Terence Tao

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**Alan Baker 1939-2018**

I'm sorry to report that Alan Baker died yesterday after a major stroke a few days ago. I knew him first when I was an undergraduate going to his number theory lectures, and later I was a colleague of his both in the Cambridge mathematics department and at Trinity College.

He became famous when he proved a far-reaching generalization of the Gelfond-Schneider theorem, which answered Hilbert's seventh problem. Hilbert's problem was the following. Suppose that a and b are two real numbers such that a is algebraic and not equal to 0 or 1, and b is irrational and also algebraic. Must a raised to the power b be transcendental (meaning that it can't be a root of a polynomial with integer coefficients)? For example, must 2 raised to the power the square root of 2 be transcendental? If I remember correctly, Hilbert believed that this was one of the hardest of his problems, but in fact it was one of the first to be solved.

One can reformulate the theorem by taking logs. It is equivalent to saying that if a_1 and a_2 are algebraic (and not 0 or 1) and b_1 is algebraic and irrational, then b_1log(a_1)-log(a_2) cannot be zero. We can express this more symmetrically by saying that if a_1 and a_2 are algebraic and b_1 and b_2 are algebraic with an irrational ratio b_1/b_2, then b_1log(a_1)+b_2log(a_2) cannot be zero.

Yet another way of expressing the result is to say that if a_1 and a_2 are algebraic numbers and their logarithms are linearly independent over the rationals, then these logarithms must in fact be linearly independent over the algebraic numbers.

Alan Baker extended this to an arbitrary number of algebraic numbers. That is, if a_1,...,a_k are algebraic and log(a_1),...,log(a_k) are linearly independent over the rationals (meaning that no non-trivial rational combination gives zero), then they are linearly independent over the algebraic numbers (meaning that no non-trivial linear combination with algebraic coefficients gives zero). This immediately implies the transcendence of all sorts of other numbers.

In fact, Baker did more: he gave a lower bound for how far away an algebraic combination of the logs of the a_i had to be from zero (in terms of various "heights", which tell you how complicated a polynomial you need to demonstrate that a number is algebraic). This had important applications to Diophantine equations.

One result I like of Baker's is an effective version of a consequence of Roth's theorem. I won't go into details, but a famous theorem of Roth implies that for every c>2 there is a constant delta>0 such that if p and q are positive integers and a is the cube root of 2, then |a - p/q| must be at least delta/q^c. Liouville's theorem, which is much easier, shows that it must be at least 1/q^3 or something similar. A problem with Roth's theorem is that it is ineffective: it does not tell you how small the constant delta needs to be. Baker managed to prove that if c=2.995 then you can take delta to be 0.000001. I like it because it is apparently such a negligible improvement over what you get from Liouville's theorem (indeed, q has to be very large before it gives any improvement at all), but that of course reflects how difficult the problem is.

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Nominations for the Breakthrough Prize and New Horizons Prize in mathematics are now open.

The Breakthrough Prize in Mathematics is a $3,000,000 prize for transformative breakthrough(s) in mathematics, with special attention to recent developments. In selecting the prize winner we will pay particular attention to results from the last 10 years, although earlier contributions may also be taken into account. This is not intended as a lifetime achievement award, but rather it is intended to recognize someone currently making outstanding contributions. There are no age limits nor nationality restrictions. In exceptional circumstances, when a prize is awarded for joint work, the committee may split the prize. The recent winners were:

2015 Ian Agol

2016 Jean Bourgain

2017 Christopher Hacon and James McKernan jointly

There are up to 3 New Horizons Prizes in Mathematics of $100,000 each for promising young researchers, who have already produced very important work. Candidates should not have been awarded a PhD before January 1st, 2008. (This requirement can be waived in exceptional circumstances, such as an interrupted career path.) There are no nationality restrictions. In exceptional circumstances, when a prize is awarded for joint work, the committee may split the prize. Previously these prizes were offered to:

2015 Larry Guth, André Neves and Peter Scholze

2016 Mohammed Abouzaid, Hugo Duminil-Copin, and jointly Ben Elias and Geordie Williamson

2018 Aaron Naber, Maryna Viazovska and jointly Zhiwei Yun and Wei Zhang.

Nominations may be submitted online at

https://breakthroughprize.org/Nominations

A short statement is required from the nominator, along with between 1 and 3 letters of reference from other experts. The closing date for nominations is April 30th. If you have questions about the nomination process, queries can be addressed to Rob Meyer (meyer@breakthroughprize.org).

