Karen Uhlenbeck

Karen Uhlenbeck
	;
Umumiy maʼlumotlar
Tavalludi	24-avgust 1942; Klivlend
Qardosh loyihalar
	Vikipediyadagi maqola Vikimaʼlumotlar sahifasi Vikiombordagi tasvirlar

Karen Keskulla Uhlenbeck (1942-yil 24-avgust) — amerikalik matematik va zamonaviy geometrik tahlilning asoschisi. U Texas Austin universitetida matematika boʻyicha faxriy professor boʻlib, u yerda Sid W. Richardson Foundation regentlari raislik lavozimini egallagan. Hozirda u Ilgʻor tadqiqotlar institutida faxriy tashrif buyuruvchi professor va Prinston universitetida tashrif buyuruvchi katta ilmiy xodim sifatida faoliyat yuritmoqda. U 2019-yilgi Abel mukofotiga sazovor boʻlgan.

Iqtiboslar

Kompakt koʻpobrazlilikdagi Laplas operatorining xos funksiyalari haqidagi eng aniq maʼlumotlar yuqori darajadagi simmetriya mavjud boʻlgan hisob-kitoblardan kelib chiqadi. Bunday holatlarda xos fazolar katta oʻlchamli boʻlishi mumkin, xos funksiyalarning nollari koʻpincha kritik nuqtalar hisoblanadi va xos funksiyalar odatda degenerat kritik nuqtalarga ega boʻladi. Biroq, bu xususiyatlarning barchasi metrikadagi kichik perturbatsiyalarga (oʻzgarishlarga) nisbatan beqarordir va shuning uchun kishining intuitsiyasini chalgʻitishi mumkin^[1].

Most explicit information on the eigenfunctions of a Laplace operator on a compact manifold comes from computations where a high degree of symmetry is present. In these cases, eigenspaces may be of large dimension, the zeros of the eigenfunctions are often critical points, and the eigenfunctions usually have degenerate critical points. However, these properties are all unstable under small perturbations of the metric, and are therefore rather misleading to one's intuition.

Oxirgi bir necha yil ichida kvant maydon nazariyasidagi kalibrlangan maydonlar (gauge theories) ustidagi tadqiqotlar nolineer elliptik differensial tenglamalarga oid baʼzi qiziqarli muammolarni keltirib chiqardi. Shunday muammolardan biri Evklid 4-oʻlchamli fazosidagi Yang-Mills maydonlarining lokal harakatidir... Bizning asosiy natijamiz lokal regulyarlik teoremasidir: 4-oʻlchamli koʻpobrazlilikdagi chekli energiyaga ega Yang-Mills maydoni yakkalangan maxsusliklarga (singularity) ega boʻlishi mumkin emas. Koʻrinib turgan nuqtaviy maxsusliklar kalibrlash almashtirishi (gauge transformation) orqali yoʻq qilinishi mumkin. Xususan, R⁴ fazosidagi bogʻlam ustida chekli energiyaga ega boʻlgan Yang-Mills maydoni R⁴ $\cup$ {∞} = S⁴. (4-oʻlchamli sfera) ustidagi silliq maydongacha davom ettirilishi mumkin^[2].

In the last several years, the study of gauge theories in quantum field theory has led to some interesting problems in nonlinear elliptic differential equations. One such problem is the local behavior of Yang-Mills fields ... over Euclidean 4-space. Our main result is a local regularity theorem: A Yang-Mills field with finite energy over a 4-manifold cannot have isolated singularities. Apparent point singularities (including singularities in the bundle) can be removed by a gauge transformation. In particular, a Yang-Mills field for a bundle over R⁴ which has finite energy may be extended to a smooth field over R⁴ $\cup$ {∞} = S⁴.

Qanday qilib kalibrlash nazariyasi (gauge theory) bor-yoʻgʻi bir necha yil ichida matematikada paydo boʻldi va muvaffaqiyatga erishdi? Fundamental matematik ingredientlar tayyor edi. Qatlamli va vektorli bogʻlamlar (bundles) hamda ularning bogʻlanishlari (connections) geometrlar tomonidan har kuni ishlatilardi. Chern-Vayl nazariyasi (hatto Chern-Simons invariantlari ham) differensial geometriya boʻyicha koʻplab magistrlik kurslarida oʻrgatilar edi. De Ram kogomologiyasi va uning garmonik formalar haqidagi Xoj (Hodge) nazariyasi orqali realizatsiyasi differensial topologiyaning standart qismlari edi. Oʻtmishga nazar tashlab aytish mumkinki, Yang-Mills tenglamalari kashf etilishini kutib yotgan edi. Shunga qaramay, matematiklar ularni oʻzlari yarata olmadilar. Kalibrlash maydonlari nazariyasi — bu asrab olingan farzanddir^[3].

How did gauge theory appear and become successful in mathematics in the space of a few years? The fundamental mathematical ingredients were in place. The basics of fibre and vector bundles and their connections were in daily use by geometers. Chern-Weil theory (and even Chern-Simons invariants) were studied in most graduate courses in differential geometry. De Rham cohomology and its realization via the Hodge theory of harmonic forms were standard items in differential topology. In hindsight, the Yang-Mills equations were waiting to be discovered. Yet mathematicians were in themselves unable to create them. Gauge field theory is an adopted child.

Manbalar

↑ (1972). "Eigenfunctions of Laplace operators". Bulletin of the American Mathematical Society 78 (6): 1073–1076. ISSN 0002-9904. DOI:10.1090/S0002-9904-1972-13117-3. (quote from p. 1073)
↑ (1979). "Removable singularities in Yang-Mills fields". Bulletin of the American Mathematical Society 1 (3): 579–581. ISSN 0273-0979. DOI:10.1090/S0273-0979-1979-14632-9.
↑ „Instantons and Their Relatives“, Proceedings of the AMS Centennial Symposium, August 1988 — 467–477-bet. (quote from p. 469)

[1] (1972). "Eigenfunctions of Laplace operators". Bulletin of the American Mathematical Society 78 (6): 1073–1076. ISSN 0002-9904. DOI:10.1090/S0002-9904-1972-13117-3. (quote from p. 1073)

[2] (1979). "Removable singularities in Yang-Mills fields". Bulletin of the American Mathematical Society 1 (3): 579–581. ISSN 0273-0979. DOI:10.1090/S0273-0979-1979-14632-9.

[3] „Instantons and Their Relatives“, Proceedings of the AMS Centennial Symposium, August 1988 — 467–477-bet. (quote from p. 469)

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