. tiny ripples gradually grew by attracting dark matter George Smoot is known as a man who saved the Big with gravity, eventually becoming big tsunamis to form Bang, and received the 2006 Nobel Prize in Physics. galaxies. Not only he saved the Big Bang, cosmology He stayed with us for a month, gave several talks for became science. us and our scientic neighbors, and was very active at We at IPMU wish to follow his footsteps. Studying our daily tea time leading discussions and inspiring our the Universe is no longer what Greek philosophers young members. He really liked IPMU, and wants to did; it is now a subject in science. And this pursuit is come back. You can read about his conservation with a big drama involving many people collaborating and our PI Naoshi Sugiyama in this volume. competing, making mistakes and working hard. We Back in the 80’s and early 90’s, cosmology was would like to understand the entire history of the said to be in“ crisis.” There was even a report in Time Universe by observations and experiments, and predict Magazine titled“ Bang! A Big Theory May Be Shot.” our future. We are lucky to be a part of this exciting The Universe now is lumpy, with stars, galaxies and scientic pursuit. clusters of galaxies. But we can also directly see the baby Universe because there is still light called CMB (cosmic microwave background) that came from the Big Bang. The problem was that the CMB looked exactly the same everywhere. How come that Director’s everywhere was the same in the baby Universe, but it Corner became so lumpy today? The observed structure did not agree with the Big Bang theory. George set out on a long journey to show that the baby Universe had seeds for structure. He used spy planes. He lost a balloon in a jungle. He convinced NASA to y his apparatus on a satellite called COBE. And he found them after two decades of search. The seeds were unimaginably small: like a millimeter- size ripple on a 100-meter deep ocean. But they were 3 F EATURE Principal Investigator Kyoji Saito Research Area：Mathematics Jugendtraum of a mathematician A number which appears when we count things §1 Kronecker’s Jugendtraum as one, two, three, ... is called a natural number. There is a phrase“ Kronecker’s Jugendtraum The collection of all natural numbers is denoted (dream of youth)” in mathematics. Leopold by N. When we want to prove a statement which Kronecker was a German mathematician who holds for all natural numbers, we use mathematical worked in the latter half of the 19th century. He induction as we learn in high school. It can be obtained his degree at the University of Berlin in proved by using induction that we can define 1845 when he was 22 years old, and after that, he addition and multiplication for elements of N (that successfully managed a bank and a farm left by his is, natural numbers) and obtain again an element of deceased uncle. When he was around 30, he came N as a result. But we cannot carry out subtraction back to mathematics with the study of algebraic in it. For example, 2－3 is not a natural number equations because he could not give up his love for anymore. Subtraction is defined for the system of mathematics. Kronecker’s Jugendtraum refers to a numbers ..., －3, －2, －1, 0, 1, 2, 3, .... We call such a series of conjectures in mathematics he had in those number an integer and denote by Z the collection days ̶ maybe more vague dreams of his, rather of integers. For Z, we have addition, subtraction, and than conjectures ̶ on subjects where the theories multiplication, but still cannot carry out division. For of algebraic equations and of elliptic functions example, －2/3 is not an integer. A number which intersect exquisitely. In the present note, I will explain is expressed as a ratio p/q of two integers (q≠0) is the dream itself and then how it is connected with called a rational number (in particular an integer my dream of the present time. is a rational number) and the collection of them is denoted by Q. Rational numbers form a system of §2 Natural numbers N, integers Z and numbers for which we have addition, subtraction, rational numbers Q multiplication, and division.＊1 Such a system of Let us review systems of numbers for explaining numbers is called a field in mathematics. Kronecker’s dream. Some technical terms and We ask whether we can measure the universe by symbols used in mathematics will appear in the rational numbers. The answer is“ no” since they still sequel and I will give some comments on them, miss two type of numbers: (1) solutions of algebraic but please skip them untill §9 and §10 if you don’t equations, (2) limits of sequences. In the follwoing understand them. §3 and §4, we consider two extensions Q and Q 4 IPMU News No. 9 March 2010 of Q, and in §6, both extensions are unied in the §4 Real numbers R complex number eld C. Analysis was sarted in modern Europe by Newton (1642-1723) and Leibniz (1646-1716) and followed Ｎ Ｑ Ｎ Ｚ by Bernoulli and Euler. It introduced the concept of ＋, × √ Ｑ approximations of unknown numbers or functions Ｚ ＋, ×,－ ＊2 R = Ｑ by sequences of known numbers or functions. lim C Ｑ ＋, ×,－,÷ n→∞ At nearly the same time in modern Japanes mathematics (called Wasan), started by Seki Takakazu §3 Algebraic numbers Q (1642(?)-1708) and developed by his student Takebe It was already noticed by ancient Greeks that Katahiro (1664-1739), approximations of certain one cannot“ measure the world” only by rational inverse trigonometric functions by a power series numbers. For example, the length of the hypotenuse and that of π by series by rational numbers were of a right-angled isosceles triangle with the short also studied. Takebe wrote“ I am not so pure as Seki, edges of length 1 is denoted by √2 (Fig. 1) and so could not capture objects at once algebraically. Greeks knew that it is not a rational number. If Instead, I have done long complicated calculations.” we express √2 by the symbol x, then it satisfies We see that Takebe moved beyond the algebraic the equation x2－2= 0. In general, a polynomial world, an area of expertise of his master Seki, and equality including an unknown number x such as understood numbers and functions which one n n－1 a0x + a1x + … + an－1 x + an =0 (a0≠0, a1, ..., an can reach only by analysis (or series). Nowadays, a are known numbers called coeffcients) is called an number which“ can be approximated as precisely algebraic equation. We call a number x an algebraic as required by rational numbers” is called a real number if it satisfies an algebraic equation with number and the whole of them is denoted by R.＊3 A rational number coeffcients. The collection of number which has an infinite decimal representation algebraic numbers, including rational numbers, is (e.g. π =3.141592 ... ) is a real number and the inverse denoted by Q. It is a field since it admits addition, is also true. Thus, numbers, which we learn in school, subtraction, multiplication, and division. Moreover are real numbers. Japanese mathematicians of the we can prove that solutions of algebraic equations time had high ability to calculate such aproximations whose coeffcients are algebraic numbers are again by using abacuses, and competed with each other Feature algebraic numbers. Referring to this property, we in their skills.
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