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[This guest post is authored by Caroline Series.]

The Chern Medal is a relatively new prize, awarded once every four years jointly by the IMU
and the Chern Medal Foundation (CMF) to an individual whose accomplishments warrant
the highest level of recognition for outstanding achievements in the field of mathematics.
Funded by the CMF, the Medalist receives a cash prize of US$250,000. In addition, each Medalist may nominate one or more organizations to receive funding totalling US$ 250,000, for the support of research, education, or other outreach programs in the field of mathematics.

Professor Chern devoted his life to mathematics, both in active research and education, and in nurturing the field whenever the opportunity arose. He obtained fundamental results in all the major aspects of modern geometry and founded the area of global differential geometry. Chern exhibited keen aesthetic tastes in his selection of problems, and the breadth of his work deepened the connections of geometry with different areas of mathematics. He was also generous during his lifetime in his personal support of the field.

Nominations should be sent to the Prize Committee Chair:  Caroline Series, email: chair@chern18.mathunion.org by 31st December 2016. Further details and nomination guidelines for this and the other IMU prizes can be found at http://www.mathunion.org/general/prizes/

[This guest post is authored by Matilde Lalin, an Associate Professor in the Département de mathématiques et de statistique at the Université de Montréal.  I have lightly edited the text, mostly by adding some HTML formatting. -T.]

Mathematicians (and likely other academics!) with small children face some unique challenges when traveling to conferences and workshops. The goal of this post is to reflect on these, and to start a constructive discussion what institutions and event organizers could do to improve the experiences of such participants.

The first necessary step is to recognize that different families have different needs. While it is hard to completely address everybody’s needs, there are some general measures that have a good chance to help most of the people traveling with young children. In this post, I will mostly focus on nursing mothers with infants ($\leq 24$ months old) because that is my personal experience. Many of the suggestions will apply to other cases such as non-nursing babies, children of single parents, children of couples of mathematicians who are interested in attending the same conference, etc..

The mother of a nursing infant that wishes to attend a conference has three options:

1. Bring the infant and a relative/friend to help caring for the infant. The main challenge in this case is to fund the trip expenses of the relative. This involves trip costs, lodging, and food. The family may need a hotel room with some special amenities such as crib, fridge, microwave, etc. Location is also important, with easy access to facilities such as a grocery store, pharmacy, etc. The mother will need to take regular breaks from the conference in order to nurse the baby (this could be as often as every three hours or so). Depending on personal preferences, she may need to nurse privately. It is convenient, thus, to make a private room available, located as close to the conference venue as possible. The relative may need to have a place to stay with the baby near the conference such as a playground or a room with toys, particularly if the hotel room is far.
2. Bring the infant and hire someone local (a nanny) to help caring for the infant. The main challenges in this case are two: finding the caregiver and paying for such services. Finding a caregiver in a place where one does not live is hard, as it is difficult to conduct interviews or get references. There are agencies that can do this for a (quite expensive) fee: they will find a professional caregiver with background checks, CPR certification, many references, etc. It may be worth it, though, as professional caregivers tend to provide high-quality services and peace of mind is priceless for the mother mathematician attending a conference. As in the previous case, the mother may have particular needs regarding the hotel room, location, and all the other facilities mentioned for Option 1.
3. Travel without the infant and pump milk regularly. This can be very challenging for the mother, the baby, and the person that stays behind taking care of the baby, but the costs of this arrangement are much lower than in Option 1 or 2 (I am ignoring the possibility that the family needs to hire help at home, which is necessary in some cases). A nursing mother away from her baby has no option but to pump her milk to prevent her from pain and serious health complications. This mother may have to pump milk very often. Pumping is less efficient than nursing, so she will be gone for longer in each break or she will have more breaks compared to a mother that travels with her baby. For pumping, people need a room which should ideally be private, with a sink, and located as close to the conference venue as possible. It is often impossible for these three conditions to be met at the same time, so different mothers give priority to different features. Some people pump milk in washrooms, to have easy access to water. Other people might prefer to pump in a more comfortable setting, such as an office, and go to the washroom to wash the breast pump accessories after. If the mother expects that the baby will drink breastmilk while she is away, then she will also have to pump milk in advance of her trip. This requires some careful planning.Many pumping mothers try to store the pumped milk and bring it back home. In this case the mother needs a hotel room with a fridge which (ideally, but hard to find) has a freezer. In a perfect world there would also be a fridge in the place where she pumps/where the conference is held.

