2023 January


18 Jan 2023: 10 am - 12 noon

Speaker:

松木 謙二 氏（パデュー大学）

Kenji Matsuki (Purdue University)

Title (first part):

爆発の操作から眺めた双有理幾何の景色
Sceneries of “Birational Geometry” from the view point of “Blowing Up”

Abstract (first part):

私個人の趣味としては、難しい話、特に題名からして技術的な用語が散りばめられていると、最初から訳が分からないので敬遠してしまいます。と言えば聞こえはいいのですが、もっと正直に言うと、能力がないので時間の無駄になりそうだ、と言い訳をして参加を躊躇ってしまいます。しかしながら、こういう偏屈な態度は、自分の視野を広げるのを妨げるだけでなく、周り回って、自分の専門の分野の“爆発”そして“双有理幾何”という非常に技術的な言葉が散りばめられた話をなんとか一般の聴衆の方に理解してもらう、と言う輪廻に追い込んでしまいました。しかし、自分が研究している時の興奮と対象の美しさは本物で、それを伝えるためにベストを尽くしたいと思います。

(入門) 代数幾何という数学の分野は、 その文字通り方程式で与えられた(代数)図形(幾何)を研究することを目的としています。y = x^2という方程式で与えられた図形が綺麗な放物線を描くのを習うのは、中学生の時に遡ります。 よく観てみると、 方程式の解が、 媒介変数表示x = t, y = t^2で与えられることも容易にわかります。一方、y^2 = x^3という方程式で与えられた図形は、原点で尖っていて放物線のように綺麗ではなく特異点を持っています。方程式の解は、今度は媒介変数表示x = t^2, y = t^3で与えられています。一見関係がなさそうに見える二つの図形ですが、実は尖点にダイナマイトを仕掛け“爆発”させてやると、綺麗な放物線に化けるのです。

実際の私の話では、この“爆発”の操作をムービーでお見せしたいと思っています。

少し高尚な言い方をすると、放物線には環C[t, t^2] = C[t]が対応し、尖点には環C[t^2, t^3]が対応しています。後者には、tが欠けているので同じ物ではありませんが、割り算を許して体にすると、両方ともC(t)と一変数の関数体になり、同じものになるという現象が観察されるのです。

図形の観点からは、尖点と放物線のようにその大分(Zariski位相による開集合)では一致しているもの、代数の観点からは、環としては違っていても同じ関数体を持つようなもの、を研究するのが“双有理幾何”であると言えます。

(双有理幾何のさまざまな景色) 私の話では、数々の双有理幾何の景色のなかで、次の五つのトピックに光を当てたいと思います。

実は、個々に独立したトピックのように見えますが、爆発の操作を中心に、お互いが有機的に絡み合っています。そんな面白さを伝えられれば、というのが最終的な願いにも似た目標です。


If you ask for my personal taste, I do not like a talk whose title is peppered with some difficult technical terms. I am already lost even before the talk starts. I make a lame excuse to myself, saying “It is probably way beyond my limited capacity, and I will waste my time not getting anything out of it.” I end up missing the talk. I don’t know how many times this negative attitude has costed me a precious opportunity to broaden my horizon. The karma, as the destiny dictates itself, forces me into the situation where I myself have to give a talk whose title is peppered with some difficult technical terms like “Birational Geometry” and “Blowing Up”, and where my job is to make myself understood by the general audience. Such as life ! However, the excitement I feel when I immerse myself into mathematics is genuine, and so is the beauty of the objects I study. I would like to do my best to convey the very excitement and beauty.

(Introduction) “Algebraic Geometry”, as the name indicates, is the subject in mathematics which studies the figures (“Geometry” part) defined by a bunch of equations (“Algebra(ic)” part). You may recall vividly that the equation y = x^2 defines a beautiful parabola from the time you were a

middle school student. Looking closer, you may also observe that the solution to y = x^2 has a parametric expression x = t, y = t^2. On the other hand, the figure, defined by the equation y^2 = x^3, is not as beautiful as the parabola, having a rugged point at the origin called a cusp singularity. The solution to y^2 = x^3 has a parametric expression x = t2, y = t3 this time. Even though these two figures seem to be unrelated, in fact, if we set a “dynamite” and apply the operation of “Blowing Up” to the cusp singularity, then it will transform into the parabola.

I would like to show a movie to demonstrate this transformation in my talk.

In a more sophisticated terminology, we say that the ring C[t, t^2] = C[t] corresponds to the parabola, whereas the ring C[t^2, t^3] corresponds to the cusp. The latter ring, missing t, is not the same ring as the former. However, if we allow the division, then they become the same field C(t) of fractions, the field of rational functions of one variable.

