Bộ môn Giải tích

04/06/2013

Bảo vệ khoá luận 2013

Filed under: Không định dạng — Thẻ:, , — doanchi @ 21:15

Hôm nay mình về Khoa để dự lễ bảo vệ khoá luận tốt nghiệp của sinh viên k54. Năm ngoái mình đã không được dự rồi, nên năm nay cũng cố gắng đến xem sao. Bộ môn Giải tích có 13 sinh viên đăng kí làm khoá luận. Tiếc là cán bộ hướng dẫn hơi tập trung (4 của thầy Châu, 3 của thầy Mậu, 3 của Chuẩn, 1 của Thu và 1 của thầy Sang). Đành rằng sinh viên tự chọn chủ đề và chọn thầy, nhưng khi mà trong bộ môn có những người không có ai làm việc cùng, lại có những người “thừa mứa” ra thì cũng nên xem lại chủ trương. Các vấn đề xoay quanh lý thuyết nửa nhóm, ứng dụng cho phương trình vi phân dưới dạng toán tử, phương trình tích phân các loại… Mình không ngồi đến cuối giờ (chắc phải 13h), nhưng chưa thấy một cái gì đó hay ho cả. Thật tiếc quá :-)

19/01/2013

Chúc mừng Quốc Anh

Filed under: Không định dạng — Thẻ:, — doanchi @ 12:13

Chúc mừng Ngô Quốc Anh đã bảo vệ thành công luận án Tiến sĩ (18/1/2013) tại NUS. Chúc mừng BMGT có thêm một Tiến sĩ. 

05/11/2012

Gần một năm

Filed under: Không định dạng — doanchi @ 16:11

Xem ra trang weblog này của BMGT không còn được mọi người quan tâm nữa. Cũng phải thôi, các bạn dều đã và đang có các mối quan tâm khác, cá nhân và tập thể, học thuật và đời sống. Đâu có thể dành thời gian cho cái hư vô này được…

03/01/2012

2011 in review

Filed under: Không toán học, Vui chơi — doanchi @ 06:16

The WordPress.com stats helper monkeys prepared a 2011 annual report for this blog.

Here’s an excerpt:

The concert hall at the Syndey Opera House holds 2,700 people. This blog was viewed about 8 800 times in 2011. If it were a concert at Sydney Opera House, it would take about 3 sold-out performances for that many people to see it.

Click here to see the complete report.

29/12/2011

Seminar liên bộ môn GT – ĐS-HH-TP, buổi thứ 2

Seminar liên bộ môn GT – ĐS-HH-TP về Giải tích trên đa tạp đã bắt đầu được 1 buổi.

Trong buổi đầu tiên, GS. NHVHưng đã thuyết trình về phép tính vi phân trên đa tạp. Những khái niệm tưởng như quen thuộc với mỗi người học giải tích cơ sở như Định lý Schwartz, phép tính vi phân cấp cao,…, khi được trình bày đối với các hàm trong các không gian định chuẩn, đòi hỏi phải được nhìn nhận thích hợp.

Buổi thứ hai sẽ được diễn ra vào 8h30, ngày 30/12/2011.

Địa điểm: 409 T3, Trường ĐHKHTN, 334 Nguyễn Trãi, Thanh Xuân, Hà Nội

Mời các bạn quan tâm tới tham dự.

22/12/2011

Seminar liên bộ môn GT – ĐS-HH-TP về GT trên đa tạp

Trong thời gian tới, Bộ môn Giải tích và Bộ môn Đại số – Hình học- Tô pô phối hợp tổ chức một seminar về “Giải tích trên đa tạp”.

Seminar dự kiến được tổ chức vào các sáng Thứ Sáu tại phòng 409 nhà T3.

Kế hoạch buổi seminar tới như sau:

Tiêu đề báo cáo: Giải tích trên đa tạp
Người trình bày: GS. TSKH Nguyễn Hữu Việt Hưng (Bộ môn Đại số – Hình Học – Tô pô )

Thời gian: 09h00, Thứ 6, ngày 23/12/2011
Địa điểm: P409, nhà T3.
Sơ lược về nội dung:
Khái niệm đa tạp là một khái niệm trung tâm của nhiều lĩnh vực trong hình học và vật lý hiện đại bởi nó cho phép ta diễn đạt và hiểu những cấu trúc phức tạp bằng những tính chất đã biết của các không gian đơn giản hơn. Ví dụ đa tạp cùng với cấu trúc khả vi cho phép ta thực hiện các phép toán vi tích phân trên nó. Một cách đơn giản ta có thể xem đa tạp khả vi như là một sự mở rộng tự nhiên của đường cong và mặt cong (đường thẳng, đường tròn là các đa tạp một chiều, mặt phẳng, mặt cầu là các đa tạp hai chiều…). Khái niệm đa tạp khả vi được sử dụng lần đầu tiên (mà không có giải thích) trong các bài giảng của Riemann vào năm 1851 và phải mất hơn một nửa thế kỷ, người ta mới đưa ra một định nghĩa chính xác cho nó…

