what AMS is not doing, but probably should be

The American Mathematical Society (AMS) is the leading professional organization for mathematicians primarily based in research-focused institutions, particularly departments of mathematics with PhD programs. It is a large organization with approximately 30,000 members, including international members, retirees, and others.

The AMS engages in many activities typical of professional organizations: it operates a publishing house, serves as a platform for conferences, lobbies the U.S. government, and awards various prizes, much like the IEEE or the ACM. However, I believe this is not enough. The AMS could, and likely should, do more to better serve the community—and perhaps to ensure its own continued relevance.

Below, I outline three areas where the AMS could make a meaningful impact. While there are certainly other opportunities, I want to begin with the most obvious ones.


Graduate Programs Ranking

One of the most straightforward opportunities for improvement is the ranking of mathematics departments, specifically their graduate programs. Currently, this ranking is notoriously handled by a commercial enterprise: U.S. News & World Report1They also rank mattresses.. Their methodology is often criticized for its lack of transparency, susceptibility to manipulation, and reliance on polling the heads of graduate programs2i.e., converting unpaid labor (a surprising common thread of this post) into their profit..

There is no compelling reason why this task should be left to an external entity. The AMS is well-positioned to create a transparent and rigorous ranking system. This would involve establishing clearly defined criteria, developing aggregation algorithms, and implementing robust data collection methods. Importantly, many of the most labor-intensive metrics—such as graduation rates, placements, and publication activity—are already gathered by the AMS. Polling the programs would also be more straightforward, as directors of graduate studies are more likely to respond to a professional, non-profit organization than to a commercial outlet with a poor reputation.

Some might argue that U.S. News has an advantage because of its outsider status, claiming this ensures objectivity. I would counter that this detachment leads to indifference, not impartiality. True objectivity and fairness come from collective action, something the AMS is fully capable of organizing.

I estimate this project could be completed within a year by a volunteer committee of mathematicians, with the support of a few IT professionals for software implementation.


Graduate admissions

Graduate admissions in mathematics departments are stressful not only for applicants but also for the departments themselves. Every fall, thousands of prospective students apply to graduate programs. Large departments, like ours, receive hundreds of applications, and committees spend weeks reviewing and ranking them.

The real challenge begins when departments are ready to extend offers. How many offers should be sent? As applicants apply to multiple schools, departments typically offer more admissions than they can accommodate, knowing that only a fraction will accept. However, this acceptance rate (yield) is highly unpredictable.

While past data offers some guidance, significant fluctuations are inevitable, and mistakes can be costly. Once an offer is made, the department takes on legal and financial obligations. If too many offers are accepted, the department’s budget could be strained; if too few, they may be left without enough research or teaching assistants.

This creates a chaotic situation. After offers go out, departments remain anxious, as applicants aren’t required to respond until the shared April 15th deadline. This uncertainty makes program planning difficult and stressful.

Fortunately, this problem is not new, and an effective solution has existed for over 70 years, in a different field though. I am talking about the Match, or National Resident Matching Program (NRMP), which is used in medical residency placements.

While the field is different, the similarities are clear: just as residency is a critical step in training doctors, PhD study plays a similar role in training research mathematicians.In preparation of practicing medical doctors, the residency is the last, critical step, not unlike PhD study in training of research mathematicians3The medical trade also shares the somewhat bizarre business model most of the departments of mathematics in the US (especially those at large public universities) are operating. Namely, we provide graduate education for free, while underpaying the student’s work (whether teaching or research assistantships), compared to what tenure-track faculty makes. Essentially the same model is deployed by the residencies: the doctors spend several years working under supervision, under a lot of pressure (in real hospital setting, addressing real medical challenges), for far smaller salaries than their established colleagues. The Match is a centralized mechanism of pairing the applicants to the hospitals that mathematical graduate education in the US can, and I believe, should, adopt.

The Match pairs applicants with hospitals based on mutual rankings—applicants rank programs, and programs rank applicants. Hospitals also indicate how many positions they plan to fill. A matching algorithm, based on the Gale-Shapley stable matching method4In fact the Match algorithm is essentially equivalent to Gale-Shapley one, and predates it., ensures an outcome, where no pairing could be simultaneously improved for both the applicant and the program.

This system has been proven to work over time. There is an enormous follow-up literature to that paper addressing many twists and subtle points in the choice of implementation and various add-ons, – see, e.g., this article. What is important to us is the existence of a robust and tested implementation of the algorithm.

While implementing a similar system for mathematics PhD programs wouldn’t be easy, it is feasible. Departments would need to collaborate with graduate colleges, and some legal challenges might arise. A centralized implementation would be essential, something that a consortium of math PhD programs may struggle to achieve on their own.

This is where the AMS, as the national society for research mathematicians, could play a pivotal role. The AMS is well-positioned to lead the effort and serve as the central hub for the implementation of such a system. Doing so would greatly improve and streamline the graduate admissions process for mathematics departments across the U.S.


Mathematical Reviews reform

Mathematical Reviews is one of the most important endeavors of the AMS. Subscriptions to its current electronic incarnation, MathSciNet account for about 42% of its total revenue (all other publishing combined is another 28%). More importantly, MathSciNet is widely relied upon by the mathematicians in their research.

MathSciNet is also under threat of losing any relevance within a decade. To begin with, there is competition, – not only from zbMATH (the electronic version of Zentralblatt für Mathematik, Math Reviews’ older sibling) which is now supported by the German government and is free to use, but also from the general indexing services, like Semantic Scholar. Unlike MathSciNet and Zentralblatt, startups like Semantic Scholar rely not on the free labor of its reviewers, but on AI, and, to be frank, TL;DR’s they produce are perfectly fine, and are free5Sure, their business model will change, but as of now they are extremely inexpensive to operate compared to the traditional outlets.. Does AMS expect the university libraries will keep subscriptions as the news about free equivalent products spreads?

For the MathSciNet to stay relevant, it needs to find ways to be useful in the changing world. They start in a good position: great name recognition, institutional knowledge, remarkably clean database of authors, but these is not the same as having a product the universities will be ready to pay for.

Search for a new business model is never an easy one. I believe the MathSciNet should start by becoming more open, to enable to data analytic explorations. Creating APIs6Application Programming Interfaces, tools that allow an application to access the database on the background. Right now, any automated download from the MathSciNet requires a manual intervention of the Executive Director…, and allowing the community to use it (for free or for a nominal fee) would go long way to creating intellectual property that will be hard to reproduce elsewhere.

Once the curious hordes get access, one could only imagine what uses they will bring to the table.

My personal favorite would be to restructure the Mathematical Subject Classification (MSC).

Currently, the MSC is a taxonomy updated once a decade, by a committee. The updates are very protective to legacy rubrics, and keep the overall structure of a tree with very few levels and a lot of children at each branching. It does have a warm, nostalgic feel, but is increasingly useless7Remember how often choosing the MSC for your paper feels like an unnecessary chore?. This slide into irrelevance happens at the background of an explosion of theoretical understanding and algorithmic development of clustering techniques. Ideally, we should be analyzing the current state of the trade8As an experiment, try finding out how many articles are published at a few top level MSC categories, and marvel at the disparities in numbers., and create clusters reflecting existing interrelationships between areas in mathematics (and beyond), not necessarily as a tree-like taxonomy.

Whether MSC, or other use cases, – it is clear to me that MathSciNet should drastically expand its interface with the mathematical and other communities. Otherwise, the operation of this mighty enterprise will turn awry and lose the name of action, and, as I mentioned, without the MSN the existing business model for the AMS as a whole just isn’t viable.


Over the past couple of decades, the AMS seemed to attempt to become a version of its engineering counterparts, introducing named lectures and fellowship programs. I feel this cannot be the way to become more (or even to stay) relevant. Rather, the AMS should serve the community, and the society in general, by introducing programs that materially help mathematical community by leveraging its status and clout in innovative ways. A few ideas outlined above might be not the best ones, but one needs to start somewhere, – to keep the American Mathematical Society the powerful institution it is now.


Update: I sent this little note to AMS, offering to publish it in the Notices. They told me that “it was decided” I need to shorten it to one third of its size. Not sure anyone will benefit from reading a thus abridged version, and I am not eager to publish one.