Monday, July 17, 2017

Maryam Mirzakhani (May 3, 1977 - July 15, 2017)

Some twenty five years ago when I was told by one of  her teachers that she is truly brilliant and exceptional I could have not imagined that I will see this terribly sad day. A shining star is turned off far too soon. Heartfelt condolences to her husband, family, and mathematics community worldwide.

از شمار دو چشم یک تن کم
وز شمار خرد هزاران بیش

"Two eyes are gone
 Thousands of minds withered"

Here is the Stanford University obituary:

Stanford mathematics Professor Maryam Mirzakhani, the first and to-date only female winner of the Fields Medal since its inception in 1936, died Friday, July 14. She had been battling breast cancer since 2013; the disease spread to her liver and bones in 2016. Mirzakhani was 40 years old.
The quadrennial Fields Medal, which Mirzakhani won in 2014, is the most prestigious award in mathematics, often equated in stature with the Nobel Prize. Mirzakhani specialized in theoretical mathematics that read like a foreign language by those outside of mathematics: moduli spaces, Teichmüller theory, hyperbolic geometry, Ergodic theory and symplectic geometry.
Mastering these approaches allowed Mirzakhani to pursue her fascination for describing the geometric and dynamic complexities of curved surfaces – spheres, doughnut shapes and even amoebas – in as great detail as possible. Her work was highly theoretical in nature, but it could have impacts concerning the theoretical physics of how the universe came to exist and, because it could inform quantum field theory, secondary applications to engineering and material science. Within mathematics, it has implications for the study of prime numbers and cryptography.
Mirzakhani joined the faculty of Stanford University in 2008, where she served as a professor of mathematics until her death.
“Maryam is gone far too soon, but her impact will live on for the thousands of women she inspired to pursue math and science,” said Stanford President Marc Tessier-Lavigne. “Maryam was a brilliant mathematical theorist, and also a humble person who accepted honors only with the hope that it might encourage others to follow her path. Her contributions as both a scholar and a role model are significant and enduring, and she will be dearly missed here at Stanford and around the world.”
Despite the breadth of applications of her work, Mirzakhani said she enjoyed pure mathematics because of the elegance and longevity of the questions she studied.
A self-professed “slow” mathematician, Mirzakhani’s colleagues describe her as ambitious, resolute and fearless in the face of problems others would not, or could not, tackle. She denied herself the easy path, choosing instead to tackle thornier issues. Her preferred method of working on a problem was to doodle on large sheets of white paper, scribbling formulas on the periphery of her drawings. Her young daughter described her mother at work as “painting.”
“You have to spend some energy and effort to see the beauty of math,” she told one reporter.
In another interview, she said of her process: “I don’t have any particular recipe [for developing new proofs]. … It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out.”
Mirzakhani was born in Tehran, Iran, and – by her own estimation – was fortunate to come of age after the Iran-Iraq war when the political, social and economic environment had stabilized enough that she could focus on her studies. She dreamed of becoming a writer, but mathematics eventually swept her away.
She attended an all-girls high school in Tehran, led by a principal unbowed by the fact that no girl had ever competed for Iran’s International Mathematical Olympiad team. Mirzakhani first gained international recognition during the 1994 and 1995 competitions. In 1994, she earned a gold medal. In 1995, she notched a perfect score and another gold medal.
After graduating college at Sharif University in Tehran, she headed to graduate school at Harvard University, where she was guided by Curtis McMullen, a fellow Fields Medal winner. At Harvard, Mirzakhani was distinguished by her determination and relentless questioning, despite the language barrier. She peppered her professors with questions in English. She jotted her notes in Farsi.
McMullen described Mirzakhani as filled with “fearless ambition.” Her 2004 dissertation was a masterpiece. In it, she solved two longstanding problems. Either solution would have been newsworthy in its own right, according to Benson Farb, a mathematician at the University of Chicago, but then Mirzakhani connected the two into a thesis described as “truly spectacular.” It yielded papers in each of the top three mathematics journals.
“The majority of mathematicians will never produce something as good,” Farb said at the time. “And that’s what she did in her thesis.”
Iranian President Hassan Rouhani said the “unprecedented brilliance of this creative scientist and modest human being, who made Iran’s name resonate in the world’s scientific forums, was a turning point in showing the great will of Iranian women and young people on the path towards reaching the peaks of glory … in various international arenas,” according to Iranian state media.
“What’s so special about Maryam, the thing that really separates her, is the originality in how she puts together these disparate pieces,” said Steven Kerckhoff at the time of her Fields Medal award. Kerckhoff is a professor at Stanford who works in the same area of mathematics. “That was the case starting with her thesis work, which generated several papers in all the top journals. The novelty of her approach made it a real tour de force.”
After earning her doctorate at Harvard, Mirzakhani accepted a position as assistant professor at Princeton University and as a research fellow at the Clay Mathematics Institute before joining the Stanford faculty.
“Maryam was a wonderful colleague,” said Ralph L. Cohen, the Barbara Kimball Browning Professor of Mathematics at Stanford. “She  not only was a brilliant and fearless researcher, but she was also a great teacher and terrific PhD adviser.  Maryam embodied what being a mathematician or scientist is all about:  the attempt to solve a problem that hadn’t been solved before, or to understand something that hadn’t been understood before.  This is driven by a deep intellectual curiosity, and there is great joy and satisfaction with every bit of success. Maryam had one of the great intellects of our time, and she was a wonderful person.  She will be tremendously missed.”
In recent years, she collaborated with Alex Eskin at the University of Chicago to answer a mathematical challenge that physicists have struggled with for a century: the trajectory of a billiard ball around a polygonal table. That investigation into this seemingly simple action led to a 200-page paper which, when it was published in 2013, was hailed as “the beginning of a new era” in mathematics and “a titanic work.”
“You’re torturing yourself along the way,” she would offer, “but life isn’t supposed to be easy.”
Mirzakhani is survived by her husband, Jan Vondrák, and a daughter, Anahita.
The university will organize a memorial service and an academic symposium in her honor in the fall, when students and faculty have returned to campus.