Seminar có mục tiêu giới thiệu một số kiến thức cơ bản về giải tích trên đa tạp, cũng như các kết quả đặc sắc của môn học này ((Đa tạp, Không gian tiếp xúc, Phân thớ tiếp xúc, Trường véctơ, Đạo hàm, Vi phân, Đạo hàm cấp cao, Không gian đối tiếp xúc, Phân thớ đối tiếp xúc, Tích phân các dạng vi phân, (Vì sao phải dùng dạng vi phân? Bỏ dạng vi phân đi mà cứ nghiên cứu tích phân của các hàm trên đa tạp thì có được không? Vì sao phải định hướng đa tạp), Công thức Stockes và những ứng dụng, Đối đồng điều De Rham, vài mối liên quan sơ khởi với các ngành lân cận như Đại số tuyến tính (định thức như là tỷ số giãn nở thể tích của đồng cấu, sự có mặt của Jacobien trong công thức đổi biến tích phân, đại số ngoài…), Tôpô Đại số, Hình học hoặc Tôpô Vi phân…). Đây là kiến thức cần thiết cho việc nghiên cứu nhiều ngành khác nhau của toán học và vật lý ví dụ như Hình học vi phân, Tô pô vi phân, Phương trình vi phân, Vật lý lý thuyết….
Tài liệu: “An Introduction to Manifolds” của Loring W. Tu, “Giải tích trên đa tạp” – M. Spivak, etc.
Kính mời các thầy cô và các anh chị nghiên cứu sinh, học viên cao học quan tâm tham dự !

16/09/2011

Seminar BMGT

Filed under: Không định dạng — bmgt @ 11:25

Beginning at 9h00
Room: 409T3

16/04/2011

Tin buồn

Filed under: Không toán học — doanchi @ 20:21

Giáo sư, Tiến sĩ, Nhà giáo Ưu tú Nguyễn Thế Hoàn

Nguyên Chủ nhiệm Bộ môn Giải tích, Khoa Toán-Cơ-Tin học, Trường Đại học Khoa học Tự nhiên Hà Nội;

đã từ trần hồi 01 giờ 30 phút, ngày 17/03/2011 (tức ngày 13/2 năm Tân Mão).

Lễ viếng được tổ chức từ 07h30 đến 9h00, ngày 23 tháng 03 năm 2011,
tại nhà tang lễ Bộ quốc phòng (số 5 Trần Thánh Tông, Hà Nội).
An táng tại công viên Vĩnh Hằng, Ba Vì, Hà Nội.

08/09/2010

QS World University Rankings Results 2010

Filed under: Không toán học, Không định dạng, Tra cứu, Trao đổi — Ngô Quốc Anh @ 21:11

Overall Rankings – Top 200

Rank 2010
Rank 2009
School Name
Country
Size
Research
Focus
Score 2010
1 2 University of Cambridge United Kingdom L VH FC 100.00
2 1 Harvard University United States L VH FC 99.18
3 3 Yale University United States M VH FC 98.68
4 4 UCL (University College London) United Kingdom L VH FC 98.54
5 9 Massachusetts Institute of Technology (MIT) United States M VH CO 98.19
6 5= University of Oxford United Kingdom L VH FC 98.16
7 5= Imperial College London United Kingdom L VH FC 97.78
8 7 University of Chicago United States M VH FC 97.52
9 10 California Institute of Technology (Caltech) United States S VH CO 96.46
10 8 Princeton University United States M VH CO 96.03
11 11 Columbia University United States L VH FC 95.99
12 12 University of Pennsylvania (UPenn) United States L VH FC 95.97
13 16 Stanford University United States L VH FC 93.62
14 14 Duke University United States L VH FC 92.29
15 19 University of Michigan United States XL VH FC 92.20
16 15 Cornell University United States L VH FC 90.44
17 13 Johns Hopkins University United States L VH FC 89.67
18 20= ETH Zurich (Swiss Federal Institute of Technology) Switzerland L VH FO 89.28
19 18 McGill University Canada L VH FC 89.25
20 17 Australian National University (ANU) Australia M VH CO 88.58
21 23 King’s College London (KCL) United Kingdom L VH FC 88.45
22 20= University of Edinburgh United Kingdom L VH FC 88.01
23 24 University of Hong Kong (HKU) Hong Kong L VH FC 87.28
24 22 University of Tokyo, The Japan L VH FC 86.74
25 25 Kyoto University Japan L VH FC 85.89
26 32= Northwestern University United States L VH FC 85.40
27 34 University of Bristol United Kingdom L VH FC 85.22
28 39 University of California, Berkeley (UCB) United States XL VH FC 85.18
29 29 University of Toronto Canada XL VH FC 84.29
30 26 University of Manchester United Kingdom XL VH FC 83.33
31 30 National University of Singapore (NUS) Singapore XL VH FC 82.78

Details: http://www.topuniversities.com/university-rankings/world-university-rankings/2010/results

20/08/2010

Chúc mừng anh Ngô Bảo Châu

Filed under: Toán học, Tra cứu, Trao đổi — Thẻ:, , , — doanchi @ 00:14

Anh Ngô Bảo Châu là người Việt Nam đầu tiên nhận Fields Medal tại ICM 2010 (19/08/2010). Anh là niềm tự hào của cả dân tộc Việt Nam, là tấm gương học tập nghiên cứu cho tuổi trẻ Việt Nam hướng tới.
Chúng em xin chúc mừng anh.
Trang wordpress blog của anh: http://thichhoctoan.wordpress.com

17/07/2010

Manifold Destiny: A legendary problem and the battle over who solved it

Filed under: Không toán học, Trao đổi — Ngô Quốc Anh @ 22:07

On the evening of June 20th, several hundred physicists, including a Nobel laureate, assembled in an auditorium at the Friendship Hotel in Beijing for a lecture by the Chinese mathematician Shing-Tung Yau. In the late nineteen-seventies, when Yau was in his twenties, he had made a series of breakthroughs that helped launch the string-theory revolution in physics and earned him, in addition to a Fields Medal—the most coveted award in mathematics—a reputation in both disciplines as a thinker of unrivalled technical power.

