## 2016 EMS prize winners announced

July 18, 2016

[Posted on july 18, 2016]

The 7th ECM has just started today in Berlin, and the prizes have been announced as follows (see the booklet, I’m just adding URLs). The 10 EMS Prizes for researchers under 35 went to :

• Peter Scholze (28, Arithmetic Geometry) from the Mathematisches Institut of the university of Bonn
• Sara Zahedi (35, Numerical Analysis of PDEs) from KTH
• Mark Braverman (32, Theoretical Computer Science) from the university of Princeton
• Vincent Calvez (35, Mathematical Biology) from ENS Lyon and CNRS
• Guido de Philippis (30, Calculus of Variations & Geometric measure Theory & PDEs) from SISSA Trieste
• James Maynard (29, Analytic Number Theory) from Magdalen College of the university of Oxford
• Péter Varjú (34, Random Walks in Groups) from the university of Cambridge
• Thomas Willwacher (33, Mathematical Physics) from ETH Zürich
• Geordie Williamson (35, Representation Theory) from the Max Planck Institut of the university of Bonn
• Hugo Duminil-Copin (30, Combinatorics & Probability & Mathematical Physics) from IHÉS (homepage still at Geneva)

And also :

• the Felix Klein Prize went to Patrice Hauret (38, Computational Solid Mechanics) from the french tire firm Michelin
• the Otto Neugebauer went to Jeremy Gray (69, History of Mathematics) from the Open University

Finally, the next ECM will take place in 2020 in Portoroz, Slovenia.

## Further july items

July 9, 2016

[Posted on july 9, 2016.]

Spotted recently :

• the results of the arXiv survey (previously discussed on this blog) are out and interesting (in particular the share of respondents below 30 is higher than I would have expected), hopefully the new search tools in particular will be ready by the end of the year
• by the way, the extra digit in the identifier still hasn’t been needed, but it is getting close reaching 09644 last month after a record of 09792 in may
• the paper that Shinichi Mochizuki produced last year has now been published, and there’s a rumor by David Hansen that the 4 original ones have been accepted
• the MSRI summer school on the McKay conjecture starts next week
• the youtube channel of CIRM has a playlist dedicated to talks related to the forthcoming 7th ECM,   while the channel of IHÉS has one for the june conference of its Trimestre Ondes Non Linéaires
• the 2016 IMO takes place next week, maybe an occasion for a new mini-polymath
• as it happens, the coldest place in the solar system is named after Charles Hermite

Temporary cubes next to Centre Pompidou-Metz

(july 2016, Public domain)

## Some july items

July 5, 2016

[Posted on july 5, 2016]

In no particular order :

• Shinichi Mochizuki has a new paper out (at the bottom of this page) which is an introduction to Inter-universal Teichmuller Theory building on analogies with the Gaussian integral ; the workshop at RIMS will take place in a fortnight
• on friday the third journée parité en mathématiques, organized by several french scientific societies, will take place at IHP, and will include some twitter activity on the Femmes et maths account (topically, this year’s agrégation results are out, with 235 Monsieur and 70 Madame for a ratio of 22,95%, while last year it was 21,89%, and the year before 21,09%)
• Hugo Duminil-Copin will become Permanent Professor at IHÉS next september
• several new papers have appeared in Discrete Analysis since its launch last february
• Jacob Lurie has been visiting several UK universities last week and this week as part of the LMS Hardy Lectureship
• the UK’s Council for the Mathematical Sciences has just released a statement regarding the Brexit issue

Square des Poètes, Paris june 2016 (Public domain).

## Consequences of Brexit on mathematics

June 30, 2016

[Posted on june 30, 2016]

Like many observers, I have been quite stunned by the result. So, even though I have no legitimaticy for this (as I’m not an academic — I do have a british PhD but to some Leavers this is a bias), I’m posting this to gather opinions on possible impacts of Brexit on mathematics : both straightforward ones and chain reactions. I’d like in particular to explore how to mitigate the effects of the negative ones.

For general issues beyond mathematics there are several relevant articles in the THES : on EU grant application, (and that earlier one) and on non-british students and academics. (Edit 02 july: see also these frightening figures for Cambridge alone). As yet, Article 50 has not been triggered, so what follows only makes sense under the hypothesis that at some point before the end of the year it will.