It is important to keep in mind that each option has its own set of challenges (even when expenses and facilities are all covered) and that different families may be restricted in their choice of options for a variety of reasons. It is therefore important that all these three options be facilitated.

As for the effect these choices have on the conference experience for the mother, Option 1 means that she has to balance her time between the conference and spending time with her relative/friend. This pressure disappears when we consider Option 2, so this option may lead to more participation in the conferences activities. In Option 3, the mother is in principle free to participate in all the conference activities, but the frequent breaks may limit the type of activity. A mother may choose different options depending on the nature of the conference.

I want to stress, for the three options, that having to make choices about what to miss in the conference is very hard. While talks are important, so are the opportunities to meet people and discuss mathematics that happen during breaks and social events. It is very difficult to balance all of this. This is particularly difficult for the pumping mother in Option 3: because she travels without her baby, she is not perceived to be a in special situation or in need of accommodation. However, this mother is probably choosing between going to the last lecture in the morning or having lunch alone, because if she goes to pump right after the last lecture, by the time she is back, everybody has left for lunch.

Here is the Hall of Fame for those organizations that are already supporting nursing mothers’ travels in mathematics:

• The Natural Sciences and Engineering Research Council of Canada (NSERC) (search for “child care”) allows to reimburse the costs of child care with Option 2 out of the mother’s grants. They will also reimburse the travel expenses of a relative with Option 1 up to the amount that would cost to hire a local caregiver.
• The ENFANT/ELEFANT conference (co-organized by Lillian Pierce and Damaris Schindler) provided a good model to follow regarding accommodation for parents with children during conferences that included funding for covering the travel costs of accompanying caretakers (the funding was provided by the Deutsche Forschungsgemeinschaft, and lactation rooms and play rooms near the conference venue (the facilities were provided by the Hausdorff Center for Mathematics).Additional information (where to go with kids, etc) was provided on site by the organizers and was made available to all participants all the time, by means of a display board that was left standing during the whole week of the conference.
• The American Institute of Mathematics (AIM) reimburses up to 500 dollars on childcare for visitors and they have some online resources that assist in finding childcare and nannies.

[UPDATED] Added a few more things to the Hall of Fame

In closing, here is a (possibly incomplete) list of resources that institutes, funding agencies, and conferences could consider providing for nursing mother mathematicians:

1. Funding (for cost associated to child care either professional or by an accompanying relative).
2. List of childcare resources (nannies, nanny agencies, drop-in childcare centre, etc).
3. Nursing rooms and playrooms near the conference venue. Nearby fridge.
4. Breaks of at least 20 minutes every 2-3 hours.
5. Information about transportation with infants. More specific, taxi and/or shuttle companies that provide infant car seats. Information regarding the law on infant seats in taxis and other public transportation.
6. Accessibility for strollers.
7. [UPDATED] A nearby playground location. (comment from Peter).

I also find it important that these resources be listed publicly in the institute/conference website. This serves a double purpose: first, it helps those in need of the resources to access them easily, and second, it contributes to make these accommodations normal, setting a good model for future events, and inspiring organizers of future events.

Finally, I am pretty sure that the options and solutions I described do not cover all cases. I would like to finish this note by inviting readers to make suggestions, share experiences, and/or pose questions about this topic.

[This guest post is authored by Ingrid Daubechies, who is the current president of the International Mathematical Union, and (as she describes below) is heavily involved in planning for a next-generation digital mathematical library that can go beyond the current network of preprint servers (such as the arXiv), journal web pages, article databases (such as MathSciNet), individual author web pages, and general web search engines to create a more integrated and useful mathematical resource. I have lightly edited the post for this blog, mostly by adding additional hyperlinks. – T.]

This guest blog entry concerns the many roles a World Digital Mathematical Library (WDML) could play for the mathematical community worldwide. We seek input to help sketch how a WDML could be so much more than just a huge collection of digitally available mathematical documents. If this is of interest to you, please read on!