Yau had since become a professor of mathematics at Harvard and the director of mathematics institutes in Beijing and Hong Kong, dividing his time between the United States and China. His lecture at the Friendship Hotel was part of an international conference on string theory, which he had organized with the support of the Chinese government, in part to promote the country’s recent advances in theoretical physics. (More than six thousand students attended the keynote address, which was delivered by Yau’s close friend Stephen Hawking, in the Great Hall of the People.) The subject of Yau’s talk was something that few in his audience knew much about: the Poincaré conjecture, a century-old conundrum about the characteristics of three-dimensional spheres, which, because it has important implications for mathematics and cosmology and because it has eluded all attempts at solution, is regarded by mathematicians as a holy grail.

Yau, a stocky man of fifty-seven, stood at a lectern in shirtsleeves and black-rimmed glasses and, with his hands in his pockets, described how two of his students, Xi-Ping Zhu and Huai-Dong Cao, had completed a proof of the Poincaré conjecture a few weeks earlier. “I’m very positive about Zhu and Cao’s work,” Yau said. “Chinese mathematicians should have every reason to be proud of such a big success in completely solving the puzzle.” He said that Zhu and Cao were indebted to his longtime American collaborator Richard Hamilton, who deserved most of the credit for solving the Poincaré. He also mentioned Grigory Perelman, a Russian mathematician who, he acknowledged, had made an important contribution. Nevertheless, Yau said, “in Perelman’s work, spectacular as it is, many key ideas of the proofs are sketched or outlined, and complete details are often missing.” He added, “We would like to get Perelman to make comments. But Perelman resides in St. Petersburg and refuses to communicate with other people.”

For ninety minutes, Yau discussed some of the technical details of his students’ proof. When he was finished, no one asked any questions. That night, however, a Brazilian physicist posted a report of the lecture on his blog. “Looks like China soon will take the lead also in mathematics,” he wrote.

Grigory Perelman is indeed reclusive. He left his job as a researcher at the Steklov Institute of Mathematics, in St. Petersburg, last December; he has few friends; and he lives with his mother in an apartment on the outskirts of the city. Although he had never granted an interview before, he was cordial and frank when we visited him, in late June, shortly after Yau’s conference in Beijing, taking us on a long walking tour of the city. “I’m looking for some friends, and they don’t have to be mathematicians,” he said. The week before the conference, Perelman had spent hours discussing the Poincaré conjecture with Sir John M. Ball, the fifty-eight-year-old president of the International Mathematical Union, the discipline’s influential professional association. The meeting, which took place at a conference center in a stately mansion overlooking the Neva River, was highly unusual. At the end of May, a committee of nine prominent mathematicians had voted to award Perelman a Fields Medal for his work on the Poincaré, and Ball had gone to St. Petersburg to persuade him to accept the prize in a public ceremony at the I.M.U.’s quadrennial congress, in Madrid, on August 22nd.

The Fields Medal, like the Nobel Prize, grew, in part, out of a desire to elevate science above national animosities. German mathematicians were excluded from the first I.M.U. congress, in 1924, and, though the ban was lifted before the next one, the trauma it caused led, in 1936, to the establishment of the Fields, a prize intended to be “as purely international and impersonal as possible.”

However, the Fields Medal, which is awarded every four years, to between two and four mathematicians, is supposed not only to reward past achievements but also to stimulate future research; for this reason, it is given only to mathematicians aged forty and younger. In recent decades, as the number of professional mathematicians has grown, the Fields Medal has become increasingly prestigious. Only forty-four medals have been awarded in nearly seventy years—including three for work closely related to the Poincaré conjecture—and no mathematician has ever refused the prize. Nevertheless, Perelman told Ball that he had no intention of accepting it. “I refuse,” he said simply.

Over a period of eight months, beginning in November, 2002, Perelman posted a proof of the Poincaré on the Internet in three installments. Like a sonnet or an aria, a mathematical proof has a distinct form and set of conventions. It begins with axioms, or accepted truths, and employs a series of logical statements to arrive at a conclusion. If the logic is deemed to be watertight, then the result is a theorem. Unlike proof in law or science, which is based on evidence and therefore subject to qualification and revision, a proof of a theorem is definitive. Judgments about the accuracy of a proof are mediated by peer-reviewed journals; to insure fairness, reviewers are supposed to be carefully chosen by journal editors, and the identity of a scholar whose pa-per is under consideration is kept secret. Publication implies that a proof is complete, correct, and original.

by Sylvia Nasar  and David Gruber

View all: http://www.newyorker.com/archive/2006/08/28/060828fa_fact2?currentPage=all

03/05/2010

Maple 14

Filed under: Không toán học, Không định dạng, Trao đổi — Ngô Quốc Anh @ 20:16
(more…)

31/12/2009

Chúc mừng năm mới 2010

Filed under: Không toán học, Trao đổi — doanchi @ 22:35

Năm 2010, khởi đầu của một cái “mười năm” nữa.

Chặng đường còn dài, mong rằng mọi người đều gắng sức.

Chúc anh em Bộ môn thật nhiều thành công.