1) Clearly one specific issue for mathematics is the health and standing of the UK math departments :

1.a) what is the share of EU-funded Postdocs and Postgraduates students there ? In particular what are the numbers in the very select places like Cambridge, Oxford, Imperial, Warwick… ?

1.b) what is the annual share of EU-funded conferences ?

How to mitigate a potentially complete stop to both funding and free movement ? Might a form of dual diplomas and dual conference system work (systematically coupling things with an EU partner that has funding)  ?

2) One possible ripple effect is the movement of academics to nearby EU countries (or Switzerland) to follow that pool of students and postdocs :

2.a) which continental places would likely benefit, and would they actively try to lure away UK-based mathematicians ?

2.b) what about the US and Canada, is brexit seen as an opportunity there to recruit UK-based mathematicians ?

3) Are there specific mathematics-related issues not mentioned above ?

(Note: the comments to this post will go to the moderation queue, which I can review about twice a day usually).

Brexit logo by swissbert on flickr

## Late april items

April 30, 2016

[Posted on april 30, 2016]

In no particular order :

• the latest issue of Gazette de Mathematiciens has just appeared, it  includes an introduction to the Berkovich line by Jérôme Poineau, and also part of an exchange between Alice Jacquet and Claire Mathieu on the work of the latter and her collaborators on the emergence of a glass ceiling in social networks (here is the paper of Mathieu et al itself, and note also a talk about this at BNF next month) among lots of other things
• Claire Voisin has very recently been elected to the newly created Chaire de Géométrie Algébrique du Collège de France
• an article from the Harvard Crimson about tenure choices in the math departement
• a curious case of nearly full Open Access : Elsevier-owned Comptes Rendus Mathématiques will henceforth provide some of its articles for free, for that the corresponding author must have a french affiliation (technically, an email at an insitution based in France) otherwise the article will stay behind a paywall for readers but with the strange and fortunate freedom to post the final pdf on the institutional server of the authors (but not on a preprint server for 3 years).
• the Spiegel has recently featured Peter Scholze (behind a paywall)

Low clouds over the lake, by claudiadea131 on flickr

## That arXiv questionnaire, and other news

April 16, 2016

[Posted on april 16, 2016]

Filling that arXiv questionnaire (now offline) was interesting. No idea how many people answered, and more importantly how biased that sample will turn out to be. I hope, for the sake of transparency, that they’ll quickly make the numbers freely available (the free-comment sections are of course private).

Also, it’s informative to see how varied opinions can be. I do agree with some points made by Izabella Laba in her blog post : no comments please (think low quality MO questions that quickly and deservedly get many downvotes, or the sometimes very irrelevant comments made by amateurs on blogs). That would surely drive lots of serious folks away. Some people agree, others don’t. On the other hand,  flagging “substantial overlap” could be useful if properly defined, IMHO.

As for citations stats and tools, taking a well-known preprint that never got formally published, by just clicking on the NASA ADS link one easily gets useful citation tools, while the blog trackbacks are suitably moderated. Not sure what could be added on top of that.

In other news :

• topically, a math.GM paper on Navier-Stokes made it to a local story
• a wonderfully clear and interesting talk by Mireille Bousquet-Mélou at CIRM (in french, but with slides in english) on plane lattice paths avoiding a quadrant, a topic related to a series of works done in the past 15 years by lots of people (and where many nice things occur, like the issue of finiteness or not of a certain group naturally associated to the path counting method)

• an interview of Manjul Bhargava in CNRS News made after the conference mentioned in the previous post
• a job ad for a mathematician in the gaming industry in Dublin

## Early april news

April 5, 2016

[Posted on april 5, 2016]

In no particular order :