“Birational Geometry” in short, is the study of the relation between two figures, like the parabola and the cusp, which are the same for the most part (more precisely, isomorphic over the dense open subset w.r.t. the Zariski topology) from a geometric point of view. From an algebraic point of view, it is the study of the relation between two rings, not isomorphic to each other, which nevertheless share the same field of fractions.

(Various Sceneries of Birational Geometry)

In my talk I would like to shed light on the following five topics:

1. Resolution of singularities

2. Mori Theory

3. Factorization of birational maps

4. Toroidalization

5. Toward the theory in positive characteristic

The above five seem to be independent from each other at first sight. They interact with each other, however, being intertwined closely in a systematic way. This systematic interaction is what has captivated my heart mathematically for more than 40 years. The ultimate goal of my talk is to convey the beating of my heart !

Title (second part):

“好きだから続けられた”数学人生の綱渡り
I could stay on the tight rope of my mathematical career, only because “I like mathematics.”

Abstract (second part):

こういう所で若い方達に話をする際には、これこれこうすると良い、という先輩としての知恵を授けるというのが常ですが、私にはそれが見当たりません。逆に、こんな無茶はしないほうが良い、という反面教師にはなれるかもしれない、人生の綱渡りの連続です。東大の学部生の時には、ヨット部で１年間１５０日江ノ島の海で合宿していました。あまりに日焼けしていたので、数学の図書の方に、“ここは数学科の方しか使えません。’’と呼び止められたのを鮮明に覚えています。大学院入試の面接も“君は取りませんが、それでも試験を受けますか？”というのが始まりの言葉でした。そんな、“落ちこぼれ”の私を救って育ててくださったのが、飯高茂先生、川又雄二郎先生です。また、将来に関する展望が皆無の状態で(数学以外の動機から)アメリカに渡った際に助けて頂いた森重文先生は真の恩人です。そんな私ですから、今日、こうして数学を生業として生活していかれること自体が幸せで夢のような毎日です。綱渡りといっても、実は何度も落下していて、その度に数々の先生方、共同研究者の方々、友人に元の綱に戻してもらったというのが本当です。ただ、“数学が好きなのでそれを続けていきたい。”という思いは、一貫していました。個人的に好きな数学と、職業としての数学の葛藤。近年、癌の闘病を通して“数学が好き”という言葉も、別の重みを持って来ました。就職難の中、次の研究職が見つけられるかどうか不安でならない、という方に、こんな奴でも“数学が好き”というだけで生きて来られた、という例があるんだと 。。。そんな話ができれば幸いです。

It is usually the case that, if one is given a chance to talk to the younger people, he would present some words of wisdom, earned through his life experience, so that they can serve as the guiding light for the younger generations to follow. In my case, however, I cannot find any such words. Why ? Because, looking back, I can firmly say that my life has been a sequence of reckless and foolhardy actions. I have been walking on a tight rope for my life. How can I find words of wisdom to follow ? The only thing I may be able to do is to become a bad example myself so that the younger generations will avoid falling into the same pitfalls. When I was an undergraduate student at Tokyo University, I spent 150 days in a year on the Pacific ocean sailing. (I was a member of the sailing club.) I was so sun-tanned that one day when I tried to go into the library of the mathematics department, a librarian stopped me, saying “Hey, you ! This library can only be used by a math. student !” My oral exam into a graduate school started with the chief examiner proclaiming “We are not going to pass you. Would you still dare to take the test ?”

At the time the word “Ochi-Kobore” was prevalent in the Japanese society. It referred to a failing student or a person, who could not be saved by the safety net and went through it. It was Prof. Shigeru Iitaka and Prof. Yujiro Kawamata who reached their hands out to an “Ochi-Kobore” like me, and saved my mathematical career. I came to the United States without any concrete mathematical plan or prospective into the future. I owe everything to Prof. Shigefumi Mori for what I am today and how happy I am, living in the U.S. and doing mathematics as my profession. Even though I said my life had been like walking on a tight rope, the truth is that I fell from the rope so many times. Every time I fell, however, it was my teachers, co-workers, and friends who put me back onto the rope.

Maybe if there is one word I can say to the younger people, it is “I wanted to continue doing mathematics, because I liked it throughout.”

Yes, I did experience some conflict between the mathematics I like to do personally and the mathematics I have to do as a profession. These days, through the battle against the cancer, the very word “I like mathematics.” carries a different weight to me. I am sure there are many younger people who would feel insecure when they worry about the possibility of finding a next job/a next research position. Through the second part of this colloquium, some of them may start whispering in their minds “A moron like Matsuki could survive through the career only because he liked mathematics ? Then why not me ?” ... If that happens, then it would make me more than happy.

Discussion theme (second part):

変化への挑戦と支援
Challenges and supports for "a change"

Feedback from participants:

(松木 謙二1)Catch-allTalk-min.pdf
(松木 謙二2)Catch-all2ndPart.pdf