Dusa McDuff

Dusa McDuff FRS
Umumiy maʼlumotlar
Tavalludi	18-oktyabr 1945 London
Qardosh loyihalar
Vikipediyadagi maqola Vikimaʼlumotlar sahifasi Vikiombordagi tasvirlar

Dusa McDuff FRS (1945-yil 18-oktyabrda tugʻilgan) — ingliz matematigi, simplektik geometriya boʻyicha tadqiqotlari bilan tanilgan.

Iqtiboslar

Soʻnggi bir necha yil ichida simplektik geometriya sohasida bir qator hayajonli oʻzgarishlar yuz berdi va koʻplab muhim, shu paytgacha erishib boʻlmaydigan muammolarni hal qilish sari ilk qadamlar qoʻyildi. Buni amalga oshirishga imkon bergan yangi usullar ham variatsion hisobdan, ham elliptik xususiy hosilali differensial operatorlar nazariyasidan kirib keldi. Ushbu maqolada Gromov elliptik usullardan foydalangan holda erishgan baʼzi natijalar tavsiflanadi, soʻngra Floer ushbu elliptik texnikalarni Morse nazariyasiga yangicha yondashuvni ishlab chiqishda qanday qoʻllaganligi koʻrsatiladi; bu yondashuv simplektik geometriya bilan bir qatorda 3 va 4 oʻlchamli koʻpxilliklar nazariyasida ham muhim amaliy ahamiyatga egadir.

The past few years have seen several exciting developments in the field of symplectic geometry, and a beginning has been made towards solving many important and hitherto inaccessible problems. The new techniques which have made this possible have come both from the calculus of variations and from the theory of elliptic partial differential operators. This paper describes some of the results that Gromov obtained using elliptic methods, and then shows how Floer applied these elliptic techniques to develop a new approach to Morse theory, which has important applications in the theory of 3- and 4-manifolds as well as in symplectic geometry.

Simplektik geometriya — bu yopiq qiyshiq-simmetrik formaning geometriyasidir. U bizga tanish boʻlgan Riman geometriyasidan tubdan farq qiladi. Muhim farqlardan biri shundaki, uning barcha tushunchalari dastlab silliq kategoriyada (masalan, differensial formalar orqali) ifodalangan boʻlsa-da, qaysidir ichki mohiyatiga koʻra ular hosilalarni oʻz ichiga olmaydi. Shunday qilib, simplektik geometriya tabiatan asosan topologik xarakterga ega. Darhaqiqat, koʻpincha simplektik topologiya haqida gapiriladi. Yana bir muhim jihati shundaki, u haqiqiy egri chiziqlarning uzunligini emas, balki kompleks egri chiziqlarning yuzasini oʻlchaydigan 2 oʻlchamli geometriyadir.

Symplectic geometry is the geometry of a closed skew-symmetric form. It turns out to be very different from the Riemannian geometry with which we are familiar. One important difference is that, although all its concepts are initially expressed in the smooth category (for example, in terms of differential forms), in some intrinsic way they do not involve derivatives. Thus symplectic geometry is essentially topological in nature. Indeed, one often talks about symplectic topology. Another important feature is that it is a 2-dimensional geometry that measures the area of complex curves instead of the length of real curves.

Soʻnggi 15 yil davomida simpektik geometriya oʻziga xos yoʻnalish sifatida shakllandi va endilikda matematikaning boy va mazmunli sohasi boʻlgan anʼanaviy Riman geometriyasi bilan teng darajada turadi. Asosiy taʼriflar matematik nuqtai nazardan juda tabiiy: bunda simmetrik emas, balki qiyshiq-simmetrik bilinear shakl ω geometriyasi oʻrganiladi. Biroq, simmetriyaning bu goʻyoki sodda oʻzgarishi tub oqibatlarga olib keladi. Masalan, bir oʻlchamli oʻlchovlar yoʻqoladi, chunki qiyshiq-simmetriya tufayli ω(v, v) = −ω(v, v) boʻladi. Nazariya ikki qiyofaga ega. Simpektik koʻpmanifoldlarda geometrik ostobyektlarning ikki turi mavjud: dinamik qurilmalarda uchraydigan gipersirtlar va Lagranj ostkoʻpmanifoldlari hamda Riman va kompleks geometriya bilan chambarchas bogʻliq boʻlgan juft oʻlchamli simpektik ostkoʻpmanifoldlar. Koʻramizki, simpektik koʻpmanifolddagi geodeziyaning analogi — J-golomorf egri chiziq deb ataluvchi ikki oʻlchamli sirt hisoblanadi.

Over the past 15 years symplectic geometry has developed its own identity, and can now stand alongside traditional Riemannian geometry as a rich and meaningful part of mathematics. The basic definitions are very natural from a mathematical point of view: one studies the geometry of a skew-symmetric bilinear form ω rather than a symmetric one. However, this seemingly innocent change of symmetry has radical effects. For example, one dimensional measurements vanish since ω(v, v) = −ω(v, v) by skew-symmetry. The theory has two faces. There are two kinds of geometric subobjects in a symplectic manifolds, hypersurfaces and Lagrangian submanifolds that appear in dynamical constructions, and even-dimensional symplectic submanifolds that are closely related to Riemannian and complex geometry. As we shall see, the analog of a geodesic in a symplectic manifold is a two-dimensional surface called a J-holomorphic curve.