• Laurent Lafforgue recently gave a colloquium (direct link to the mp4 file) on Grothendieck Toposes in Nantes “based on his conversations with Olivia Caramello” (he is announcing a long common text in the process of being written, to appear on his website at some point)
• registration opened a few days ago for the summer school at Institut Fourier on Geometric Analysis, Metric Geometry and Topology
• a workshop on Geometric Langlands and a conjecture of Fargues is taking place this week in Oberwolfach, while a workshop on the work of Vincent Lafforgue will take place at AIM next december
• a recent and highly interesting AI paper that takes into account what makes things look natural to humans (inner sense of what is physically possible or not, and first-person view of the world) by Brenden M. Lake, Tomer D. Ullman, Joshua B. Tenenbaum and Samuel J. Gershman ; the fact that these gentlemen work at high-profile institutions will hopefully turn this paper into a kind of benchmark along which other folks will want to test their systems
• Manjul Bhargava gave several talks in Paris yesterday, including one on Ramanujan’s work (the movie is announced for september in France, but will be released in just a few weeks in several countries), as part of the UNESCO’s International Conference on the Zero

Blue in the shades, by coniferconifer on flickr

## Poisson, Guinand, Meyer

March 27, 2016

[Posted on march 27, 2016]

A recent highly interesting paper by Yves Meyer (PNAS paywalled, local version at ENS Cachan, and seminar notes) constructs explicitly new Poisson-type summation formulas (building on previous little known work of Andrew-Paul Guinand  and an existence result of Nir Lev and Aleksander Olevskii) : the big difference with Poisson summation is that the new formulas do not have support on a lattice but only on a locally finite set (and then provide new examples of crystalline measures).

Since these new results involve some arithmetic (see below) I’ve asked over at MO whether this was known to number theorists, but there hasn’t been any immediate answer, so perhaps not and there’s probably room for interesting further work on the topic.

To state things very explicitely (for my own benefit, but also just for the beauty of it), here are the formulas taken directly from Meyer’s paper :

Poisson (Dirac comb case): on a lattice $\Gamma\subset\mathbb{R}^n$ and its dual $\Gamma^*$ we have for any function $f$ in the Schwartz class $\mathcal{S}(\mathbb{R}^n)$ that

$\displaystyle \mbox{vol}(\Gamma) \sum_{\gamma\in\Gamma}f(\gamma) = \sum_{\eta \in\Gamma^*}\widehat{f}(\eta)$

Poisson (corollary of Dirac comb case) : for every $\alpha,\beta \in\mathbb{R}^n$ we have (in terms of distributions to make the comb more explicit still)

$\displaystyle \mbox{vol}(\Gamma) \sum_{\gamma\in\Gamma +\alpha} e^{2i\pi\beta .\gamma}\delta_{\gamma} = e^{2i\pi\alpha .\beta} \sum_{\eta \in\Gamma^* +\beta} e^{2i\pi\alpha .\eta}\delta_{\eta}$

Guinand : define for any $n\in\mathbb{N}$ the number of sums of three squares that equal to $n$ by $r_3(n)$ (by Legendre’s theorem this is possible only for those $n$ not of the form $4^j(8k+7)$). Then introducing Guinand’s distribution (acting on functions of the variable $t$)

$\displaystyle \sigma := -2\frac{d}{dt}\delta_0 + \sum_{n=1}^{+\infty} \frac{r_3(n)}{\sqrt{n}} (\delta_{\sqrt{n}}-\delta_{-\sqrt{n}})$

then we have $\langle \sigma ,f\rangle = \langle -i\sigma ,\widehat{f}\rangle$.

Meyer (first example) : introducing the function $\chi$ on $\mathbb{N}$ (this is a clash of notation with Dirichlet characters) by $\chi(n)=-\frac{1}{2}$ when $n\not\equiv 0\pmod{4}$, $\chi (n)=4$ when $n\equiv 4\pmod{16}$ and $\chi (n)= 0$ when $n\equiv 0\pmod{16}$ then with the distribution

$\displaystyle \tau := \sum_{n=1}^{+\infty} \frac{\chi(n)r_3(n)}{\sqrt{n}} (\delta_{\frac{\sqrt{n}}{2}}-\delta_{-\frac{\sqrt{n}}{2}})$

we have $\langle \tau ,f\rangle = \langle -i\tau ,\widehat{f}\rangle$.

The support of $\sigma$ and $\tau$ are thus defined as subsets of $\{\pm\sqrt{n}|n\in\mathbb{N}\}$ and $\{\pm\frac{\sqrt{n}}{2}|n\in\mathbb{N}\}$ by their respective arithmetic conditions, and thus are definitely not equally spaced lattice points.