The “we” seeking input are the Committee on Electronic Information and Communication (CEIC) of the International Mathematical Union (IMU), and a special committee of the US National Research Council (NRC), charged by the Sloan Foundation to look into this matter. In the US, mathematicians may know the Sloan Foundation best for the prestigious early-career fellowships it awards annually, but the foundation plays a prominent role in other disciplines as well. For instance, the Sloan Digital Sky Survey (SDSS) has had a profound impact on astronomy, serving researchers in many more ways than even its ambitious original setup foresaw. The report being commissioned by the Sloan Foundation from the NRC study group could possibly be the basis for an equally ambitious program funded by the Sloan Foundation for a WDML with the potential to change the practice of mathematical research as profoundly as the SDSS did in astronomy. But to get there, we must formulate a vision that, like the original SDSS proposal, imagines at least some of those impacts. The members of the NRC committee are extremely knowledgeable, and have been picked judiciously so as to span collectively a wide range of expertise and connections. As president of the IMU, I was asked to co-chair this committee, together with Clifford Lynch, of the Coalition for Networked InformationPeter Olver, chair of the IMU’s CEIC, is also a member of the committee. But each of us is at least a quarter century older than the originators of MathOverflow or the ArXiv when they started. We need you, internet-savvy, imaginative, social-networking, young mathematicians to help us formulate the vision that may inspire the creation of a truly revolutionary WDML!

Some history first.  Several years ago, an international initiative was started to create a World Digital Mathematical Library. The website for this library, hosted by the IMU, is now mostly a “ghost” website — nothing has been posted there for the last seven years. [It does provide useful links, however, to many sites that continue to be updated, such as the European Mathematical Information Service, which in turn links to many interesting journals, books and other websites featuring electronically available mathematical publications. So it is still worth exploring …] Many of the efforts towards building (parts of) the WDML as originally envisaged have had to grapple with business interests, copyright agreements, search obstructions, metadata secrecy, … and many an enterprising, idealistic effort has been slowly ground down by this. We are still dealing with these frustrations — as witnessed by, e.g., the CostofKnowledge initiative. They are real, important issues, and will need to be addressed.

[This article was guest authored by Frank Morgan, the vice president of the American Mathematical Society.]
The American Mathematical Society (AMS) has launched a new blog

[This post is authored by Timothy Chow.]

I recently had a frustrating experience with a certain out-of-print mathematics text that I was interested in.  A couple of used copies were listed at over \$150 a pop on Bookfinder.com, but that was more than I was willing to pay.  I wrote to the American Mathematical Society asking if they were interested in bringing the book back into print.  To their credit, they took my request seriously, and solicited the opinions of some other mathematicians.

Unfortunately, these referees all said that the field in question was not active, and in any case there was a more recent text that was a better reference.  So the AMS rejected my proposal.  I have to say that I was surprised, because the referees did not back up their opinions with any facts, and I knew that in addition to the high price that the book commanded on the used-book market, there was some circumstantial evidence that it was in demand.  A MathSciNet search confirmed my belief that, contrary to what the referees had said, the field was most definitely active.  Plus, another text on the same subject that Dover had recently brought back into print had a fine Amazon sales rank (much higher than that of the recent text cited by the referees).

A colleague then suggested that maybe I should instead contact the author directly, asking him to regain the copyright from the publisher.  The author could then make the book available on his website or pursue print-on-demand options, if conventional publishers were not interested. I tried this, but was again surprised to discover that the author thought it was not worth the trouble to get the copyright back, let alone to make the text available.  Again the argument was that, allegedly, nobody was interested in the book.

In both cases I was frustrated because I did not know how to find other people who were interested in the same book, to prove to the AMS or the author that there were in fact many of us who wanted to see the book back in print.

Now for the good news.  After hearing my story, Klaus Schmid promptly set up a prototype website at

Anyone can go to this site and suggest a book, or vote for books that others have suggested.  This is precisely the kind of information that I believe would have greatly helped me argue my case.  Of course, the site works only if people know about it, so if you like the idea, please spread the word to your friends and colleagues.

It might be that a better long-term solution than Schmid’s site is to convince a bookselling website to tally votes of this sort, because such a site will catch users “red-handed” searching for an out-of-print book.  I have tried to contact some sites with this suggestion; so far, Booksprice.com and Fetchbook.info have said that they like the idea and may eventually implement it.  In the meantime, hopefully Schmid’s site will  become a useful tool in its own right.

Let me conclude with a question.  What else can we be doing to increase the availability of out-of-print books, especially those that are still copyrighted?  Several people have told me that the solution is for authors to regain the copyrights to their out-of-print books and make their books available themselves, but authors are often too busy (if they are not deceased!).  What can we do to help in such situations?

[This post is authored by Gil Kalai, who has kindly “guest blogged” this week’s “open problem of the week”. – T.]

The entropy-influence conjecture seeks to relate two somewhat different measures as to how a boolean function has concentrated Fourier coefficients, namely the total influence and the entropy.

