Blog Archive

22 July 2015

Two great losses



Below is a link to the blog by Alain Connes on the passing of Uffe Haagerup. I had the good fortune of meeting Prof Haagerup in 2002. His paper, Random matrices, free probability and the invariant subspace problem relative to a von Neumann Algebra, served as encouragement for me to study random matrices and application to denoising correlation matrix estimators for financial markets (and complex systems more generally) - it was a pivot for a switch from theory to applications of functional analysis at that time.


  








Thanks for the generous and gentle way in which you communicated with novices.

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http://noncommutativegeometry.blogspot.com/2015/07/two-great-losses.html: Two great losses
It is with incommensurable sadness that we learned of the death of two great figures of the fields of operator algebras and mathematical physics.
Daniel Kastler died on July 4-th in his house in Bandol.
Uffe Haagerup died on July 5-th in a tragic accident while swimming near his summer house in Denmark.

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Here is Prof Connes' blog on Haagerup: http://noncommutativegeometry.blogspot.com/2015/07/uffe-haagerup.html

Uffe Haagerup was a wonderful man, with a perfect kindness and openness of mind, and a mathematician of incredible power and insight. 
His whole career is a succession of amazing achievements and of decisive and extremely influential contributions to the field of operator algebras, C*-algebras and von Neumann algebras. 
His first work (1973-80) concerned the theory of weights and more generally the modular theory of Tomita-Takesaki. Uffe  Haagerup began by solving a key open question, showing that semi-finite weights are indeed supremum of families of normal states. This allowed him to develop the standard form of von Neumann algebras, a basic result used over and over since then.  I remember vividly his first appearance in the field of operator algebras and the striking elegance, clarity and strength of his contributions.
He was the first to introduce the Lp-spaces associated to a type III von Neumann algebra. These new spaces have since then received a lot of attention both from operator algebraists and specialists of Banach spaces and operator spaces.
In a remarkable paper published in Inventiones at the end of the seventies, Haagerup was able to analyse the operator norm in the C*-algebra of the free group, to control it by suitable Sobolev norms in spite of the exponential growth of the group and to prove in particular that in spite of its lack of nuclearity the reduced C*-algebra has the Banach space approximation property. This turned out in the long run to be a breakthrough of major importance. It was extended by Haagerup and his collaborators to any discrete subgroup of a simple real rank one Lie-group. One corollary is the property RD of rapid decay which allows one to define the analogue of the Harish-Chandra algebra of smooth elements in the general context of Gromov hyperbolic groups.  This highly non-trivial result of Haagerup turned out to be the technical key in the proof of the Novikov conjecture for such groups. Moreover the study of discrete groups with the Haagerup property continues to play a major role in geometric group theory.  The associated approximation property for the corresponding factors of type II_1 (called the Haagerup property) also played a major role in the solution by Popa of the long-standing problem of factors N non-isomorphic to M_2(N), and of exhibiting a factor with trivial fundamental group. 
The next fundamental contribution of Haagerup is to the classification of injective factors (1983-87). In my work on the subject I had left one case completely open (in 76) and the classification was thus incomplete. After several years of extremely hard work Haagerup was able to settle this question by proving that there is up to isomorphism only one hyperfinite factor of type III_1. This is a wonderful achievement.
Then Haagerup turned to the theory of subfactors, and, once again, was able to make fundamental contributions such as his construction of new irreducible subfactors of small index > 4, and outclass the best specialists of the subject created by Vaughan Jones at the beginning of the eighties. 
Another key contribution of Haagerup is to Voiculescu's Free Probability and to random matrices which he was able to apply very successfully to the theory of C*-algebras and of factors of type II1. For instance he proved that, in all II1 factors fulfilling the approximate embedding property (which is true in all known cases) every operator T with a non-trivial Brown measure has a non-trivial closed invariant subspace affiliated with the von Neumann algebra generated by T. 
Uffe continued at the same relentless pace to produce amazing contributions and the few mentioned above only give a glimpse of his most impressive collection of breakthrough achievements.
Uffe Haagerup was a marvelous mathematician, well-known to operator algebraists, and to the general community of analysts.  From my own perspective an analyst is characterized by the ability of having ”direct access to the infinite” and Uffe Haagerup possessed that quality to perfection. His disparition is a great loss for all of us. 