Sunday, April 16, 2017

Great Thanks!

Let me express my heartfelt thanks to the organizers of the Shanghai 2017 NCG-event

as well as to each participant. Your presence and talks were the most wonderful gift showing so many lively facets of mathematics and physics displaying the vitality of NCG! The whole three weeks were a pleasure due to the wonderful hospitality of our hosts, Xiaoman Chen, Guoliang Yu and Yi-Jun Yao. It is great that the many talks of the three workshops have been recorded in video which will be put online after being edited into appropriate format and upon approval of each speaker. For now the list of abstracts (which will be updated when I get the missing abstracts) is available here, with a few last minute nuances which I will mention:
Farzad Fathizadeh could not come because of obvious worries on getting back in US after a trip abroad in the DT times. So his talk on the second day of the first workshop was delivered by Masoud Khalkhali. On the second week, Dennis Sullivan gave a wonderful improvised talk (not the same as in the abstract) on the subject of "Sobolev manifolds" i.e. manifolds in which the natural regularity of functions is to have a first derivative in L^p where the critical case is when p=d is the dimension of the manifold. Dennis was as he used to be in IHES, making key comments in every talk on the second week. His generosity with his time and devotion to "understanding" showed at their best. There were many many interactions during these three weeks and it will take time to digest them, but modernity does have some advantages and the recording of the talks should help a lot!  For sure there was a general feeling of ongoing excitement to see how the field of NCG has developed and branched so diversely after 40 years, and each of the words connected to this special event went straight to my heart.
ps: Here is the "histoire courte" with Jacques Dixmier, so remarkably well elaborated by Anne Papillault and Jean-François Dars, which was shown on the large screen at the banquet on 01/04
and here is a link to the photos Jean-François took on April first !