29/10/2009

Mathematics Subject Classification: MSC2010

Filed under: Không toán học, Tra cứu, Trao đổi — Ngô Quốc Anh @ 16:53

Mathematical Reviews (MR) and Zentralblatt MATH (Z-MATH) collaborate in maintaining the Mathematics Subject Classification (MSC), which is used by these reviewing services and many others to categorize items in the mathematical sciences literature. The MSC has undergone a general revision, with some additions, changes, and corrections, to create MSC2010, the successor to MSC2000, the scheme for the past 10 years. MR and Z-MATH carefully considered input received from the community in recent years, especially since the announcement of the projected revision in December 2006, and used it in the preparation of their joint MSC revision. As anticipated, there are no changes at the two-digit level but refinements have been made at the three- and five-digit levels. With July 2009 MR and Z-MATH started to use MSC2010 as their classification scheme.

MR and Z-MATH welcome and encourage community adoption of MSC2010. Comments can be submitted through the Web form found at http://msc2010.org/feedback or by email to feedback@msc2010.org. All information about MSC2010 is jointly shared by MR and Z-MATH.

The Editors and their staffs wish to express their gratitude to the numerous members of the community for their assistance in this lengthy revision process.

Bernd Wegner, Editor-in-Chief, Z-MATH

Download MSC2010 (PDF)

More information: http://www.ams.org/mathscinet/msc/msc2010.html

Source: http://www.zentralblatt-math.org/msc/en/

20/09/2009

e*Calendar 4.0: Quyển lịch Bloc bỏ túi dành cho người Việt

Filed under: Không toán học, Tra cứu, Trao đổi, Vui chơi — Ngô Quốc Anh @ 14:52

Mùa Xuân sắp đến, đây cũng là lúc mọi người trang hoàng nhà cửa và mua cho mình một quyển lịch Bloc mới. Tuy nhiên, có một quyển lịch Bloc bạn có thể đem về dùng mà không cần phải trả bất cứ chi phí nào, đó là e*Calendar 4.0, một quyển lịch treo trên desktop với giao diện tiếng Việt thân thiện.

Những tính năng hấp dẫn của e*Calendar 4.0:

  • Tra cứu âm dương lịch từ năm 1901 đến 3001 (1100 năm).
  • Tờ lịch ngày (Bloc) thiết kế theo hình dáng của bloc treo tường thông dụng, có đủ tháng, ngày, giờ âm lịch theo can chi. Cửa sổ xem trăng cho biết chính xác mức độ tròn của mặt trăng hiện hành.
  • Cho phép chọn một ngày bất kỳ trong khoảng thời gian 1100 năm bằng vài thao tác chuột hoặc phím bấm.
  • Hiển thị các ngày lễ, tết, kỷ niệm, sinh nhật… Cho phép người sử dụng tự định nghĩa và sửa đổi những ngày đặc biệt.
  • Cài đặt hệ thống lịch hẹn với tính năng tự động báo giờ.
  • Sổ tay ghi chép.
  • Tùy biến ảnh nền của cuốn lịch.
  • Tùy biến các câu thơ, thành ngữ, tục ngữ… theo định dạng HTML để hiển thị trên Bloc…

Download:

Có 2 bản

Bản full (~70 MB): Đầy đủ các tính năng

Bản mini (~10 MB): Không có hình nền, cách ngôn…

→ Sau đó, tải về tính năng cộng thêm nào mình thích:
I. Các bước cài thêm hình nền cho bloc:

1. Download 1 hoặc nhiều bộ hình theo sở thích:

2. Thi hành file vừa download về (Ví dụ “Art.exe”)
3. Nhấp nút BROWSE, chỉ đến thư mục …\imgBloc (mặc định là C:\Program Files\Enter PVH\eCalendar 4.0\imgBloc)
4. Nhấp chuột phải lên biểu tượng “Calendar” trên Taskbar, chọn “Thiết lập cấu hình…”II. Các bước tùy biến dòng chữ (Cách ngôn, ngạn ngữ…) trên bloc:1. Download Text on bloc: RS / UL / FF
2. Xả nén vào thư mục …\HTML (mặc định là C:\Program Files\Enter PVH\eCalendar 4.0\HTML)
3. Dùng các phần mềm chuyên dụng (như FrontPage…) để chỉnh sửa các file *.htm theo ý thích.
—————————————————
(*) Do sơ suất trong đóng gói, khi gọi trợ giúp CT sẽ báo lỗi không tìm thấy file: eCalendar.chm
→ Cách khắc phục:
Sau khi cài đặt xong, vào thư mục trợ giúp (mặc định là: “C:\Program Files\Enter PVH\eCalendar 4.0\Help\“) đổi tên file Calendar.chm thành ECalendar.chm.
Theo Echip và PVH’s Weblog

10/09/2009

When is next Thursday?

Filed under: Không toán học, Trao đổi, Vui chơi — Ngô Quốc Anh @ 22:21

Yesterday one of my colleagues circulated an email about a future event, specifying the time as “just before the lab meeting next Thursday”. It set off a whole bundle of confusion (does she mean “The next Thursday we will experience”, or “Thursday of next week”?) and got me thinking about this kind of reference to time.