16 July 2015

Books to read

http://theadvisorcambodia.com/2015/06/all-for-nought/

All for nought

Posted On Jun 20, 2015
The concept of ‘zero’ is perhaps the most paradoxical theory of mankind. At once nothing and everything, it is the basis of the numerical system incorporated into most Westernised societies.  Amir D. Aczel investigates its origins in his latest book, Finding Zero.
Finding ZeroThe concept of zero, nought, or 0, is considered to be one of the highest intellectual achievements of mankind, and almost certainly the greatest conceptual leap in the history of mathematics.
As science writer and mathematician Amir Aczel puts it, “zero is not only a concept of nothingness, which allows us to do arithmetic well, and to algebraically define negative numbers… zero enables our base-10 number system to work, so that the same 10 numerals can be used over and over again, at different positions in a number.”
Finding Zero describes Aczel’s lifelong quest to discover from whence our number system came.
In the West, the Roman numeral system was used up until around the 13th century, yet didn’t have a zero, and it was incredibly cumbersome. Before that, the Sumerians and Babylonians counted in base 60 (which is still in use today for telling time and measuring angle), but also lacked a zero. The Mayan civilization of Central America used a glyph for a zero in some of their more complicated calendars, but its use varied, and it never made it out of Central America, so it can’t be related to “our” zero.
Aczel’s quest for the first zero takes him around the globe. Along the way, he encounters Indiana Jones-esque artefacts with splendid names, like the Ishango Bone, the Aztec Stone of the Sun, the Bakhshali Manuscript and the Nana Ghat Inscriptions. He travels to Thailand, Laos and Vietnam, and to India, where he studies a stone inscription found at the Chatturbhuja temple in Gwalior, which has been dated to 876 AD, and was long thought to be the planet’s first zero.
Eventually, his research leads him to Cambodia in search of an inscription originally discovered by French scholar George Coedès in 1929. The stone was found among the ruins of a 7th century temple at Sambor-on-Mekong, in present-day Kratie.
The key phrase on the stone is a date marker: “The Chaka era reached 605 in the year of the waning moon,” and can be dated to 683 AD, making it a full two centuries earlier than the Indian zero.
The stone with the zero (actually a dot), known prosaically as K-127, was moved to Phnom Penh, then to Siem Reap in 1969, when it disappeared. Many thought that the Khmer Rouge, with their hatred of culture and learning, had destroyed it. Nevertheless, Aczel tracks it down, rescues it from some entertainingly stupid Italian archaeologists, and sees it consigned, quite properly, to the National Museum (it’s worth noting, however, that K-127 is not on display at the museum, and no one seems to have much of a clue about where it is).
Along the way, Aczel considers whether Buddhist philosophical concepts about being and nothingness could have pushed Eastern thinkers toward developing the concept of zero, and decides that they probably did.
Aczel has written some 20 books on scientific topics. His prolific output and range of interests might make some of his mistakes in Finding Zero forgivable (e.g. Siem Reap does not translate to “Siam Victorious,” and you’d have had to have lived a very sheltered life to describe Phnom Penh’s National Museum as “one of the finest museums in the world”). And while it’s unlikely that “writes like a mathematician” will ever be seen as a huge compliment, Finding Zero is at least clearly and efficiently written, and tells an engaging and important story.

03 July 2015

Women solving challenges


Here are links to 2 interesting and inspiring videos  on women in chess and a link to scholarships for women from Africa and developing countries:


http://www.bbc.com/news/world-africa-21351317
Making the move from slum child to chess champion
6 February 2013 
Phiona Mutesi grew up in the slums of Kampala, Uganda, but her life changed when she walked into a local chess club which was offering free meals. Ten years on she is one of the best young female chess players in Africa.


http://www.bbc.com/news/world-us-canada-32618139
Girl, 11, is youngest US chess master
12 May 2015
For centuries chess has been a small game for big thinkers. Each player gets just 16 pieces to play on a board with 64 little squares. The numbers start out simple, but the outcomes are virtually limitless. No surprise, then, that it takes most people years to get any good - let alone become what the chess world calls a "master". Well, Carissa Yip has upended all those assumptions - for a start she's only 11 years old, and the word master doesn't really fit.


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2016 List of Scholarships for African women and Developing Countries

http://www.afterschoolafrica.com/255/international-scholarship-for-women-in/