We begin by defining the total influence. Let $\{-1,+1\}^n$ be the discrete cube, i.e. the set of $\pm 1$ vectors $(x_1,\ldots,x_n)$ of length n. A boolean function is any function $f: \{-1,+1\}^n \to \{-1,+1\}$ from the discrete cube to {-1,+1}. One can think of such functions as “voting methods”, which take the preferences of n voters (+1 for yes, -1 for no) as input and return a yes/no verdict as output. For instance, if n is odd, the “majority vote” function $\hbox{sgn}(x_1+\ldots+x_n)$ returns +1 if there are more +1 variables than -1, or -1 otherwise, whereas if $1 \leq k \leq n$, the “$k^{th}$ dictator” function returns the value $x_k$ of the $k^{th}$ variable.

We give the cube $\{-1,+1\}^n$ the uniform probability measure $\mu$ (thus we assume that the n voters vote randomly and independently). Given any boolean function f and any variable $1 \leq k \leq n$, define the influence $I_k(f)$ of the $k^{th}$ variable to be the quantity

$I_k(f) := \mu \{ x \in \{-1,+1\}^n: f(\sigma_k(x)) \neq f(x) \}$

where $\sigma_k(x)$ is the element of the cube formed by flipping the sign of the $k^{th}$ variable. Informally, $I_k(f)$ measures the probability that the $k^{th}$ voter could actually determine the outcome of an election; it is sometimes referred to as the Banzhaf power index. The total influence I(f) of f (also known as the average sensitivity and the edge-boundary density) is then defined as

$I(f) := \sum_{k=1}^n I_k(f).$

Thus for instance a dictator function has total influence 1, whereas majority vote has total influence comparable to $\sqrt{n}$. The influence can range between 0 (for constant functions +1, -1) and n (for the parity function $x_1 \ldots x_k$ or its negation). If f has mean zero (i.e. it is equal to +1 half of the time), then the edge-isoperimetric inequality asserts that $I(f) \geq 1$ (with equality if and only if there is a dictatorship), whilst the Kahn-Kalai-Linial (KKL) theorem asserts that $I_k(f) \gg \frac{\log n}{n}$ for some k. There is a result of Friedgut that if $I(f)$ is bounded by A (say) and $\varepsilon > 0$, then f is within a distance $\varepsilon$ (in $L^1$ norm) of another boolean function g which only depends on $O_{A,\varepsilon}(1)$ of the variables (such functions are known as juntas).

[This post is authored by Alexandre Borovik. An extended version of this post, with further links and background information, can be found on Alexandre’s blog.]

I had never in my life seen an arrested blackboard encircled by a police tape, still with some group theory problems on it.

But this had actually happened when the Turkish authorities closed
down the Mathematical Summer School
run by Ali Nesin for Turkish undergraduate students; Ali Nesin may face prosecution for an Orwellian offence of “education without permission“. Please sign the online petition in his support.
Read the rest of this entry »

[This post is authored by Emmanuel Kowalski.]

This post may be seen as complementary to the post “The parity problem in sieve theory“. In addition to a survey of another important sieve technique, it might be interesting as a discussion of some of the foundational issues which were discussed in the comments to that post.

Many readers will certainly have heard already of one form or another of the “large sieve inequality”. The name itself is misleading however, and what is meant by this may be something having very little, if anything, to do with sieves. What I will discuss are genuine sieve situations.

The framework I will describe is explained in the preprint arXiv:math.NT/0610021, and in a forthcoming Cambridge Tract. I started looking at this first to have a common setting for the usual large sieve and a “sieve for Frobenius” I had devised earlier to study some arithmetic properties of families of zeta functions over finite fields. Another version of such a sieve was described by Zywina (“The large sieve and Galois representations”, preprint), and his approach was quite helpful in suggesting more general settings than I had considered at first. The latest generalizations more or less took life naturally when looking at new applications, such as discrete groups.

Unfortunately (maybe), there will be quite a bit of notation involved; hopefully, the illustrations related to the classical case of sieving integers to obtain the primes (or other subsets of integers with special multiplicative features) will clarify the general case, and the “new” examples will motivate readers to find yet more interesting applications of sieves.

[This post is authored by Gil Kalai, who has kindly “guest blogged” this week’s “open problem of the week”. – T.]