There are quite a few ways to express a future day of the week: my own variant of English makes a strong distinction between “This Thursday” and “Next Thursday”. The former refers to the next Thursday that will be experienced, while “Next Thursday” is the Thursday that follows “This Thursday”. This is in addition to the simple “Thursday” which is essentially synonymous with “This Thursday”. “This” and “Next” when used with days don’t seem to work the same as “This” and “Next” in other contexts (I would use “This bus” only if it can be seen, otherwise “The next bus” to refer to the bus-equivalent of “This Thursday”), and there are additional constraints. For example, if today is Wednesday (which it is not), it doesn’t sound correct to say “This Thursday” when “Tomorrow” is a possibility (unless I have lost track of which day it is [sadly this is a fairly common occurrence]). So in this circumstance “This Thursday” has been replaced by “Tomorrow” while “Next Thursday” remains “Thursday of next week”. And it also gets awkward once Thursday of a particular week has passed; if today is Friday, “this Thursday” used in a future tense then means “Thursday of next week” (“this Thursday” may also be used in the past tense in order to mean “The previous Thursday”; fortunately English verbs allow this ambiguity to be avoided), but “next Thursday” is much more ambiguous (it could mean “Thursday of next week”, although I still typically use it to mean “the second Thursday in the future”. But the use of “next” for a day 13 days in the future may be a bit much). My distinction between “This” and “Next” does not depend on the boundary between weeks; I would still use “This Monday” to refer to the upcoming Monday even if today is Thursday (which it is not), and “Next Monday” to refer to the following one.

However, other English speakers do not typically use “This Thursday” as I do (I also occasionally use “This coming Thursday” or “This past Thursday”, but this kind of disambiguation is not really necessary). Hence the confusion arising from my cow-orker’s email (She meant “Next Thursday” in the sense in which I use it, but other colleagues misinterpreted it as meaning “This Thursday”). This may be because British English uses “next” differently, thanks to the “week” expression. UK “Thursday week” apparently has the same meaning as my “Next Thursday”, and UK “Next Thursday” has the same meaning as my “This Thursday” (one of OED’s definitions of “week” is “Seven days after the day specified”). Here’s an instance of someone who ran into the next/week problem (The blogger’s user info suggests that this is also a US/UK translation difference); and here is a discussion related to learning English as a second language. It’s unclear to me whether such expressions also apply for a day that has just passed (if today is Wednesday [which it is not], is “Tuesday week” six or 13 days in the future?). Or expressions like “Next Tuesday week” which just make my head spin.

Source: http://newpics.org/david/WhenIsNextThursday.aspx

17/05/2009

Maple 13

Filed under: Không toán học, Tra cứu, Trao đổi — Ngô Quốc Anh @ 02:20

13/05/2009

L’Oise, a sunny day

Filed under: Không định dạng — doanchi @ 23:05

loise

I try to test the utile of posting from email. That’s good, I hope.

22/03/2009

Birds and Frogs, part 4

Filed under: Không toán học, Tra cứu, Trao đổi — Ngô Quốc Anh @ 00:51

Weak Chaos

If von Neumann had known about chaos when he spoke in Amsterdam, one of the unsolved problems that he might have talked about was weak chaos. The problem of weak chaos is still unsolved fifty years later. The problem is to understand why chaotic motions often remain bounded and do not cause any violent instability. A good example of weak chaos is the orbital motions of the planets and satellites in the solar system. It was discovered only recently that these motions are chaotic. This was a surprising discovery, upsetting the traditional picture of the solar system as the prime example of orderly stable motion. The mathematician Laplace two hundred years ago thought he had proved that the solar system is stable. It now turns out that Laplace was wrong. Accurate numerical integrations of the orbits show clearly that neighboring orbits diverge exponentially. It seems that chaos is almost universal in the world of classical dynamics.

Chaotic behavior was never suspected in the solar system before accurate long-term integrations were done, because the chaos is weak. Weak chaos means that neighboring trajectories diverge exponentially but never diverge far. The divergence begins with exponential growth but afterwards remains bounded. Because the chaos of the planetary motions is weak, the solar system can survive for four billion years. Although the motions are chaotic, the planets never wander far from their customary places, and the system as a whole does not fly apart. In spite of the prevalence of chaos, the Laplacian view of the solar system as a perfect piece of clockwork is not far from the truth.

We see the same phenomena of weak chaos in the domain of meteorology. Although the weather in New Jersey is painfully chaotic, the chaos has firm limits. Summers and winters are unpredictably mild or severe, but we can reliably predict that the temperature will never rise to 45 degrees Celsius or fall to minus 30, extremes that are often exceeded in India or in Minnesota. There is no conservation law of physics that forbids temperatures from rising as high in New Jersey as in India, or from falling as low in New Jersey as in Minnesota. The weakness of chaos has been essential to the long-term survival of life on this planet. Weak chaos gives us a challenging variety of weather while protecting us from fluctuations so severe as to endanger our existence. Chaos remains mercifully weak for reasons that we do not understand. That is another unsolved problem for young frogs in the audience to take home. I challenge you to understand the reasons why the chaos observed in a great diversity of dynamical systems is generally weak.

The subject of chaos is characterized by an abundance of quantitative data, an unending supply of beautiful pictures, and a shortage of rigorous theorems. Rigorous theorems are the best way to give a subject intellectual depth and precision. Until you can prove rigorous theorems, you do not fully understand the meaning of your concepts. In the field of chaos I know only one rigorous theorem, proved by Tien-Yien Li and Jim Yorke in 1975 and published in a short paper with the title, “Period Three Implies Chaos”, [4]. The Li-Yorke paper is one of the immortal gems in the literature of mathematics. Their theorem concerns nonlinear maps of an interval onto itself. The successive positions of a point when the mapping is repeated can be considered as the orbit of a classical particle. An orbit has period N if the point returns to its original position after N mappings. An orbit is defined to be chaotic, in this context, if it diverges from all periodic orbits. The theorem says that if a single orbit with period three exists, then chaotic orbits also exist. The proof is simple and short. To my mind, this theorem and its proof throw more light than a thousand beautiful pictures on the basic nature of chaos. The theorem explains why chaos is prevalent in the world. It does not explain why chaos is so often weak. That remains a task for the future. I believe that weak chaos will not be understood in a fundamental way until we can prove rigorous theorems about it.