The Breakthrough Prize in Mathematics is a $3,000,000 prize for transformative breakthrough(s) in mathematics, with special attention to recent developments. In selecting the prize winner we will pay particular attention to results from the last 10 years, although earlier contributions may also be taken into account. This is not intended as a lifetime achievement award, but rather it is intended to recognize someone currently making outstanding contributions. There are no age limits nor nationality restrictions. In exceptional circumstances, when a prize is awarded for joint work, the committee may split the prize. The recent winners were:

2015 Ian Agol

2016 Jean Bourgain

2017 Christopher Hacon and James McKernan jointly

There are up to 3 New Horizons Prizes in Mathematics of $100,000 each for promising young researchers, who have already produced very important work. Candidates should not have been awarded a PhD before January 1st, 2008. (This requirement can be waived in exceptional circumstances, such as an interrupted career path.) There are no nationality restrictions. In exceptional circumstances, when a prize is awarded for joint work, the committee may split the prize. Previously these prizes were offered to:

2015 Larry Guth, André Neves and Peter Scholze

2016 Mohammed Abouzaid, Hugo Duminil-Copin, and jointly Ben Elias and Geordie Williamson

2018 Aaron Naber, Maryna Viazovska and jointly Zhiwei Yun and Wei Zhang.

Nominations may be submitted online at

https://breakthroughprize.org/Nominations

A short statement is required from the nominator, along with between 1 and 3 letters of reference from other experts. The closing date for nominations is April 30th. If you have questions about the nomination process, queries can be addressed to Rob Meyer (meyer@breakthroughprize.org).

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Applications for these four-year postdocs at ANU close on Nov 22.

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Congratulations to Jean for a well deserved award.

Jean Bourgain, Institute for Advanced Study, will receive the 2018 Steele Prize for Lifetime Achievement for the breadth of his contributions made in the advancement of mathematics. He'll receive the prize at the 2018 Joint Mathematics Meetings in San Diego. http://bit.ly/2zvSW5O (Photo: Cliff Moore, IAS.)

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Applications for this postdoctoral fellowship at Oxford close on Dec 8. (Via Ben Green.)

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Representation theory as gendered dystopia. Expressed through the medium of dance.

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Congratulations to Emmanuel! My one foray into applied mathematics was my work with him on the foundations of compressed sensing, and remains one of my favourite collaborations.

EDIT: LA Times profile at http://www.latimes.com/science/sciencenow/la-sci-sn-macarthur-genius-candes-20171010-story.html

EDIT: LA Times profile at http://www.latimes.com/science/sciencenow/la-sci-sn-macarthur-genius-candes-20171010-story.html

Candès wins MacArthur.

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Another untimely death. I never actually met Voevodsky in person, and there isn't much overlap between his areas of mathematics and mine, but I wish I had had the opportunity to interact with him while he was alive.

I've just learned that Vladamir Voevodsky has died. He was a brilliant and generous mathematician in the prime of his career (and quite humble, as well, in his own way). His pioneering work in Univalent mathematics and Homotopy Type Theory defined a new set of questions and problems (and understandings!) for a whole generation of students of type theory and constructive mathematics, myself included.

His provocations to type theorists to turn their object of study into something of "proper mathematical interest" were very important as well, and his work on B- and C-systems has been very inspiring. He made great strides in recent years, and I am sorry he will not be around to complete this very ambitious program.

A very accessible and motivating talk was given to undergraduates in 2014: https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_08_ASC_lecture.pdf

Those with some greater knowledge of type theory may be interested in his plenary lecture at RDP in 2015: https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2015_06_30_RDP.pdf

His provocations to type theorists to turn their object of study into something of "proper mathematical interest" were very important as well, and his work on B- and C-systems has been very inspiring. He made great strides in recent years, and I am sorry he will not be around to complete this very ambitious program.

A very accessible and motivating talk was given to undergraduates in 2014: https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_08_ASC_lecture.pdf

Those with some greater knowledge of type theory may be interested in his plenary lecture at RDP in 2015: https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2015_06_30_RDP.pdf

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A one-year postdoc (with possibility of extension) in the area of harmonic analysis and discrete geometry is now open for applications at Bar-Ilan University. (via Nir Lev)

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Applications are now open for the quantitative linear algebra program to be held from Mar 19 - Jun 15 2018 at the Institute for Pure and Applied Mathematics (IPAM) here at UCLA. (I am one of the organisers of this program and will be in residence for the duration.)

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