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I recently came across this question on MathOverflow asking if there are any polynomials of two variables with rational coefficients, such that the map is a bijection. The answer to this question is almost surely “no”, but it is remarkable how hard this problem resists any attempt at rigorous proof. (MathOverflow users with enough privileges to see deleted answers will find that there are no fewer than seventeen deleted attempts at a proof in response to this question!)
On the other hand, the one surviving response to the question does point out this paper of Poonen which shows that assuming a powerful conjecture in Diophantine geometry known as the BombieriLang conjecture (discussed in this previous post), it is at least possible to exhibit polynomials which are injective.
I believe that it should be possible to also rule out the existence of bijective polynomials if one assumes the BombieriLang conjecture, and have sketched out a strategy to do so, but filling in the gaps requires a fair bit more algebraic geometry than I am capable of. So as a sort of experiment, I would like to see if a rigorous implication of this form (similarly to the rigorous implication of the ErdosUlam conjecture from the BombieriLang conjecture in my previous post) can be crowdsourced, in the spirit of the polymath projects (though I feel that this particular problem should be significantly quicker to resolve than a typical such project).
Here is how I imagine a BombieriLangpowered resolution of this question should proceed (modulo a large number of unjustified and somewhat vague steps that I believe to be true but have not established rigorously). Suppose for contradiction that we have a bijective polynomial . Then for any polynomial of one variable, the surface
has infinitely many rational points; indeed, every rational lifts to exactly one rational point in . I believe that for “typical” this surface should be irreducible. One can now split into two cases:
 (a) The rational points in are Zariski dense in .
 (b) The rational points in are not Zariski dense in .
Consider case (b) first. By definition, this case asserts that the rational points in are contained in a finite number of algebraic curves. By Faltings’ theorem (a special case of the BombieriLang conjecture), any curve of genus two or higher only contains a finite number of rational points. So all but finitely many of the rational points in are contained in a finite union of genus zero and genus one curves. I think all genus zero curves are birational to a line, and all the genus one curves are birational to an elliptic curve (though I don’t have an immediate reference for this). These curves all can have an infinity of rational points, but very few of them should have “enough” rational points that their projection to the third coordinate is “large”. In particular, I believe
 (i) If is birational to an elliptic curve, then the number of elements of of height at most should grow at most polylogarithmically in (i.e., be of order .
 (ii) If is birational to a line but not of the form for some rational , then then the number of elements of of height at most should grow slower than (in fact I think it can only grow like ).
I do not have proofs of these results (though I think something similar to (i) can be found in Knapp’s book, and (ii) should basically follow by using a rational parameterisation of with nonlinear). Assuming these assertions, this would mean that there is a curve of the form that captures a “positive fraction” of the rational points of , as measured by restricting the height of the third coordinate to lie below a large threshold , computing density, and sending to infinity (taking a limit superior). I believe this forces an identity of the form
for all . Such identities are certainly possible for some choices of (e.g. for arbitrary polynomials of one variable) but I believe that the only way that such identities hold for a “positive fraction” of (as measured using height as before) is if there is in fact a rational identity of the form
for some rational functions with rational coefficients (in which case we would have and ). But such an identity would contradict the hypothesis that is bijective, since one can take a rational point outside of the curve , and set , in which case we have violating the injective nature of . Thus, modulo a lot of steps that have not been fully justified, we have ruled out the scenario in which case (b) holds for a “positive fraction” of .
This leaves the scenario in which case (a) holds for a “positive fraction” of . Assuming the BombieriLang conjecture, this implies that for such , any resolution of singularities of fails to be of general type. I would imagine that this places some very strong constraints on , since I would expect the equation to describe a surface of general type for “generic” choices of (after resolving singularities). However, I do not have a good set of techniques for detecting whether a given surface is of general type or not. Presumably one should proceed by viewing the surface as a fibre product of the simpler surface and the curve over the line . In any event, I believe the way to handle (a) is to show that the failure of general type of implies some strong algebraic constraint between and (something in the spirit of (1), perhaps), and then use this constraint to rule out the bijectivity of by some further ad hoc method.