String Theorists

I would like to say a few words about string theory. Few words, because I know very little about string theory. I never took the trouble to learn the subject or to work on it myself. But when I am at home at the Institute for Advanced Study in Princeton, I am surrounded by string theorists, and I sometimes listen to their conversations. Occasionally I understand a little of what they are saying. Three things are clear. First, what they are doing is first-rate mathematics. The leading pure mathematicians, people like Michael Atiyah and Isadore Singer, love it. It has opened up a whole new branch of mathematics, with new ideas and new problems. Most remarkably, it gave the mathematicians new methods to solve old problems that were previously unsolvable. Second, the string theorists think of themselves as physicists rather than mathematicians. They believe that their theory describes something real in the physical world. And third, there is not yet any proof that the theory is relevant to physics. The theory is not yet testable by experiment. The theory remains in a world of its own, detached from the rest of physics. String theorists make strenuous efforts to deduce consequences of the theory that might be testable in the real world, so far without success.

My colleagues Ed Witten and Juan Maldacena and others who created string theory are birds, flying high and seeing grand visions of distant ranges of mountains. The thousands of humbler practitioners of string theory in universities around the world are frogs, exploring fine details of the mathematical structures that birds first saw on the horizon. My anxieties about string theory are sociological rather than scientific. It is a glorious thing to be one of the first thousand string theorists, discovering new connections and pioneering new methods. It is not so glorious to be one of the second thousand or one of the tenth thousand. There are now about ten thousand string theorists scattered around the world. This is a dangerous situation for the tenth thousand and perhaps also for the second thousand. It may happen unpredictably that the fashion changes and string theory becomes unfashionable. Then it could happen that nine thousand string theorists lose their jobs. They have been trained in a narrow specialty, and they may be unemployable in other fields of science.

Why are so many young people attracted to string theory? The attraction is partly intellectual. String theory is daring and mathematically elegant. But the attraction is also sociological. String theory is attractive because it offers jobs. And why are so many jobs offered in string theory? Because string theory is cheap. If you are the chairperson of a physics department in a remote place without much money, you cannot afford to build a modern laboratory to do experimental physics, but you can afford to hire a couple of string theorists. So you offer a couple of jobs in string theory, and you have a modern physics department. The temptations are strong for the chairperson to offer such jobs and for the young people to accept them. This is a hazardous situation for the young people and also for the future of science. I am not saying that we should discourage young people from working in string theory if they find it exciting. I am saying that we should offer them alternatives, so that they are not pushed into string theory by economic necessity.

Finally, I give you my own guess for the future of string theory. My guess is probably wrong. I have no illusion that I can predict the future. I tell you my guess, just to give you something to think about. I consider it unlikely that string theory will turn out to be either totally successful or totally useless. By totally successful I mean that it is a complete theory of physics, explaining all the details of particles and their interactions. By totally useless I mean that it remains a beautiful piece of pure mathematics. My guess is that string theory will end somewhere between complete success and failure. I guess that it will be like the theory of Lie groups, which Sophus Lie created in the nineteenth century as a mathematical framework for classical physics. So long as physics remained classical, Lie groups remained a failure. They were a solution looking for a problem. But then, fifty years later, the quantum revolution transformed physics, and Lie algebras found their proper place. They became the key to understanding the central role of symmetries in the quantum world. I expect that fifty or a hundred years from now another revolution in physics will happen, introducing new concepts of which we now have no inkling, and the new concepts will give string theory a new meaning. After that, string theory will suddenly find its proper place in the universe, making testable statements about the real world. I warn you that this guess about the future is probably wrong. It has the virtue of being falsifiable, which according to Karl Popper is the hallmark of a scientific statement. It may be demolished tomorrow by some discovery coming out of the Large Hadron Collider in Geneva.

Manin Again

To end this talk, I come back to Yuri Manin and his book Mathematics as Metaphor. The book is mainly about mathematics. It may come as a surprise to Western readers that he writes with equal eloquence about other subjects such as the collective unconscious, the origin of human language, the psychology of autism, and the role of the trickster in the mythology of many cultures. To his compatriots in Russia, such many-sided interests and expertise would come as no surprise. Russian intellectuals maintain the proud tradition of the old Russian intelligentsia, with scientists and poets and artists and musicians belonging to a single community. They are still today, as we see them in the plays of Chekhov, a group of idealists bound together by their alienation from a superstitious society and a capricious government. In Russia, mathematicians and composers and filmproducers talk to one another, walk together in the snow on winter nights, sit together over a bottle of wine, and share each others’ thoughts.

Manin is a bird whose vision extends far beyond the territory of mathematics into the wider landscape of human culture. One of his hobbies is the theory of archetypes invented by the Swiss psychologist Carl Jung. An archetype, according to Jung, is a mental image rooted in a collective unconscious that we all share. The intense emotions that archetypes carry with them are relics of lost memories of collective joy and suffering. Manin is saying that we do not need to accept Jung’s theory as true in order to find it illuminating.