… Gelfand matematika haqida goʻyo u sheʼriyatdek soʻzlab, meni hayratga soldi. U menga fon Neyman nimani amalga oshirishga harakat qilganini va uning ishlarining ortida qanday gʻoyalar yotganini tushuntirishga urindi. Bu men uchun haqiqiy kashfiyot boʻldi — matematika haqida shunday tarzda gapirish mumkin ekan. U shunchaki mavhum va chiroyli qurilma emas, balki ayrim asosiy hodisalarni anglashga boʻlgan intilishdan kelib chiqadi; bu hodisalar qandaydir gʻoya yoki nazariya orqali ifodalanishga harakat qilinadi. Agar uni bir yoʻl bilan aniq ifodalab boʻlmasa, boshqa yoʻl sinab koʻriladi. Agar u ham ish bermasa, butunlay boshqacha yondashuv orqali oldinga siljishga uruniladi. Gʻoyalar va savollardan iborat chuqur, uzluksiz ichki oqim mavjud.

… Gelfand amazed me by talking of mathematics as if it were poetry. He tried to explain to me what von Neumann had been trying to do and what the ideas were behind his work. That was a revelation for me — that one could talk about mathematics that way. It is not just some abstract and beautiful construction but is driven by the attempt to understand certain basic phenomena that one tries to capture in some idea or theory. If you canʼt quite express it one way, you try another. If that doesnʼt quite work, you try to get further by some completely different approach. There is a whole undercurrent of ideas and questions.

My Life (1998)

	Yaqinda ikki xil holatda (erkak) matematiklar mendan butunlay begʻaraz ohangda shunday deb soʻrashdi: „Ammo siz, albatta, hech qachon kamsitishga duch kelmagansiz-ku?“
	On two different occasions recently, (male) mathematicians asked me in all innocence: But you surely never suffered any discrimination?

Men hech qachon kamsitishga uchradimmi? Kamsitishning ikki turi mavjud: aniq (explicit) va noaniq (implicit). Koʻpincha, aniq kamsitish menga katta taʼsir koʻrsatmagan. Biroq, retrospektiv nazarda, noaniq kamsitish — masalan, men postdoktor sifatida kollej hayotida qatnasha olmaganim uchun juda izolyatsiyalangan boʻlishim — hamda oʻzimdagi ichki misoginiya sezilarli taʼsir koʻrsatgan, garchi oʻsha paytda buni deyarli sezmagan boʻlsam ham. Yana bir muhim omil, va u men uchun sezilarli edi, keng tarqalgan, lekin ochiq koʻrinmagan: Britaniyada oʻsha paytda ayollar professional olim boʻlishi juda kam uchrardi, asosan ilm-fan (ayniqsa „aniq“ fanlar, gumanitar fanlariga nisbatan) juda nofemina ish deb hisoblanganligi sababli. … Bugungi kunda, ayollarning matematika sohasida ishtirokiga boʻlgan koʻpchilik sezilarli toʻsiqlar olib tashlangan boʻlsa ham, ichki, juda kuchli va noaniq toʻsiqlar hali ham mavjud, masalan, stereotip xavfi yoki imposter sindromi kabi hodisalarda namoyon boʻladi. Odamlar Cambridgeda ayollar diplom olmasligi (ular birinchi marta 1948-yilda olgan)ni butunlay normal deb qabul qilishga olib keladigan qarashlar juda kuchli va tashqi toʻsiq olib tashlangan zahotiyoq yoʻqolmaydi. … 1960-yillarda, albatta, akademik hayot ayollar uchun emas degan fikrning juda koʻrinadigan namoyon boʻlishlari mavjud edi. Oʻsha paytda, AQSHdagi koʻpchilik Ivy League universitetlari ayollarni qabul qilmasdi, va Britaniyada eng obroʻli universitetlardagi (Oxford va Cambridge) deyarli barcha kollejlar faqat bir jinsga mansub edi.

Was I ever discriminated against? There are two kinds of discrimination: explicit and implicit. For the most part, explicit discrimination did not affect me much. However, in retrospect, implicit discrimination—for example, the fact that I was so isolated as a postdoc because I could not share in college life—as well as my own internalized misogyny, did have a significant effect, though I hardly noticed this at the time. Another important factor, and one that I was aware of, was pervasive but not overt: it was very rare that women became professional scientists in Britain at the time, largely because science (and particularly “hard” as opposed to “life” science) was considered such a very unfeminine thing to do. ... These days, when most of the obvious barriers to womenʼs participation in mathematics have been removed, there still remain very strong and insidious internal barriers, shown in such phenomena as stereotype threat or imposter syndrome. The prejudices that lead to people accepting as completely normal that women should not get degrees at Cambridge (they first could get Cambridge degrees in 1948) are very strong and do not disappear immediately when the external barrier is removed. ... In the 1960s there were, of course, very visible manifestations of the idea that academic life is not for women. At the time, most Ivy League universities in the States did not admit women, and in Britain almost all the colleges at the most prestigious universities (Oxford and Cambridge) were single sex.