The Polymath15 paper “Effective approximation of heat flow evolution of the Riemann function, and a new upper bound for the de BruijnNewman constant“, submitted to Research in the Mathematical Sciences, has just been uploaded to the arXiv. This paper records the mix of theoretical and computational work needed to improve the upper bound on the de BruijnNewman constant . This constant can be defined as follows. The function
where is the Riemann function
has a Fourier representation
where is the superexponentially decaying function
The Riemann hypothesis is equivalent to the claim that all the zeroes of are real. De Bruijn introduced (in different notation) the deformations
of ; one can view this as the solution to the backwards heat equation starting at . From the work of de Bruijn and of Newman, it is known that there exists a real number – the de BruijnNewman constant – such that has all zeroes real for and has at least one nonreal zero for . In particular, the Riemann hypothesis is equivalent to the assertion . Prior to this paper, the best known bounds for this constant were
with the lower bound due to Rodgers and myself, and the upper bound due to Ki, Kim, and Lee. One of the main results of the paper is to improve the upper bound to
At a purely numerical level this gets “closer” to proving the Riemann hypothesis, but the methods of proof take as input a finite numerical verification of the Riemann hypothesis up to some given height (in our paper we take ) and converts this (and some other numerical verification) to an upper bound on that is of order . As discussed in the final section of the paper, further improvement of the numerical verification of RH would thus lead to modest improvements in the upper bound on , although it does not seem likely that our methods could for instance improve the bound to below without an infeasible amount of computation.
We now discuss the methods of proof. An existing result of de Bruijn shows that if all the zeroes of lie in the strip , then ; we will verify this hypothesis with , thus giving (1). Using the symmetries and the known zerofree regions, it suffices to show that
whenever and .
For large (specifically, ), we use effective numerical approximation to to establish (2), as discussed in a bit more detail below. For smaller values of , the existing numerical verification of the Riemann hypothesis (we use the results of Platt) shows that
for and . The problem though is that this result only controls at time rather than the desired time . To bridge the gap we need to erect a “barrier” that, roughly speaking, verifies that
for , , and ; with a little bit of work this barrier shows that zeroes cannot sneak in from the right of the barrier to the left in order to produce counterexamples to (2) for small .
To enforce this barrier, and to verify (2) for large , we need to approximate for positive . Our starting point is the RiemannSiegel formula, which roughly speaking is of the shape
where , is an explicit “gamma factor” that decays exponentially in , and is a ratio of gamma functions that is roughly of size . Deforming this by the heat flow gives rise to an approximation roughly of the form
where and are variants of and , , and is an exponent which is roughly . In particular, for positive values of , increases (logarithmically) as increases, and the two sums in the RiemannSiegel formula become increasingly convergent (even in the face of the slowly increasing coefficients ). For very large values of (in the range for a large absolute constant ), the terms of both sums dominate, and begins to behave in a sinusoidal fashion, with the zeroes “freezing” into an approximate arithmetic progression on the real line much like the zeroes of the sine or cosine functions (we give some asymptotic theorems that formalise this “freezing” effect). This lets one verify (2) for extremely large values of (e.g., ). For slightly less large values of , we first multiply the RiemannSiegel formula by an “Euler product mollifier” to reduce some of the oscillation in the sum and make the series converge better; we also use a technical variant of the triangle inequality to improve the bounds slightly. These are sufficient to establish (2) for moderately large (say ) with only a modest amount of computational effort (a few seconds after all the optimisations; on my own laptop with very crude code I was able to verify all the computations in a matter of minutes).
The most difficult computational task is the verification of the barrier (3), particularly when is close to zero where the series in (4) converge quite slowly. We first use an Euler product heuristic approximation to to decide where to place the barrier in order to make our numerical approximation to as large in magnitude as possible (so that we can afford to work with a sparser set of mesh points for the numerical verification). In order to efficiently evaluate the sums in (4) for many different values of , we perform a Taylor expansion of the coefficients to factor the sums as combinations of other sums that do not actually depend on and and so can be reused for multiple choices of after a onetime computation. At the scales we work in, this computation is still quite feasible (a handful of minutes after software and hardware optimisations); if one assumes larger numerical verifications of RH and lowers and to optimise the value of accordingly, one could get down to an upper bound of assuming an enormous numerical verification of RH (up to height about ) and a very large distributed computing project to perform the other numerical verifications.
This post can serve as the (presumably final) thread for the Polymath15 project (continuing this post), to handle any remaining discussion topics for that project.
[This post is collectively authored by the ICM structure committee, whom I am currently chairing – T.]
The International Congress of Mathematicians (ICM) is widely considered to be the premier conference for mathematicians. It is held every four years; for instance, the 2018 ICM was held in Rio de Janeiro, Brazil, and the 2022 ICM is to be held in Saint Petersburg, Russia. The most highprofile event at the ICM is the awarding of the 10 or so prizes of the International Mathematical Union (IMU) such as the Fields Medal, and the lectures by the prize laureates; but there are also approximately twenty plenary lectures from leading experts across all mathematical disciplines, several public lectures of a less technical nature, about 180 more specialised invited lectures divided into about twenty section panels, each corresponding to a mathematical field (or range of fields), as well as various outreach and social activities, exhibits and satellite programs, and meetings of the IMU General Assembly; see for instance the program for the 2018 ICM for a sample schedule. In addition to these official events, the ICM also provides more informal networking opportunities, in particular allowing mathematicians at all stages of career, and from all backgrounds and nationalities, to interact with each other.
For each Congress, a Program Committee (together with subcommittees for each section) is entrusted with the task of selecting who will give the lectures of the ICM (excluding the lectures by prize laureates, which are selected by separate prize committees); they also have decided how to appropriately subdivide the entire field of mathematics into sections. Given the prestigious nature of invitations from the ICM to present a lecture, this has been an important and challenging task, but one for which past Program Committees have managed to fulfill in a largely satisfactory fashion.
Nevertheless, in the last few years there has been substantial discussion regarding ways in which the process for structuring the ICM and inviting lecturers could be further improved, for instance to reflect the fact that the distribution of mathematics across various fields has evolved over time. At the 2018 ICM General Assembly meeting in Rio de Janeiro, a resolution was adopted to create a new Structure Committee to take on some of the responsibilities previously delegated to the Program Committee, focusing specifically on the structure of the scientific program. On the other hand, the Structure Committee is not involved with the format for prize lectures, the selection of prize laureates, or the selection of plenary and sectional lecturers; these tasks are instead the responsibilities of other committees (the local Organizing Committee, the prize committees, and the Program Committee respectively).
The first Structure Committee was constituted on 1 Jan 2019, with the following members:

 Terence Tao [Chair from 15 Feb, 2019]
 Carlos Kenig [IMU President (from 1 Jan 2019), ex officio]
 Nalini Anantharaman
 Alexei Borodin
 Annalisa Buffa
 Hélène Esnault [from 21 Mar, 2019]
 Irene Fonseca
 János Kollár [until 21 Mar, 2019]
 Laci Lovász [Chair until 15 Feb, 2019]
 Terry Lyons
 Stephane Mallat
 Hiraku Nakajima
 Éva Tardos
 Peter Teichner
 Akshay Venkatesh
 Anna Wienhard
As one of our first actions, we on the committee are using this blog post to solicit input from the mathematical community regarding the topics within our remit. Among the specific questions (in no particular order) for which we seek comments are the following:
 Are there suggestions to change the format of the ICM that would increase its value to the mathematical community?
 Are there suggestions to change the format of the ICM that would encourage greater participation and interest in attending, particularly with regards to junior researchers and mathematicians from developing countries?
 What is the correct balance between research and exposition in the lectures? For instance, how strongly should one emphasize the importance of good exposition when selecting plenary and sectional speakers? Should there be “Bourbaki style” expository talks presenting work not necessarily authored by the speaker?
 Is the balance between plenary talks, sectional talks, and public talks at an optimal level? There is only a finite amount of space in the calendar, so any increase in the number or length of one of these types of talks will come at the expense of another.
 The ICM is generally perceived to be more important to pure mathematics than to applied mathematics. In what ways can the ICM be made more relevant and attractive to applied mathematicians, or should one not try to do so?
 Are there structural barriers that cause certain areas or styles of mathematics (such as applied or interdisciplinary mathematics) or certain groups of mathematicians to be underrepresented at the ICM? What, if anything, can be done to mitigate these barriers?
Of course, we do not expect these complex and difficult questions to be resolved within this blog post, and debating these and other issues would likely be a major component of our internal committee discussions. Nevertheless, we would value constructive comments towards the above questions (or on other topics within the scope of our committee) to help inform these subsequent discussions. We therefore welcome and invite such commentary, either as responses to this blog post, or sent privately to one of the members of our committee. We would also be interested in having readers share their personal experiences at past congresses, and how it compares with other major conferences of this type. (But in order to keep the discussion focused and constructive, we request that comments here refrain from discussing topics that are out of the scope of this committee, such as suggesting specific potential speakers for the next congress, which is a task instead for the 2022 ICM Program Committee.)
While talking mathematics with a postdoc here at UCLA (March Boedihardjo) we came across the following matrix problem which we managed to solve, but the proof was cute and the process of discovering it was fun, so I thought I would present the problem here as a puzzle without revealing the solution for now.
The problem involves word maps on a matrix group, which for sake of discussion we will take to be the special orthogonal group of real matrices (one of the smallest matrix groups that contains a copy of the free group, which incidentally is the key observation powering the BanachTarski paradox). Given any abstract word of two generators and their inverses (i.e., an element of the free group ), one can define the word map simply by substituting a pair of matrices in into these generators. For instance, if one has the word , then the corresponding word map is given by
for . Because contains a copy of the free group, we see the word map is nontrivial (not equal to the identity) if and only if the word itself is nontrivial.
Anyway, here is the problem:
Problem. Does there exist a sequence of nontrivial word maps that converge uniformly to the identity map?
To put it another way, given any , does there exist a nontrivial word such that for all , where denotes (say) the operator norm, and denotes the identity matrix in ?
As I said, I don’t want to spoil the fun of working out this problem, so I will leave it as a challenge. Readers are welcome to share their thoughts, partial solutions, or full solutions in the comments below.
This is the eleventh research thread of the Polymath15 project to upper bound the de BruijnNewman constant , continuing this post. Discussion of the project of a nonresearch nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
There are currently two strands of activity. One is writing up the paper describing the combination of theoretical and numerical results needed to obtain the new bound . The latest version of the writeup may be found here, in this directory. The theoretical side of things have mostly been written up; the main remaining tasks to do right now are
 giving a more detailed description and illustration of the two major numerical verifications, namely the barrier verification that establishes a zerofree region for for , and the Dirichlet series bound that establishes a zerofree region for ; and
 giving more detail on the conditional results assuming more numerical verification of RH.
Meanwhile, several of us have been exploring the behaviour of the zeroes of for negative ; this does not directly lead to any new progress on bounding (though there is a good chance that it may simplify the proof of ), but there have been some interesting numerical phenomena uncovered, as summarised in this set of slides. One phenomenon is that for large negative , many of the complex zeroes begin to organise themselves near the curves
(An example of the agreement between the zeroes and these curves may be found here.) We now have a (heuristic) theoretical explanation for this; we should have an approximation
in this region (where are defined in equations (11), (15), (17) of the writeup, and the above curves arise from (an approximation of) those locations where two adjacent terms , in this series have equal magnitude (with the other terms being of lower order).
However, we only have a partial explanation at present of the interesting behaviour of the real zeroes at negative t, for instance the surviving zeroes at extremely negative values of appear to lie on the curve where the quantity is close to a halfinteger, where
The remaining zeroes exhibit a pattern in coordinates that is approximately 1periodic in , where
A plot of the zeroes in these coordinates (somewhat truncated due to the numerical range) may be found here.
We do not yet have a total explanation of the phenomena seen in this picture. It appears that we have an approximation
where is the nonzero multiplier
and
The derivation of this formula may be found in this wiki page. However our initial attempts to simplify the above approximation further have proven to be somewhat inaccurate numerically (in particular giving an incorrect prediction for the location of zeroes, as seen in this picture). We are in the process of using numerics to try to resolve the discrepancies (see this page for some code and discussion).
This is the tenth “research” thread of the Polymath15 project to upper bound the de BruijnNewman constant , continuing this post. Discussion of the project of a nonresearch nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
Most of the progress since the last thread has been on the numerical side, in which the various techniques to numerically establish zerofree regions to the equation have been streamlined, made faster, and extended to larger heights than were previously possible. The best bound for now depends on the height to which one is willing to assume the Riemann hypothesis. Using the conservative verification up to height (slightly larger than) , which has been confirmed by independent work of Platt et al. and GourdonDemichel, the best bound remains at . Using the verification up to height claimed by GourdonDemichel, this improves slightly to , and if one assumes the Riemann hypothesis up to height the bound improves to , contingent on a numerical computation that is still underway. (See the table below the fold for more data of this form.) This is broadly consistent with the expectation that the bound on should be inversely proportional to the logarithm of the height at which the Riemann hypothesis is verified.
As progress seems to have stabilised, it may be time to transition to the writing phase of the Polymath15 project. (There are still some interesting research questions to pursue, such as numerically investigating the zeroes of for negative values of , but the writeup does not necessarily have to contain every single direction pursued in the project. If enough additional interesting findings are unearthed then one could always consider writing a second paper, for instance.
Below the fold is the detailed progress report on the numerics by Rudolph Dwars and Kalpesh Muchhal.
This is the ninth “research” thread of the Polymath15 project to upper bound the de BruijnNewman constant , continuing this post. Discussion of the project of a nonresearch nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
We have now tentatively improved the upper bound of the de BruijnNewman constant to . Among the technical improvements in our approach, we now are able to use Taylor expansions to efficiently compute the approximation to for many values of in a given region, thus speeding up the computations in the barrier considerably. Also, by using the heuristic that behaves somewhat like the partial Euler product , we were able to find a good location to place the barrier in which is larger than average, hence easier to keep away from zero.
The main remaining bottleneck is that of computing the Euler mollifier bounds that keep bounded away from zero for larger values of beyond the barrier. In going below we are beginning to need quite complicated mollifiers with somewhat poor tail behavior; we may be reaching the point where none of our bounds will succeed in keeping bounded away from zero, so we may be close to the natural limits of our methods.
Participants are also welcome to add any further summaries of the situation in the comments below.
Just a quick announcement that Dustin Mixon and Aubrey de Grey have just launched the Polymath16 project over at Dustin’s blog. The main goal of this project is to simplify the recent proof by Aubrey de Grey that the chromatic number of the unit distance graph of the plane is at least 5, thus making progress on the HadwigerNelson problem. The current proof is computer assisted (in particular it requires one to control the possible 4colorings of a certain graph with over a thousand vertices), but one of the aims of the project is to reduce the amount of computer assistance needed to verify the proof; already a number of such reductions have been found. See also this blog post where the polymath project was proposed, as well as the wiki page for the project. Nontechnical discussion of the project will continue at the proposal blog post.
I am recording here some notes on a nice problem that Sorin Popa shared with me recently. To motivate the question, we begin with the basic observation that the differentiation operator and the position operator in one dimension formally obey the commutator equation
where is the identity operator and is the commutator. Among other things, this equation is fundamental in quantum mechanics, leading for instance to the Heisenberg uncertainty principle.
The operators are unbounded on spaces such as . One can ask whether the commutator equation (1) can be solved using bounded operators on a Hilbert space rather than unbounded ones. In the finite dimensional case when are just matrices for some , the answer is clearly negative, since the lefthand side of (1) has trace zero and the righthand side does not. What about in infinite dimensions, when the trace is not available? As it turns out, the answer is still negative, as was first worked out by Wintner and Wielandt. A short proof can be given as follows. Suppose for contradiction that we can find bounded operators obeying (1). From (1) and an easy induction argument, we obtain the identity
for all natural numbers . From the triangle inequality, this implies that
Iterating this, we conclude that
for any . Bounding and then sending , we conclude that , which clearly contradicts (1). (Note the argument can be generalised without difficulty to the case when lie in a Banach algebra, rather than be bounded operators on a Hilbert space.)
It was observed by Popa that there is a quantitative version of this result:
Theorem 1 Let such that
Proof: By multiplying by a suitable constant and dividing by the same constant, we may normalise . Write with . Then the same induction that established (2) now shows that
and hence by the triangle inequality
We divide by and sum to conclude that
giving the claim.
Again, the argument generalises easily to any Banach algebra. Popa then posed the question of whether the quantity can be replaced by any substantially larger function of , such as a polynomial in . As far as I know, the above simple bound has not been substantially improved.
In the opposite direction, one can ask for constructions of operators that are not too large in operator norm, such that is close to the identity. Again, one cannot do this in finite dimensions: has trace zero, so at least one of its eigenvalues must outside the disk , and therefore for any finitedimensional matrices .
However, it was shown in 1965 by Brown and Pearcy that in infinite dimensions, one can construct operators with arbitrarily close to in operator norm (in fact one can prescribe any operator for as long as it is not equal to a nonzero multiple of the identity plus a compact operator). In the above paper of Popa, a quantitative version of the argument (based in part on some earlier work of Apostol and Zsido) was given as follows. The first step is to observe the following Hilbert space version of Hilbert’s hotel: in an infinite dimensional Hilbert space , one can locate isometries obeying the equation
where denotes the adjoint of . For instance, if has a countable orthonormal basis , one could set
and
where denotes the linear functional on . Observe that (4) is again impossible to satisfy in finite dimension , as the lefthand side must have trace while the righthand side has trace .
Multiplying (4) on the left by and right by , or on the left by and right by , then gives
From (4), (5) we see in particular that, while we cannot express as a commutator of bounded operators, we can at least express it as the sum of two commutators:
We can rewrite this somewhat strangely as
and hence there exists a bounded operator such that
Moving now to the Banach algebra of matrices with entries in (which can be equivalently viewed as ), a short computation then gives the identity
for some bounded operator whose exact form will not be relevant for the argument. Now, by Neumann series (and the fact that have unit operator norm), we can find another bounded operator such that
and then another brief computation shows that
Thus we can express the operator as the commutator of two operators of norm . Conjugating by for any , we may then express as the commutator of two operators of norm . This shows that the righthand side of (3) cannot be replaced with anything that blows up faster than as . Can one improve this bound further?
This is the seventh “research” thread of the Polymath15 project to upper bound the de BruijnNewman constant , continuing this post. Discussion of the project of a nonresearch nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
The most recent news is that we appear to have completed the verification that is free of zeroes when and , which implies that . For very large (for instance when the quantity is at least ) this can be done analytically; for medium values of (say when is between and ) this can be done by numerically evaluating a fast approximation to and using the argument principle in a rectangle; and most recently it appears that we can also handle small values of , in part due to some new, and significantly faster, numerical ways to evaluate in this range.
One obvious thing to do now is to experiment with lowering the parameters and and see what happens. However there are two other potential ways to bound which may also be numerically feasible. One approach is based on trying to exclude zeroes of in a region of the form , and for some moderately large (this acts as a “barrier” to prevent zeroes from flowing into the region at time , assuming that they were not already there at time ). This require significantly less numerical verification in the aspect, but more numerical verification in the aspect, so it is not yet clear whether this is a net win.
Another, rather different approach, is to study the evolution of statistics such as over time. One has fairly good control on such quantities at time zero, and their time derivative looks somewhat manageable, so one may be able to still have good control on this quantity at later times . However for this approach to work, one needs an effective version of the Riemannvon Mangoldt formula for , which at present is only available asymptotically (or at time ). This approach may be able to avoid almost all numerical computation, except for numerical verification of the Riemann hypothesis, for which we can appeal to existing literature.
Participants are also welcome to add any further summaries of the situation in the comments below.
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