More than thirty years ago, the singer Monique Morelli made a recording of songs with words by Pierre MacOrlan. One of the songs is La Ville Morte, the dead city, with a haunting melody tuned to Morelli’s deep contralto, with an accordion singing counterpoint to the voice, and with verbal images of extraordinary intensity. Printed on the page, the words are nothing special:

“En pénétrant dans la ville morte,
Je tenait Margot par le main…
Nous marchions de la nécropole,
Les pieds brisés et sans parole,
Devant ces portes sans cadole,
Devant ces trous indéfinis,
Devant ces portes sans parole
Et ces poubelles pleines de cris”.

“As we entered the dead city, I held Margot by the hand…We walked from the graveyard on our bruised feet, without a word, passing by these doors without locks, these vaguely glimpsed holes, these doors without a word, these garbage cans full of screams.”

I can never listen to that song without a  disproportionate intensity of feeling. I often ask myself why the simple words of the song seem to resonate with some deep level of unconscious memory, as if the souls of the departed are speaking through Morelli’s music. And now unexpectedly in Manin’s book I find an answer to my question. In his chapter, “The Empty City Archetype”, Manin describes how the archetype of the dead city appears again and again in the creations of architecture, literature, art and film, from ancient to modern times, ever since human beings began to congregate in cities, ever since other human beings began to congregate in armies to ravage and destroy them. The character who speaks to us in MacOrlan’s song is an old soldier who has long ago been part of an army of occupation. After he has walked with his wife through the dust and ashes of the dead city, he hears once more:

“Chansons de charme d’un clairon
Qui fleurissait une heure lointaine
Dans un rêve de garnison”.

“The magic calls of a bugle that came to life for an hour in an old soldier’s dream”.

The words of MacOrlan and the voice of Morelli seem to be bringing to life a dream from our collective unconscious, a dream of an old soldier wandering through a dead city. The concept of the collective unconscious may be as mythical as the concept of the dead city. Manin’s chapter describes the subtle light that these two possibly mythical concepts throw upon each other. He describes the collective unconscious as an irrational force that powerfully pulls us toward death and destruction. The archetype of the dead city is a distillation of the agonies of hundreds of real cities that have been destroyed since cities and marauding armies were invented. Our only way of escape from the insanity of the collective unconscious is a collective consciousness of sanity, based upon hope and reason. The great task that faces our contemporary civilization is to create such a collective consciousness.

References

  1. M. J. Bertin et al., Pisot and Salem Numbers, Birkhäuser Verlag, Basel, 1992.
  2. M. L. Cartwright and J. E. Littlewood, On nonlinear differential equations of the second order, I, Jour. London Math. Soc. 20 (1945), 180–189.
  3. Freeman Dyson, Prof. Hermann Weyl, For.Mem.R.S., Nature 177 (1956), 457–458.
  4. Tien-Yien Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985–992.
  5. Yuri I. Manin, Mathematics as Metaphor: Selected Essays, American Mathematical Society, Providence, Rhode Island, 2007. [The Russian version is: Manin, Yu. I., Matematika kak Metafora, Moskva, Izdatyelstvo MTsNMO, 2008.]
  6. Andrew M. Odlyzko, Primes, quantum chaos and computers, in Number Theory, Proceedings of a Symposium, National Research Council, Washington DC, 1990, pp. 35–46.
  7. Hermann Weyl, Gravitation und elektrizität, Sitz. König. Preuss. Akad. Wiss. 26 (1918), 465–480.
  8. ——— , Elektron und gravitation, Zeits. Phys. 56 (1929), 350–352.
  9. ——— , Selecta, Birkhäuser Verlag, Basel, 1956, p. 192.
  10. Chen Ning Yang, Integral formalism for gauge fields, Phys. Rev. Letters 33 (1974), 445–447.
  11. Chen Ning Yang and Robert L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954), 191–195.
  12. ——— , Hermann Weyl’s contribution to physics, in Hermann Weyl, 1885–1985, (K. Chandrasekharan, ed.), Springer-Verlag, Berlin, 1986, p. 19.

Adapted from NOTICES OF AMS 2/2009

Birds and Frogs, part 3

Filed under: Không toán học, Toán học, Tra cứu, Trao đổi — Ngô Quốc Anh @ 00:39

John von Neumann

Another important figure in twentieth century mathematics was John von Neumann. Von Neumann was a frog, applying his prodigious technical skill to solve problems in many branches of mathematics and physics. He began with the foundations of mathematics. He found the first satisfactory set of axioms for set-theory, avoiding the logical paradoxes that Cantor had encountered in his attempts to deal with infinite sets and infinite numbers. Von Neumann’s axioms were used by his bird friend Kurt Gödel a few years later to prove the existence of undecidable propositions in mathematics. Gödel’s theorems gave birds a new vision of mathematics. After Gödel, mathematics was no longer a single structure tied together with a unique concept of truth, but an archipelago of structures with diverse sets of axioms and diverse notions of truth. Gödel showed that mathematics is inexhaustible. No matter which set of axioms is chosen as the foundation, birds can always find questions that those axioms cannot answer.

Von Neumann went on from the foundations of mathematics to the foundations of quantum mechanics. To give quantum mechanics a firm mathematical foundation, he created a magnificent theory of rings of operators. Every observable quantity is represented by a linear operator, and the peculiarities of quantum behavior are faithfully represented by the algebra of operators. Just as Newton invented calculus to describe classical dynamics, von Neumann invented rings of operators to describe quantum dynamics.

Von Neumann made fundamental contributions to several other fields, especially to game theory and to the design of digital computers. For the last ten years of his life, he was deeply involved with computers. He was so strongly interested in computers that he decided not only to study their design but to build one with real hardware and software and use it for doing science. I have vivid memories of the early days of von Neumann’s computer project at the Institute for Advanced Study in Princeton. At that time he had two main scientific interests, hydrogen bombs and meteorology. He used his computer during the night for doing hydrogen bomb calculations and during the day for meteorology. Most of the people hanging around the computer building in daytime were meteorologists. Their leader was Jule Charney. Charney was a real meteorologist, properly humble in dealing with the inscrutable mysteries of the weather, and skeptical of the ability of the computer to solve the mysteries. John von Neumann was less humble and less skeptical. I heard von Neumann give a lecture about the aims of his project. He spoke, as he always did, with great confidence. He said, “The computer will enable us to divide the atmosphere at any moment into stable regions and unstable regions. Stable regions we can predict. Unstable regions we can control.” Von Neumann believed that any unstable region could be pushed by a judiciously applied small perturbation so that it would move in any desired direction. The small perturbation would be applied by a fleet of airplanes carrying smoke generators, to absorb sunlight and raise or lower temperatures at places where the perturbation would be most effective. In particular, we could stop an incipient hurricane by identifying the position of an instability early enough, and then cooling that patch of air before it started to rise and form a vortex. Von Neumann, speaking in 1950, said it would take only ten years to build computers powerful enough to diagnose accurately the stable and unstable regions of the atmosphere. Then, once we had accurate diagnosis, it would take only a short time for us to have control. He expected that practical control of the weather would be a routine operation within the decade of the 1960s.

Von Neumann, of course, was wrong. He was wrong because he did not know about chaos. We now know that when the motion of the atmosphere is locally unstable, it is very often chaotic. The word “chaotic” means that motions that start close together diverge exponentially from each other as time goes on. When the motion is chaotic, it is unpredictable, and a small perturbation does not move it into a stable motion that can be predicted. A small perturbation will usually move it into another chaotic motion that is equally unpredictable. So von Neumann’s strategy for controlling the weather fails. He was, after all, a great mathematician but a mediocre meteorologist.

Edward Lorenz discovered in 1963 that the solutions of the equations of meteorology are often chaotic. That was six years after von Neumann died. Lorenz was a meteorologist and is generally regarded as the discoverer of chaos. He discovered the phenomena of chaos in the meteorological context and gave them their modern names. But in fact I had heard the mathematician Mary Cartwright, who died in 1998 at the age of 97, describe the same phenomena in a lecture in Cambridge in 1943, twenty years before Lorenz discovered them. She called the phenomena by different names, but they were the same phenomena. She discovered them in the solutions of the van der Pol equation which describe the oscillations of a nonlinear amplifier, [2]. The van der Pol equation was important in World War II because nonlinear amplifiers fed power to the transmitters in early radar systems. The transmitters behaved erratically, and the Air Force blamed the manufacturers for making defective amplifiers. Mary Cartwright was asked to look into the problem. She showed that the manufacturers were not to blame. She showed that the van der Pol equation was to blame. The solutions of the van der Pol equation have precisely the chaotic behavior that the Air Force was complaining about. I heard all about chaos from Mary Cartwright seven years before I heard von Neumann talk about weather control, but I was not far-sighted enough to make the connection. It never entered my head that the erratic behavior of the van der Pol equation might have something to do with meteorology. If I had been a bird rather than a frog, I would probably have seen the connection, and I might have saved von Neumann a lot of trouble. If he had known about chaos in 1950, he would probably have thought about it deeply, and he would have had something important to say about it in 1954.

Von Neumann got into trouble at the end of his life because he was really a frog but everyone expected him to fly like a bird. In 1954 there was an International Congress of Mathematicians in Amsterdam. These congresses happen only once in four years and it is a great honor to be invited to speak at the opening session. The organizers of the Amsterdam congress invited von Neumann to give the keynote speech, expecting him to repeat the act that Hilbert had performed in Paris in 1900. Just as Hilbert had provided a list of unsolved problems to guide the development of mathematics for the first half of the twentieth century, von Neumann was invited to do the same for the second half of the century. The title of von Neumann’s talk was announced in the program of the congress. It was “Unsolved Problems in Mathematics: Address by Invitation of the Organizing Committee”. After the congress was over, the complete proceedings were published, with the texts of all the lectures except this one. In the proceedings there is a blank page with von Neumann’s name and the title of his talk. Underneath, it says, “No manuscript of this lecture was available.”

What happened? I know what happened, because I was there in the audience, at 3:00 p.m. on Thursday, September 2, 1954, in the Concertgebouw concert hall. The hall was packed with mathematicians, all expecting to hear a brilliant lecture worthy of such a historic occasion. The lecture was a huge disappointment. Von Neumann had probably agreed several years earlier to give a lecture about unsolved problems and had then forgotten about it. Being busy with many other things, he had neglected to prepare the lecture. Then, at the last moment, when he remembered that he had to travel to Amsterdam and say something about mathematics, he pulled an old lecture from the 1930s out of a drawer and dusted it off. The lecture was about rings of operators, a subject that was new and fashionable in the 1930s. Nothing about unsolved problems. Nothing about the future. Nothing about computers, the subject that we knew was dearest to von Neumann’s heart. He might at least have had something new and exciting to say about computers. The audience in the concert hall became restless. Somebody said in a voice loud enough to be heard all over the hall, “Aufgewärmte Suppe”, which is German for “warmed-up soup”. In 1954 the great majority of mathematicians knew enough German to understand the joke. Von Neumann, deeply embarrassed, brought his lecture to a quick end and left the hall without waiting for questions.

Adapted from NOTICES OF AMS 2/2009

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