Meyer (second example) : with the distribution

$\displaystyle \rho := 2\pi\delta_{\frac{1}{2}} +2\pi\delta_{-\frac{1}{2}} + \sum_{n=1}^{+\infty} \frac{\sin(\pi\sqrt{n})r_3(n)}{\sqrt{n}} (\delta_{\frac{\sqrt{n}}{2}+\frac{1}{2}}+\delta_{\frac{\sqrt{n}}{2}-\frac{1}{2}}+\delta_{-\frac{\sqrt{n}}{2}+\frac{1}{2}}+\delta_{-\frac{\sqrt{n}}{2}-\frac{1}{2}} )$

we have $\langle \rho ,f\rangle = \langle \rho ,\widehat{f}\rangle$ (very nice!).

There are several other examples in Meyer’s paper, as well as higher-dimensional constructions (that I haven’t absorbed yet, so I’ll stop here).

Update (march 27): two relevant papers I’ve just found

• On the Number of Primitive Representations of Integers as Sums of Squares by Shaun Cooper and Michael Hirschhorn published in Ramanujan J (2007) 13:7–25, which in particular provides the explicit formula $r_3(n)=\sum_{d^2|n}r_3^p\big ( \frac{n}{d^2}\big )$ where the function $r_3^p$ is in turn explicited ($p$ is a label standing for ‘primitive’)
• Irregular Poisson Type Summation by Yu. Lyabarskii and W.R. Madych, published in SAMPLING THEORY IN SIGNAL AND IMAGE PROCESSING Vol. 7, No. 2, May 2008, pp. 173-186, which does prove a Poisson-type formula with irregularly spaced sampling points (but if I understand well the examples they mention at the end show it is still different from the results of Guinand and Meyer, to be confirmed)

## Sphere packings and other news

March 23, 2016

[Posted on 23 march, 2016]

Apparently there have been recently two great advances on sphere packings in dimension higher than 3 :

• a paper by Maryna Viasovska establishing the densest packing in dimension 8
• another paper by Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko and Maryna Viazovska solving the 24-dimensional case (with the Leech lattice)

In other news :

Oranges, by dncnH on flickr

## Early march items

March 5, 2016

[Posted on march 5, 2016]

For immediate consumption :

• the latest issue of the Newsletter of the EMS is out, with several outstanding introductory pieces, a great read.   There’s also an article by Fabian Müller and Olaf Teschke who have measured the amount of published papers that are on the arXiv thanks to new tools provided by zbMATH. Among other things, they mention that  “for the publication year 2014, about 55% of research in algebraic geometry, algebraic topology and K-Theory is available through the arXiv but only about 10% in numerical mathematics and 1% in mathematics history or mathematics education“. I’m slightly surprised that the algebraic geometry figure isn’t into the 70-80% territory.
• as most people know by now, Discrete Analysis, the arXiv orverlay journal managed by Sir Timothy Gowers aimed at challenging for-profit publishers, is now online (and has lots of nice features)
• CNRS medals in mathematics have been awarded : a Silver Medal to Isabelle Gallagher (PDEs, in particular Navier-Stokes), and a Bronze Medal to Simon Riche (Geometric Representation Theory)
• the 2015-2016 Cours Peccot will be given by Nicolas Curien (next may at Collège de France) and by Marco Robalo (TBA)
• the 2016 Abel Prize is due to be announced soon, on march 15
• Olivia Caramello has chosen to update her public controversy with category theorists
• a 177 pages paper by Tye Lidman and Ciprian Manolescu identifies two Seiberg-Witten Floer homology theories. To the outsider, this looks very much like the length and depth of papers that cement the status of a future Fields Medalist. (As an anecdote, one can notice a math.SE question in the bibliography — yes, math.SE not MO).
• a word of caution taken from the website of the IHP conference in honor of Margulis’ 70th birthday next june, which applies to all other conferences in France in that period : “Due to the European Football Championship (EURO 2016) to be held in France, June 10th-July 10th 2016, participants are encouraged to book their hotel room a long time in advance“.

Centre Pompidou Metz and the moon, winter 2016. Public domain.