This is a problem in discrete and convex geometry. It seeks to quantify the intuitively obvious fact that large convex bodies are so “fat” that they cannot avoid “detection” by a small number of observation points. More precisely, we fix a dimension d and make the following definition (introduced by Haussler and Welzl):

• Definition: Let $X \subset {\Bbb R}^d$ be a finite set of points, and let $0 < \epsilon < 1$. We say that a finite set $Y \subset {\Bbb R}^d$ is a weak $\epsilon$-net for X (with respect to convex bodies) if, whenever B is a convex body which is large in the sense that $|B \cap X| > \epsilon |X|$, then B contains at least one point of Y. (If Y is contained in X, we say that Y is a strong $\epsilon$-net for X with respect to convex bodies.)

For example, in one dimension, if $X = \{1,\ldots,N\}$, and $Y = \{ \epsilon N, 2 \epsilon N, \ldots, k \epsilon N \}$ where k is the integer part of $1/\epsilon$, then Y is a weak $\epsilon$-net for X with respect to convex bodies. Thus we see that even when the original set X is very large, one can create a $\epsilon$-net of size as small as $O(1/\epsilon)$. Strong $\epsilon$-nets are of importance in computational learning theory, and are fairly well understood via Vapnik-Chervonenkis (or VC) theory; however, the theory of weak $\epsilon$-nets is still not completely satisfactory.

One can ask what happens in higher dimensions, for instance when X is a discrete cube $X = \{1,\ldots,N\}^d$. It is not too hard to cook up $\epsilon$-nets of size $O_d(1/\epsilon^d)$ (by using tools such as Minkowski’s theorem), but in fact one can create $\epsilon$-nets of size as small as $O( \frac{1}{\epsilon} \log \frac{1}{\epsilon} )$ simply by taking a random subset of X of this cardinality and observing that “up to errors of $\epsilon$“, the total number of essentially different ways a convex body can meet X grows at most polynomially in $1/\epsilon$. (This is a very typical application of the probabilistic method.) On the other hand, since X can contain roughly $1/\epsilon$ disjoint convex bodies, each of which contains at least $\epsilon$ of the points in X, we see that no $\epsilon$-net can have size much smaller than $1/\epsilon$.

Now consider the situation in which X is now an arbitrary finite set, rather than a discrete cube. More precisely, let $f(\epsilon,d)$ be the least number such that every finite set X possesses at least one weak $\epsilon$-net for X with respect to convex bodies of cardinality at most $f(\epsilon,d)$. (One can also replace the finite set X with an arbitrary probability measure; the two formulations are equivalent.) Informally, f is the least number of “guards” one needs to place to prevent a convex body from covering more than $\epsilon$ of any given territory.

• Problem 1: For fixed d, what is the correct rate of growth of f as $\epsilon \to 0$?

[This post is authored by Ben Green, who has kindly “guest blogged” this week’s “open problem of the week”. – T.]

In an earlier blog post Terry discussed Freiman’s theorem. The name of Freiman is attached to a growing body of theorems which take some rather “combinatorial” hypothesis, such that the sumset |A+A| of some set A is small, and deduce from it rather “algebraic” information (such that A is contained in a subspace or a grid).

The easiest place to talk about Freiman’s theorem is in the finite field model ${\Bbb F}_2^n$ (see my survey article on this subject for a full discussion). Here it was shown by Ruzsa that if |A+A| is at most $K |A|$ then A is contained in a subspace of size no more than about $2^{K^4}|A|$. The exponent has been improved a few times since Ruzsa’s paper, the best result currently in print being due to Sanders, who obtains an upper bound of $2^{K^{3/2}\log K}|A|$. Terry and I are in the process of writing a paper which obtains $2^{2K + o(K)}|A|$, which is best possible in view of the example $A := \{e_1,...,e_m\}$ where $m := 2K + O(1)$; this set has doubling roughly K but is not contained in a subspace of dimension smaller than 2K.

This result has an air of finality (except for the true nature of the o(K) term, which represents an interesting open problem). This is something of an illusion, however. Even using this theorem, one loses an exponential every time one tries to transition between “combinatorial” structure and “algebraic” structure and back again. Indeed if one knows that A is contained in a subspace of size $2^{2K}|A|$ then the strongest assertion one can make about the doubling of A is that it is at most $2^{2K}$.

The Polynomial Freiman-Ruzsa conjecture (PFR), in ${\Bbb F}_2^n$, hypothesises a more precise structure theorem for sets with small doubling. Using this
conjecture, one may flit back and forth between combinatorial and algebraic structure with only polynomial losses. Ruzsa attributes the conjecture to
Marton: it states that if A has doubling at most K then A is contained in the union of $K^{O(1)}$ translates of some subspace H of size at most |A|.