Monday, April 10, 2017


It is with immense emotion and sorrow that we learnt the sudden death of David Goss. 
He was our dear friend, a joyful supporter of our field and a constant source of inspiration through his great work, of remarkable originality and depth, on function field arithmetics.
We are profoundly saddened by this tragic loss.
 David will remain in our heart, we shall miss him dearly.

Tuesday, February 7, 2017

Connes 70

I am happy to report that to  celebrate Alain Connes' 70th birthday, 3 conferences on noncommutative geometry and its interactions with different fields are planned to take place in Shanghai, China. Students, postdocs, young faculty and all those interested in the subject are encouraged to participate. Please check the  Conference webpage for more details.

Tuesday, January 3, 2017

Gamma functions and nonarchimedean analysis

Happy New Year!

I view blog writing as a great opportunity to reach out to members of the mathematics community and especially the younger members; so in this sense blog writing is, for me, very similar to writing for Math Reviews. I have enjoyed doing both for many years (and many many years for MR!). Recently I wrote a review for MR on the paper ``Twisted characteristic p zeta functions'' written
by Bruno Angles, Tuan Ngo Dac and Floric Ribeiro Tavares (``MR Number: MR3515815''). I am attaching the review here with the permission of Math Reviews. You can find it, in preprint form, here with the original (with live hyperlinks to papers) on the MR site.

The paper being reviewed makes some demands of the reader. But the devoted reader will be rewarded with an early view of a beautiful new world. Those readers familiar with Drinfeld modules know that they exist in incredible profusion: One starts with a smooth projective, geometrically connected curve X over the finite field F_q with q elements. Then one chooses a fixed closed point \infty of X and defines the algebra A to be the Dedekind domain of functions regular away from \infty; so plays the role of the integers Z in the Drinfeld theory. One instance of such an A is, of course, the ring F_q[\theta] which is, like Z, Euclidean, and indeed most of the work done so far
is concentrated on this particular A as it is both easy to work with and very similar to classical arithmetic. However, ultimately, the theory should work for general A just as the theory of Drinfeld modules (and generalizations) does. As general A is very far from factorial, one can imagine that many interesting issues arise (and the paper being reviewed discusses them from an axiomatic viewpoint).

Of course, the theory of the zeta function is intimately connected with the theory of the Gamma function and so one should also expect analogs of Gamma functions to appear in the characteristic p theory with the correct one being given decades ago by Greg Anderson and Dinesh Thakur in the polynomial case. Their function appears firstly as an element of the Tate algebra of functions inconverging on the closed unit disc. One fascinating aspect of the paper being reviewed is that this Tate algebra is replaced by Tate algebras created out of the general rings A (and so lie inside curves of
higher genus as opposed to the affine line). This is the beautiful new world I mentioned above…..

Sunday, July 24, 2016

A motivic product formula

The classical product formula for number fields is a fundamental tool in arithmetic. In 1993, Pierre Colmez published a truly inspired generalization of this to the case of Grothendieck's motives. In turn, this spring Urs Hartl and Rajneesh Kumar Singh put an equally inspired manuscript on the arXiv devoted to translating Colmez into the theory of Drinfeld modules and the like. Underneath the mountains of terminology there is a fantastic similarity between these two beautiful papers and I have created a blog to bring this to the attention of the community. Please see:

Tuesday, June 14, 2016

What is a functional equation?

Like all number theorists I am fascinated (to say the least) with the functional equation of 
classical  L-series. Years ago, I came up with a simple characterization of functional equations basically using only complex conjugation. This point being that, via a canonical change of variables (going back to Riemann), such L-series are, up to a nonzero scalar, given by real power series with the expectation that the zeroes are also real. In characteristic p the best one can hope is also that the zeroes will be as rational as the coefficients (though this statement needs to be modified to take care of standard factorizations as well as the great generality of Drinfeld's base rings A).

For those interested, a two page pdf can be found at the following link: