close

Category Archive

You are currently browsing the category archive for the ‘opinion’ category.

On the proposed rule changes to the administration of federal grants

9 June, 2026 in opinion | Tags: , , | by | 6 comments

The United States Office of Management and Budget (OMB) has proposed a vast and radical set of rule changes to how federal grants from all funding agencies are administered.  (A summary of the key changes, by a former Senior Program Officer at the National Institutes for Health, can be found here.) This is no mere tinkering at the edges of existing policy; many basic principles, such as the central role of peer review in grant-making decisions, are seriously compromised by the proposed rules, while the administrative burden of complying with grant rules are significantly increased, and hamstring the ability of funded scientists to react to new developments and forge new collaborations.

There is much to discuss in these proposals; see for instance this post by Karen Saxe (vice president for Government Relations at the American Mathematical Society), this news item on the response from the astronomy community, this op-ed from Ars Technica, this article from the New York Times, this article from Science, or this story from CNN. I will focus here on just one of the impacts, regarding the need to maintain agility and flexibility in a competitive and rapidly changing environment.  

Some types of research, particularly those closest to industrial or other real-world applications, can be planned in a predictable fashion, in which the timelines for hitting key milestones are clear, and schedules for events can be planned years in advance.  However, basic research — of which pure mathematics is a quintessential example — expects (almost by definition) to discover previously unknown directions and connections that cannot be predicted perfectly at the time a research project is proposed.  Many of the most striking breakthroughs in such subjects come from uncovering such expected developments and rapidly capitalizing on them – for instance, by quickly organizing seminars, workshops, or conferences on a suddenly “hot” topic.  

To give just one example of this sort of serendipitous discovery, a significant portion of the foundational theory of compressed sensing was initiated from a chance meeting in 2004 between myself, Emmanuel Candes (a statistician) and Justin Romberg (an electrical engineer) at a program at the Institute for Pure and Applied Mathematics (IPAM) on multiscale geometry.  This theory – has led to notable accelerations and other improvements to a range of technologies, from MRI scans to radio interferometry to electron microscopy.  The three of us, as well as the IPAM program we participated in, were all funded by grants from the National Science Foundation (NSF), but the extraordinarily fruitful collaboration was not fully anticipated in any of the proposals.  (Disclosure: I now serve as director of special projects at IPAM.)

This is the type of fortuitous interaction that would be severely impacted by the proposed rule changes. Consider for instance Section 200.432 of the Code of Federal Regulations, which concerns the use of grant funds to support conference costs:

A conference means an event whose primary purpose is to disseminate technical information beyond the recipient or subrecipient and is necessary and reasonable for successful performance under the Federal award. Allowable conference costs may include the rental of facilities, speakers’ fees, attendance fees, costs of meals and refreshments, local transportation, and other items incidental to such conferences unless further restricted by the terms and conditions of the Federal award. 

As just one of many significant rule changes proposed is the following addendum to the above text:

OMB proposes to expand § 200.432 to add a requirement that costs for attending conferences are allowable only if participation in the conference is expressly approved by the agency and included in the terms and conditions of the award. The revision would clarify that recipients are not authorized to attend conferences using Federal funds that do not serve to advance program outcomes. 

This rule change would limit conference activity support to pre-approved plans that followed the scheduled objectives in the original proposal, which is written some time before the research takes place.  However, it is the nature of novel research (particularly in fundamental sciences such as mathematics) to have serendipitous opportunities emerge that were not anticipated in the original grant proposal, such as an unexpected and exciting new connection between the problem one was initially studying, and another subfield of math or science that had previously been thought to be unrelated.  Being able to react quickly to such developments, either by attending or organizing an event around them, or by inviting key researchers to visit, is essential to keep up with such breakthroughs.  Requiring bureaucratic pre-approval in these circumstances would significantly hinder the ability for funded scientists to competitively take advantage of these opportunities.

An illustrative example would be the 2011 IPAM program on Navigating Chemical Compound Spaces. The premise sounded like a pie-in-the-sky idea: to develop computational tools to be able to somehow travel through the almost infinite space of all possible chemical compounds in the search for a compound we need — be it to create a novel drug, a better solar cell, or stronger glass. At the time, even with projected advances in computer power, accurate prediction of chemical properties of materials was seen as a distant dream. Simulating a simple protein for even a few milliseconds with existing methods would require weeks of time and an astronomical energy budget.  In addition to experts on computational mathematics and materials science, the program involved a group of people who worked in a then obscure subject called machine learning (whose practical applications at the time involved such feats as deciphering human-written zip codes).  Attending such a program might be regarded as out of scope for many material scientists. Yet the outcome of the program was the realization that machine learning methods could be used to learn and model the forces that govern electronic structure, molecular interactions, and ultimately determine chemical properties of materials through much faster and efficient computation. This idea was incredibly fruitful and literally changed the way electronic structure computations are done.  AlphaFold has become Nobel prize winning work, and AI is being used to discover new drugs. Now, 15 years later, scientists are building labs to literally navigate the chemical compound space, assisted by AI, a descendant of old machine-learning computations approaches.    

These examples also illustrate the time scales involved in fundamental research and in bringing it to the point where its application becomes an engineering endeavor.  Fundamental research means playing the long game, leveraging the richness and unpredictability of scientific discovery. It is not something a private company would fund, but it is the engine behind the continued technological transformation whose fruits we all enjoy.  It means taking risks, going in directions that are mere hunches and educated guesses, and going there only with the expectation to find new and surprising things. But it is necessary for technological progress.   

Importantly, it is unrealistic to expect that every conference attendance will result in a major and unexpected connection or breakthrough. At times, there is a slow accumulation of knowledge that suddenly produces unexpected results. It is important to understand that fundamental research operates on scales of years and decades.  The ultimate effect of attending a conference cannot always be known in advance, making the pre-approval process difficult to manage. This brings in a related point: the risk-averse nature of the proposed rules. We all know that making breakthroughs requires risk-taking; behind every successful project stand several that failed. Sometimes, communication of what failed is as useful (or more!) as communication of what succeeded, and this kind of information gets shared in informal settings at workshops and conferences. 

The willingness to take risks and move in unexpected directions has always been a particular strength of this country, both in science and elsewhere, as exemplified for instance by the Defence Advanced Research Projects Agency (DARPA)’s willingness to experiment with emerging technologies such as the internet, GPS systems, or high-energy lasers, long before they could be proven to be viable.  The additional regulatory burdens of these proposed rule changes would cripple this capability and set back the nation’s scientific competitiveness and leadership with the technologies of the future.  I encourage all stakeholders (whether individuals or organizations) to submit public comments on the proposal on the OMB site (the public comment period extends until July 13). You can also submit through the Stand Up for Science site.

(Thanks to Kevin Klowden and Dima Shylakhtenko for feedback on an initial version of this post.)

The elephant in the room

8 November, 2024 in opinion | Tags: , , | by | 90 comments

The day after the election, I found myself struggling with how to approach the complex analysis class I was teaching. Could I ignore the (almost literal) elephant in the room? Would my students be in the right mental state to learn math? Would I be in the right mental state to teach it?

I opened with the statement that usually in math we have the luxury of working in abstractions far removed from the real world. We are familiar with addressing mathematical problems with the (inessential) connections to the real world stripped away, leaving only the essential features to focus one’s attention. An election, for instance, might be treated as the outcome of N people, each of which has a probability p of voting for one candidate, and 1-p for another… and one can then try to analyze the problem from a dispassionate mathematical perspective. This type of mindset can be illuminating in many contexts. Real world events have real consequences, however, and in light of an event as consequential as the last election, a math lecture on contour integration or the central limit theorem may seem meaningless.

But there is one precious thing mathematics has, that almost no other field currently enjoys: a consensus on what the ground truth is, and how to reach it. Because of this, even the strongest differences of opinion in mathematics can eventually be resolved, and mistakes realized and corrected. This consensus is so strong, we simply take it for granted: a solution is correct or incorrect, a theorem is proved or not proved, and when a problem is solved, we simply move on to the next one. This is, sadly, not a state of affairs elsewhere. But if my students can learn from this and carry these skills— such as distinguishing an overly simple but mathematically flawed “solution” from a more complex, but accurate actual solution—to other spheres that have more contact with the real world, then my math lectures have consequence. Even—or perhaps, especially—in times like these.

What are the odds, II: the Venezuelan presidential election

2 August, 2024 in math.PR, opinion | Tags: , | by | 82 comments

In a previous blog post, I discussed how, from a Bayesian perspective, learning about some new information {E} can update one’s perceived odds {{\mathbb P}(H_1) / {\mathbb P}(H_0)} about how likely an “alternative hypothesis” {H_1} is, compared to a “null hypothesis” {H_0}. The mathematical formula here is

\displaystyle \frac{{\mathbb P}(H_1|E)}{{\mathbb P}(H_0|E)} = \frac{{\mathbb P}(E|H_1)}{{\mathbb P}(E|H_0)} \frac{{\mathbb P}(H_1)}{{\mathbb P}(H_0)}. \ \ \ \ \ (1)

Thus, provided one has
  • (i) A precise formulation of the null hypothesis {H_0} and the alternative hypothesis {H_1}, and the new information {E};
  • (ii) A reasonable estimate of the prior odds {{\mathbb P}(H_1) / {\mathbb P}(H_0)} of the alternative hypothesis {H_1} being true (compared to the null hypothesis {H_0});
  • (iii) A reasonable estimate of the probability {{\mathbb P}(E|H_0)} that the event {E} would occur under the null hypothesis {H_0}; and
  • (iv) A reasonable estimate of the probability {{\mathbb P}(E|H_1)} that the event {E} would occur under the alternative hypothesis {H_1},
one can then obtain an estimate of the posterior odds {{\mathbb P}(H_1|E) / {\mathbb P}(H_0|E)} of the alternative hypothesis being true after observing the event {E}. Ingredient (iii) is usually the easiest to compute, but is only one of the key inputs required; ingredients (ii) and (iv) are far trickier, involve subjective non-mathematical considerations, and as discussed in the previous post, depend rather crucially on ingredient (i). For instance, selectively reporting some relevant information for {E}, and withholding other key information from {E}, can make the final mathematical calculation {{\mathbb P}(H_1|E) / {\mathbb P}(H_0|E)} misleading with regard to the “true” odds of {H_1} being true compared to {H_1} based on all available information, even if the calculation was technically accurate with regards to the partial information {E} provided. Also, the computation of the relative odds of two competing hypotheses {H_0} and {H_1} can become somewhat moot if there is a third hypothesis {H_2} that ends up being more plausible than either of these two hypotheses. Nevertheless, the formula (1) can still lead to useful conclusions, albeit ones that are qualified by the particular assumptions and estimates made in the analysis.

At a qualitative level, the Bayesian identity (1) is telling us the following: if an alternative hypothesis {H_1} was already somewhat plausible (so that the prior odds {{\mathbb P}(H_1) / {\mathbb P}(H_0)} was not vanishingly small), and the observed event {E} was significantly more likely to occur under hypothesis {H_1} than under {H_0}, then the hypothesis {H_1} becomes significantly more plausible (in that the posterior odds {{\mathbb P}(H_1|E) / {\mathbb P}(H_0|E)} become quite elevated). This is quite intuitive, but as discussed in the previous post, a lot hinges on how one is defining the alternative hypothesis {H_1}.

In the previous blog post, this calculation was initially illustrated with the following choices of {H_0}, {H_1}, and {E} (thus fulfilling ingredient (i)):

  • {E} was the event that the October 1, 2022 PSCO Grand Lotto in the Phillippines drew the numbers {9, 18, 27, 36, 45, 54} (that is to say, consecutive multiples if {9}), though not necessarily in that order;
  • {H_0} was the null hypothesis that the lottery was fair and the numbers were drawn uniformly at random (without replacement) from the set {\{1,\dots,55\}}; and
  • {H_1} was the alternative hypothesis that the lottery was rigged by some corrupt officials for their personal gain.
In this case, ingredient (iii) can be computed mathematically and precisely:

\displaystyle {\mathbb P}(E|H_0) = \frac{1}{\binom{55}{6}} = \frac{1}{28,989,675}. \ \ \ \ \ (2)

 And, with a not inconceivable level of cynicism about the integrity of the lottery, the prior odds (ingredient (ii)) can be argued to be non-negligible. However, ingredient (iv) was nearly impossible to estimate: indeed, as argued in that post, there is no reason to suspect that {{\mathbb P}(E|H_1)} is much larger than the tiny probability (2), and in fact it could well be smaller (since would likely be in the interest of corrupt lottery officials to not draw attention to their activities). So, despite the very small numerical value of the probability (2), this did not lead to any significant increase in the odds of the alternative hypothesis. In the previous blog post, several other variants {H'_1}, {H''_1}, {H'''_1} of the alternative hypothesis {H_1} were also discussed; the conclusion was that while some choices of alternative hypothesis could lead to elevated probabilities for ingredient (iv), they came at the cost of significantly reducing the prior odds in ingredient (ii), and so no alternative hypothesis was located which ended up being significantly more plausible than the null hypothesis {H_0} after observing the event {E}.

In this post, I would like to run the same analysis on a numerical anomaly in the recent Venezuelan presidential election of June 28, 2024. Here are the officially reported vote totals for the two main candidates, incumbent president Nicolás Maduro and opposition candidate Edmundo González, in the election:

  • Maduro: 5,150,092 votes
  • González: 4,445,978 votes
  • Other: 462,704 votes
  • Total: 10,058,774 votes.
The numerical anomaly is that if one multiplies the total number of voters {10,058,774} by the round percentages {51.2\%}, {44.2\%}, {4.6\%}, one recovers exactly the above vote counts after rounding to the nearest integer:

\displaystyle 51.2\% \times 10,058,774 = 5,150,092.288

\displaystyle 44.2\% \times 10,058,774 = 4,445,978.108

\displaystyle 4.6\% \times 10,058,774 = 462,703.604.

Let us try to apply the above Bayesian framework to this situation, bearing in mind the caveats that this analysis is only strong as the inputs supplied and assumptions made (for instance, to simplify the discussion, we will not also discuss information from exit polling, which in this case gave significantly different predictions from the percentages above).

The first step (ingredient (i)) is to formulate the null hypothesis {H_0}, the alternative hypothesis {H_1}, and the event {E}. Here is one possible choice:

  • {E} is the event that the reported vote total for Maduro, González, and Other are all equal to the nearest integer of the total number of voters, multiplied by a round percentage with one decimal point (i.e., an integer multiple of {0.1\%}).
  • {H_0} is the null hypothesis that the vote totals were reported accurately (or with only inconsequential inaccuracies).
  • {H_1} is the alternative hypothesis that the vote totals were manipulated by officials from the incumbent administration.

Ingredient (ii) – the prior odds that {H_1} is true over {H_0} – is highly subjective, and an individual’s estimation of (ii) would likely depend on, or at least be correlated with, their opinion of the current Venezulan administration. Discussion of this ingredient is therefore more political than mathematical, and I will not attempt to quantify it further here. Now we turn to (iii), the estimation of the probability {{\mathbb P}(E|H_0)} that {E} occurs given the hypothesis {H_0}. This cannot be computed exactly without a precise probabilistic model of the voting electorate, but let us make a rough order of magnitude calculation as follows. One can focus on the anomaly just for the number of votes received by Maduro and González, since if both of these counts were the nearest integer to a round percentages then just from simple subtraction the number of votes for “other” would also be forced to also be the nearest integer from a round percentage, possibly plus or minus one due to carries, so up to a factor of two or so we can ignore the latter anomaly. As a simple model, suppose that the voting percentages for Maduro and González were distributed more or less uniformly in some square {[p-\varepsilon,p+\varepsilon] \times [q-\varepsilon,q+\varepsilon]}, where {p, q} are some proportions not too close to either {0} or {1}, and {\varepsilon} is some reasonably large margin of error (the exact values of these parameters will end up not being too important, nor will the specific shape of the distribution; indeed, the shape and size of the square here only impacts the analysis through the area {(2\varepsilon)^2} of the square, and even this quantity cancels itself out in the end). Thus, the number of votes for Maduro is distributed in an interval of length about {2\varepsilon N}, where {N = 10,058,774} is the number of voters, and similarly for González, so the total number of different outcomes here is {(2\varepsilon N)^2}, and by our model we have a uniform distribution amongst all these outcomes. On the other hand, the total number of attainable round percentages for Maduro is about {(2\varepsilon) / 0.1\% = 1000 \times 2\varepsilon}, and similarly for González, so our estimate for {{\mathbb P}(E|H_0)} is

\displaystyle {\mathbb P}(E|H_0) \approx \frac{(1000 \times 2\varepsilon)^2}{(2\varepsilon N)^2} = (1000/N)^2 \approx 10^{-8}.

This looks quite unlikely! But we are not done yet, because we also need to estimate {{\mathbb P}(E|H_1)}, the probability that the event {E} would occur under the alternative hypothesis {H_1}. Here one has to be careful, because while it could happen under hypothesis {H_1} that the vote counts were manipulated to be exactly the nearest integer to a round percentage, this is not the only outcome under this hypothesis, and indeed one could argue that it would not be in the interest of an administration to generate such a striking numerical anomaly. But one can create a reasonable chain of events with which to estimate (from below) this probability by a kind of “Drake equation“. Consider the following variants of {H_1}:
  • {H'_1}: {H_1} is true, and the administration directs election officials to report vote outcomes with some explicitly preferred (round) percentages, regardless of the actual election results.
  • {H''_1}: {H'_1} is true, and the election officials dutifully generate a report by multiplying these preferred percentages by the total number {N} of voters, and rounding to the nearest integer, without any attempt to disguise their actions.
By the chain rule for conditional probability, one has a lower bound

\displaystyle {\mathbb P}(E|H_1) \geq {\mathbb P}(E, H''_1, H'_1|H_1) = {\mathbb P}(E|H''_1) {\mathbb P}(H''_1|H'_1) {\mathbb P}(H'_1|H_1).

Inserting this into (1), we obtain our final lower bound:

\displaystyle \frac{{\mathbb P}(H_1|E)}{{\mathbb P}(H_0|E)} \gtrapprox 10^8 \times {\mathbb P}(E|H''_1) {\mathbb P}(H''_1|H'_1) {\mathbb P}(H'_1|H_1) \frac{{\mathbb P}(H_1)}{{\mathbb P}(H_0)}.

This is about as far as one can get purely with mathematical analysis. Beyond this, one has to make some largely subjective estimations for each of the remaining probabilities and odds in this formula. As mentioned, the prior odds {{\mathbb P}(H_1)/{\mathbb P}(H_0)} will likely depend on the individual making this calculation, and will not be discussed further here. The remaining question then is how large the probabilities {{\mathbb P}(H'_1|H_1)}, {{\mathbb P}(H''_1|H'_1)}, and {{\mathbb P}(E|H''_1)} are. In other words:
  • If one assumes that the administration wishes to manipulate the vote totals, how likely is it a priori (i.e., without being aware of the anomaly {E}) that they would do so by explictly selecting preferred round percentages and then requesting that election officials report these percentages?
  • If one assumes that election officials are being ordered to report vote totals to reflect a preferred round percentage, how likely is it a priori that they would follow the orders without question, and performing simple rounding instead of any more sophisticated numerical manipulation?
  • If one assumes that election officials did indeed follow the orders as above, how likely is it a priori that the report would be published as is without any concerns raised by other officials or observers?
If one’s estimate of the product of these three probabilities multiplies to be significantly greater than {10^{-8}}, then we can conclude that the event {E} has indeed significantly raised the odds of some sort of voting manipulation present. The scenario described above is somewhat plausible, especially in light of the anomaly {E}, and so certainly the posterior probabilities {{\mathbb P}(H'_1|H_1,E)}, {{\mathbb P}(H''_1|H'_1,E)} seem quite large (and the posterior probability {{\mathbb P}(E|H''_1,E)} is of course equal to {1}). But it is important here to avoid confirmation bias and work only with a priori probabilities – roughly speaking, the probabilities that one would assign to such events on July 27, 2024, before knowledge of the anomaly {E} came to light. Nevertheless, even after accounting for confirmation bias, I think it is plausible that the above product of a priori probabilities is indeed significantly larger than {10^{-8}} (for instance, to assign probabilities somewhat randomly, this would be the case of each of the conditional probabilities {{\mathbb P}(H'_1|H_1)}, {{\mathbb P}(H''_1|H'_1)} and {{\mathbb P}(E|H''_1)} all exceed {1\%}), giving credence to the theory of the election report being manipulated (though it is possible that the manipulation could occur through a third hypothesis {H_2} not covered by the original two hypotheses, such as a software glitch). If one adds in additional information beyond the purely numerical anomaly {E}, such as the fact that the reported totals were not broken down further by voting district (which would be less likely under hypothesis {H_0} than hypothesis {H_1}), and that exit polls gave significantly different results from the reported totals (which is again less likely under hypothesis {H_0} than hypothesis {H_1}), the evidence for voting irregularities becomes quite significant.

One can contrast this analysis with that of the Phillipine lottery in the original post. In both cases the probability {{\mathbb P}(E|H_0)} of the observed event under the null hypothesis was extremely small. However, in the case of the Venezuelan election, there is a plausible causal chain {H_1 \implies H'_1 \implies H''_1 \implies E} that leads to an elevated probability {{\mathbb P}(E|H_1)} of the observed event under the alternative hypothesis, whereas in the case of the lottery, only extremely implausible chains could be constructed that would lead to the specific outcome of a multiples-of-9 lottery draw for that specific lottery on that specific date.

Would it be possible to create a tool to automatically diagram papers?

18 February, 2023 in opinion | by | 54 comments

This is a somewhat experimental and speculative post. This week I was at the IPAM workshop on machine assisted proof that I was one of the organizers of. We had an interesting and diverse range of talks, both from computer scientists presenting the latest available tools to formally verify proofs or to automate various aspects of proof writing or proof discovery, as well as mathematicians who described their experiences using these tools to solve their research problems. One can find the videos of these talks on the IPAM youtube channel; I also posted about the talks during the event on my Mathstodon account. I am of course not the most objective person to judge, but from the feedback I received it seems that the conference was able to successfully achieve its aim of bringing together the different communities interested in this topic.

As a result of the conference I started thinking about what possible computer tools might now be developed that could be of broad use to mathematicians, particularly those who do not have prior expertise with the finer aspects of writing code or installing software. One idea that came to mind was a potential tool to could take, say, an arXiv preprint as input, and return some sort of diagram detailing the logical flow of the main theorems and lemmas in the paper. This is currently done by hand by authors in some, but not all, papers (and can often also be automatically generated from formally verified proofs, as seen for instance in the graphic accompanying the IPAM workshop, or this diagram generated from Massot’s blueprint software from a manually inputted set of theorems and dependencies as a precursor to formalization of a proof [thanks to Thomas Bloom for this example]). For instance, here is a diagram that my co-author Rachel Greenfeld and I drew for a recent paper:

BERJAYA

This particular diagram incorporated a number of subjective design choices regarding layout, which results to be designated important enough to require a dedicated box (as opposed to being viewed as a mere tool to get from one box to another), and how to describe each of these results (and how to colour-code them). This is still a very human-intensive task (and my co-author and I went through several iterations of this particular diagram with much back-and-forth discussion until we were both satisfied). But I could see the possibility of creating an automatic tool that could provide an initial “first approximation” to such a diagram, which a human user could then modify as they see fit (perhaps using some convenient GUI interface, for instance some variant of the Quiver online tool for drawing commutative diagrams in LaTeX).

As a crude first attempt at automatically generating such a diagram, one couuld perhaps develop a tool to scrape a LaTeX file to locate all the instances of the theorem environment in the text (i.e., all the formally identified lemmas, corollaries, and so forth), and for each such theorem, locate a proof environment instance that looks like it is associated to that theorem (doing this with reasonable accuracy may require a small amount of machine learning, though perhaps one could just hope that proximity of the proof environment instance to the theorem environment instance suffices in many cases). Then identify all the references within that proof environment to other theorems to start building the tree of implications, which one could then depict in a diagram such as the above. Such an approach would likely miss many of the implications; for instance, because many lemmas might not be proven using a formal proof environment, but instead by some more free-flowing text discussion, or perhaps a one line justification such as “By combining Lemma 3.4 and Proposition 3.6, we conclude”. Also, some references to other results in the paper might not proceed by direct citation, but by more indirect justifications such as “invoking the previous lemma, we obtain” or “by repeating the arguments in Section 3, we have”. Still, even such a crude diagram might still be helpful, both as a starting point for authors to make an improved diagram, or for a student trying to understand a lengthy paper to get some initial idea of the logical structure.

More advanced features might be to try to use more of the text of the paper to assign some measure of importance to individual results (and then weight the diagram correspondingly to highlight the more important results), to try to give each result a natural language description, and to somehow capture key statements that are not neatly encapsulated in a theorem environment instance, but I would imagine that such tasks should be deferred until some cruder proof-of-concept prototype can be demonstrated.

Anyway, I would be interested to hear opinions about whether this idea (or some modification thereof) is (a) actually feasible with current technology (or better yet, already exists in some form), and (b) of interest to research mathematicians.

Request for comments from the ICM Structure Committee

17 July, 2022 in admin, opinion | Tags: | by | 25 comments

[This post is collectively authored by the ICM structure committee, whom I am currently chairing – T.]

The ICM structure committee is responsible for the preparation of the Scientific Program of the International Congress of Mathematicians (ICM). It decides the structure of the Scientific Program, in particular,

  • the number of plenary lectures,
  • the sections and their precise definition,
  • the target number of talks in each section,
  • other kind of lectures, and
  • the arrangement of sections.

(The actual selection of speakers and the local organization of the ICM are handled separately by the Program Committee and Organizing Comittee respectively.)

Our committee can also propose more radical changes to the format of the congress, although certain components of the congress, such as the prize lectures and satellite events, are outside the jurisdiction of this committee. For instance, in 2019 we proposed the addition of two new categories of lectures, “special sectional lectures” and “special plenary lectures”, which are broad and experimental categories of lectures that do not fall under the traditional format of a mathematician presenting their recent advances in a given section, but can instead highlight (for instance) emerging connections between two areas of mathematics, or present a “big picture” talk on a “hot topic” from an expert with the appropriate perspective. These new categories made their debut at the recently concluded virtual ICM, held on July 6-14, 2022.

Over the next year or so, our committee will conduct our deliberations on proposed changes to the structure of the congress for the next ICM (to be held in-person in Philadelphia in 2026) and beyond. As part of the preparation for these deliberations, we are soliciting feedback from the general mathematics community (on this blog and elsewhere) on the current state of the ICM, and any proposals to improve that state for the subsequent congresses; we had issued a similar call on this blog back in 2019. This time around, of course, the situation is complicated by the extraordinary and exceptional circumstances that led to the 2022 ICM being moved to a virtual platform on short notice, and so it is difficult for many reasons to hold the 2022 virtual ICM as a model for subsequent congresses. On the other hand, the scientific program had already been selected by the 2022 ICM Program Committee prior to the invasion of Ukraine, and feedback on the content of that program will be of great value to our committee.

Among the specific questions (in no particular order) for which we seek comments are the following:

  1. Are there suggestions to change the format of the ICM that would increase its value to the mathematical community?
  2. 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?
  3. The special sectional and special plenary lectures were introduced in part to increase the emphasis on the quality of exposition at ICM lectures. Has this in fact resulted in a notable improvement in exposition, and should any alternations be made to the special lecture component of the ICM?
  4. Is the balance between plenary talks, sectional talks, special plenary and 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.
  5. 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?
  6. 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 under-represented at the ICM?  What, if anything, can be done to mitigate these barriers?
  7. The recently concluded virtual ICM had a sui generis format, in which the core virtual program was supplemented by a number of physical “overlay” satellite events. Are there any positive features of that format which could potentially be usefully adapted to such congresses? For instance, should there be any virtual or hybrid components at the next ICM?

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. Comments that are specific to the recently concluded virtual ICM can be made instead at this blog post.)

The International Congress of Mathematicians (ICM 2022) will now be fully virtual. How best to utilise this format?

26 February, 2022 in non-technical, opinion | Tags: | by | 68 comments

[Note: while I am chair of the ICM Structure Committee, this blog post is not an official request from this committee, as events are still moving too rapidly to proceed at present via normal committee deliberations. We are however discussing these matters and may issue a more formal request in due course. -T.]

The International Mathematical Union has just made the following announcement concerning the International Congress of Mathematicians (ICM) that was previously scheduled to be held in St. Petersburg, Russia in July.

Decision of the Executive Committee of the IMU on the upcoming ICM 2022 and IMU General Assembly

On 26 February 2022, the Executive Committee of the International Mathematical Union (IMU) decided that:

1. The International Congress of Mathematicians (ICM) 2022 will take place as a fully virtual event, hosted outside Russia but following the original time schedule planned for Saint Petersburg.
2. Participation in the virtual ICM event will be free of charge.
3. The IMU General Assembly (GA) will take place as an in-person event outside Russia.
4. A prize ceremony will be held the day after the IMU GA, at the same venue as the IMU GA, for the awarding of the 2022 IMU prizes.
5. The dates for the ICM and the GA will remain unaltered.
6. We will return with further practical information regarding the two events.

An expanded version of the announcement can be found here. (See also this addendum.)

While I am not on the IMU Executive Committee and thus not privy to their deliberations, I have been in contact with several members of this committee and I support their final decision on these matters.

As we have all experienced during the COVID-19 pandemic, virtual conferences can be rather variable in quality, but there certainly are ways to make the experience more positive for both the speakers and participants. In the interest of maximizing the benefits that this meeting can still produce, I would like to invite readers of this blog to share any experiences they have had with very large virtual conferences, and any opinions on what types of virtual events were effective and engaging.

One idea that has been suggested to me has been to have (either unofficial, semi-official, or official) regional ICM hosting events at various places worldwide where mathematicians could gather in person to view ICM talks that would be streamed online (and perhaps some ICM speakers from that area could give talks in person in such locations). This would be very nonstandard, of course, but could be one way to salvage some of the physical ICM experience, and perhaps also a way to symbolically support the spirit of the Congress. I would be interested to get some feedback on this proposal.

Finally, I would like to request that comments to this post remain focused on the upcoming virtual ICM. Broader political issues are very much worth discussing at present, but there are other venues for such discussion, and as per my usual blog policy any off-topic comments may be subject to deletion.

A statement from mathematics department faculty at Stanford and MIT

8 February, 2021 in non-technical, opinion | Tags: | by | 35 comments

[The following statement is signed by several mathematicians at Stanford and MIT in support of one of their recently admitted graduate students, and I am happy to post it here on my blog. -T]

We were saddened and horrified to learn that Ilya Dumanski, a brilliant young mathematician who has been admitted to our graduate programs at Stanford and MIT, has been imprisoned in Russia, along with several other mathematicians, for participation in a peaceful demonstration. Our thoughts are with them. We urge their rapid release, and failing that, that they be kept in humane conditions. A petition in their support has been started at

https://www.ipetitions.com/petition/a-call-for-immediate-release-of-arrested-students/

Signed,

Roman Bezrukavnikov (MIT)
Alexei Borodin (MIT)
Daniel Bump (Stanford)
Sourav Chatterjee (Stanford)
Otis Chodosh (Stanford)
Ralph Cohen (Stanford)
Henry Cohn (MIT)
Brian Conrad (Stanford)
Joern Dunkel (MIT)
Pavel Etingof (MIT)
Jacob Fox (Stanford)
Michel Goemans (MIT)
Eleny Ionel (Stanford)
Steven Kerckhoff (Stanford)
Jonathan Luk (Stanford)
Eugenia Malinnikova (Stanford)
Davesh Maulik (MIT)
Rafe Mazzeo (Stanford)
Haynes Miller (MIT)
Ankur Moitra (MIT)
Elchanan Mossel (MIT)
Tomasz Mrowka (MIT)
Bjorn Poonen (MIT)
Alex Postnikov (MIT)
Lenya Ryzhik (Stanford)
Paul Seidel (MIT)
Mike Sipser (MIT)
Kannan Soundararajan (Stanford)
Gigliola Staffilani (MIT)
Nike Sun (MIT)
Richard Taylor (Stanford)
Ravi Vakil (Stanford)
Andras Vasy (Stanford)
Jan Vondrak (Stanford)
Brian White (Stanford)
Zhiwei Yun (MIT)

BERJAYA

On the recently removed paper from the New York Journal of Mathematics

11 September, 2018 in non-technical, opinion | by | 247 comments

In the last week or so there has been some discussion on the internet about a paper (initially authored by Hill and Tabachnikov) that was initially accepted for publication in the Mathematical Intelligencer, but with the editor-in-chief of that journal later deciding against publication; the paper, in significantly revised form (and now authored solely by Hill), was then quickly accepted by one of the editors in the New York Journal of Mathematics, but then was removed from publication after objections from several members on the editorial board of NYJM that the paper had not been properly refereed or was within the scope of the journal; see this statement by Benson Farb, who at the time was on that board, for more details.  Some further discussion of this incident may be found on Tim Gowers’ blog; the most recent version of the paper, as well as a number of prior revisions, are still available on the arXiv here.

For whatever reason, some of the discussion online has focused on the role of Amie Wilkinson, a mathematician from the University of Chicago (and who, incidentally, was a recent speaker here at UCLA in our Distinguished Lecture Series), who wrote an email to the editor-in-chief of the Intelligencer raising some concerns about the content of the paper and suggesting that it be published alongside commentary from other experts in the field.  (This, by the way, is not uncommon practice when dealing with a potentially provocative publication in one field by authors coming from a different field; for instance, when Emmanuel Candès and I published a paper in the Annals of Statistics introducing what we called the “Dantzig selector”, the Annals solicited a number of articles discussing the selector from prominent statisticians, and then invited us to submit a rejoinder.)    It seems that the editors of the Intelligencer decided instead to reject the paper.  The paper then had a complicated interaction with NYJM, but, as stated by Wilkinson in her recent statement on this matter as well as by Farb, this was done without any involvement from Wilkinson.  (It is true that Farb happens to also be Wilkinson’s husband, but I see no reason to doubt their statements on this matter.)

I have not interacted much with the Intelligencer, but I have published a few papers with NYJM over the years; it is an early example of a quality “diamond open access” mathematics journal.  It seems that this incident may have uncovered some issues with their editorial procedure for reviewing and accepting papers, but I am hopeful that they can be addressed to avoid this sort of event occurring again.

 

 

Open thread for mathematicians on the immigration executive order

31 January, 2017 in math.HO, non-technical, opinion | by | 147 comments

The self-chosen remit of my blog is “Updates on my research and expository papers, discussion of open problems, and other maths-related topics”.  Of the 774 posts on this blog, I estimate that about 99% of the posts indeed relate to mathematics, mathematicians, or the administration of this mathematical blog, and only about 1% are not related to mathematics or the community of mathematicians in any significant fashion.

This is not one of the 1%.

Mathematical research is clearly an international activity.  But actually a stronger claim is true: mathematical research is a transnational activity, in that the specific nationality of individual members of a research team or research community are (or should be) of no appreciable significance for the purpose of advancing mathematics.  For instance, even during the height of the Cold War, there was no movement in (say) the United States to boycott Soviet mathematicians or theorems, or to only use results from Western literature (though the latter did sometimes happen by default, due to the limited avenues of information exchange between East and West, and former did occasionally occur for political reasons, most notably with the Soviet Union preventing Gregory Margulis from traveling to receive his Fields Medal in 1978 EDIT: and also Sergei Novikov in 1970).    The national origin of even the most fundamental components of mathematics, whether it be the geometry (γεωμετρία) of the ancient Greeks, the algebra (الجبر) of the Islamic world, or the Hindu-Arabic numerals 0,1,\dots,9, are primarily of historical interest, and have only a negligible impact on the worldwide adoption of these mathematical tools. While it is true that individual mathematicians or research teams sometimes compete with each other to be the first to solve some desired problem, and that a citizen could take pride in the mathematical achievements of researchers from their country, one did not see any significant state-sponsored “space races” in which it was deemed in the national interest that a particular result ought to be proven by “our” mathematicians and not “theirs”.   Mathematical research ability is highly non-fungible, and the value added by foreign students and faculty to a mathematics department cannot be completely replaced by an equivalent amount of domestic students and faculty, no matter how large and well educated the country (though a state can certainly work at the margins to encourage and support more domestic mathematicians).  It is no coincidence that all of the top mathematics department worldwide actively recruit the best mathematicians regardless of national origin, and often retain immigration counsel to assist with situations in which these mathematicians come from a country that is currently politically disfavoured by their own.

Of course, mathematicians cannot ignore the political realities of the modern international order altogether.  Anyone who has organised an international conference or program knows that there will inevitably be visa issues to resolve because the host country makes it particularly difficult for certain nationals to attend the event.  I myself, like many other academics working long-term in the United States, have certainly experienced my own share of immigration bureaucracy, starting with various glitches in the renewal or application of my J-1 and O-1 visas, then to the lengthy vetting process for acquiring permanent residency (or “green card”) status, and finally to becoming naturalised as a US citizen (retaining dual citizenship with Australia).  Nevertheless, while the process could be slow and frustrating, there was at least an order to it.  The rules of the game were complicated, but were known in advance, and did not abruptly change in the middle of playing it (save in truly exceptional situations, such as the days after the September 11 terrorist attacks).  One just had to study the relevant visa regulations (or hire an immigration lawyer to do so), fill out the paperwork and submit to the relevant background checks, and remain in good standing until the application was approved in order to study, work, or participate in a mathematical activity held in another country.  On rare occasion, some senior university administrator may have had to contact a high-ranking government official to approve some particularly complicated application, but for the most part one could work through normal channels in order to ensure for instance that the majority of participants of a conference could actually be physically present at that conference, or that an excellent mathematician hired by unanimous consent by a mathematics department could in fact legally work in that department.

With the recent and highly publicised executive order on immigration, many of these fundamental assumptions have been seriously damaged, if not destroyed altogether.  Even if the order was withdrawn immediately, there is no longer an assurance, even for nationals not initially impacted by that order, that some similar abrupt and major change in the rules for entry to the United States could not occur, for instance for a visitor who has already gone through the lengthy visa application process and background checks, secured the appropriate visa, and is already in flight to the country.  This is already affecting upcoming or ongoing mathematical conferences or programs in the US, with many international speakers (including those from countries not directly affected by the order) now cancelling their visit, either in protest or in concern about their ability to freely enter and leave the country.  Even some conferences outside the US are affected, as some mathematicians currently in the US with a valid visa or even permanent residency are uncertain if they could ever return back to their place of work if they left the country to attend a meeting.  In the slightly longer term, it is likely that the ability of elite US institutions to attract the best students and faculty will be seriously impacted.  Again, the losses would be strongest regarding candidates that were nationals of the countries affected by the current executive order, but I fear that many other mathematicians from other countries would now be much more concerned about entering and living in the US than they would have previously.

It is still possible for this sort of long-term damage to the mathematical community (both within the US and abroad) to be reversed or at least contained, but at present there is a real risk of the damage becoming permanent.  To prevent this, it seems insufficient for me for the current order to be rescinded, as desirable as that would be; some further legislative or judicial action would be needed to begin restoring enough trust in the stability of the US immigration and visa system that the international travel that is so necessary to modern mathematical research becomes “just” a bureaucratic headache again.

Of course, the impact of this executive order is far, far broader than just its effect on mathematicians and mathematical research.  But there are countless other venues on the internet and elsewhere to discuss these other aspects (or politics in general).  (For instance, discussion of the qualifications, or lack thereof, of the current US president can be carried out at this previous post.) I would therefore like to open this post to readers to discuss the effects or potential effects of this order on the mathematical community; I particularly encourage mathematicians who have been personally affected by this order to share their experiences.  As per the rules of the blog, I request that “the discussions are kept constructive, polite, and at least tangentially relevant to the topic at hand”.

Some relevant links (please feel free to suggest more, either through comments or by email):

It ought to be common knowledge that Donald Trump is not fit for the presidency of the United States of America

4 June, 2016 in non-technical, opinion | Tags: | by | 506 comments

In logic, there is a subtle but important distinction between the concept of mutual knowledge – information that everyone (or almost everyone) knows – and common knowledge, which is not only knowledge that (almost) everyone knows, but something that (almost) everyone knows that everyone else knows (and that everyone knows that everyone else knows that everyone else knows, and so forth).  A classic example arises from Hans Christian Andersens’ fable of the Emperor’s New Clothes: the fact that the emperor in fact has no clothes is mutual knowledge, but not common knowledge, because everyone (save, eventually, for a small child) is refusing to acknowledge the emperor’s nakedness, thus perpetuating the charade that the emperor is actually wearing some incredibly expensive and special clothing that is only visible to a select few.  My own personal favourite example of the distinction comes from the blue-eyed islander puzzle, discussed previously here, here and here on the blog.  (By the way, I would ask that any commentary about that puzzle be directed to those blog posts, rather than to the current one.)

I believe that there is now a real-life instance of this situation in the US presidential election, regarding the following

Proposition 1.  The presumptive nominee of the Republican Party, Donald Trump, is not even remotely qualified to carry out the duties of the presidency of the United States of America.

Proposition 1 is a statement which I think is approaching the level of mutual knowledge amongst the US population (and probably a large proportion of people following US politics overseas): even many of Trump’s nominal supporters secretly suspect that this proposition is true, even if they are hesitant to say it out loud.  And there have been many prominent people, from both major parties, that have made the case for Proposition 1: for instance Mitt Romney, the Republican presidential nominee in 2012, did so back in March, and just a few days ago Hillary Clinton, the likely Democratic presidential nominee this year, did so in this speech:

I highly recommend watching the entirety of the (35 mins or so) speech, followed by the entirety of Trump’s rebuttal.

However, even if Proposition 1 is approaching the status of “mutual knowledge”, it does not yet seem to be close to the status of “common knowledge”: one may secretly believe that Trump cannot be considered as a serious candidate for the US presidency, but must continue to entertain this possibility, because they feel that others around them, or in politics or the media, appear to be doing so.  To reconcile these views can require taking on some implausible hypotheses that are not otherwise supported by any evidence, such as the hypothesis that Trump’s displays of policy ignorance, pettiness, and other clearly unpresidential behaviour are merely “for show”, and that behind this facade there is actually a competent and qualified presidential candidate; much like the emperor’s new clothes, this alleged competence is supposedly only visible to a select few.  And so the charade continues.

I feel that it is time for the charade to end: Trump is unfit to be president, and everybody knows it.  But more people need to say so, openly.

Important note: I anticipate there will be any number of “tu quoque” responses, asserting for instance that Hillary Clinton is also unfit to be the US president.  I personally do not believe that to be the case (and certainly not to the extent that Trump exhibits), but in any event such an assertion has no logical bearing on the qualification of Trump for the presidency.  As such, any comments that are purely of this “tu quoque” nature, and which do not directly address the validity or epistemological status of Proposition 1, will be deleted as off-topic.  However, there is a legitimate case to be made that there is a fundamental weakness in the current mechanics of the US presidential election, particularly with the “first-past-the-post” voting system, in that (once the presidential primaries are concluded) a voter in the presidential election is effectively limited to choosing between just two viable choices, one from each of the two major parties, or else refusing to vote or making a largely symbolic protest vote. This weakness is particularly evident when at least one of these two major choices is demonstrably unfit for office, as per Proposition 1.  I think there is a serious case for debating the possibility of major electoral reform in the US (I am particularly partial to the Instant Runoff Voting system, used for instance in my home country of Australia, which allows for meaningful votes to third parties), and I would consider such a debate to be on-topic for this post.  But this is very much a longer term issue, as there is absolutely no chance that any such reform would be implemented by the time of the US elections in November (particularly given that any significant reform would almost certainly require, at minimum, a constitutional amendment).

 

Archives