I recently talked about (in https://mathstodon.xyz/@tao/113910070146861518) how solutions in a dynamical systems can be roughly divided into an effective dynamics regime, where simplifying principles such as linearity can be reasonably good approximations, and the more complicated regime of no effective dynamics, in which the behavior can be significantly more nonlinear. For instance, in linear regimes, applying a force in one direction, if followed swiftly by an equal and opposite force in the other direction, will get the state roughly back to where one started; but the same is not true in nonlinear regimes. (If one pulls a spring too far in one direction, one can end up with a broken spring, with no way to return to the initial state, regardless of how one tries to push the spring back in the opposite direction.) (1/7)
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But actually there are real-world situations where even nonlinear dynamical systems are not a sufficiently accurate model to describe the behavior, and even more complicated mathematical models, such as mean field games, are needed. This is particularly the case where individual agents in the system plan their actions not just based on the current state of their system, but on their predictions of where this state may be headed in the future.
One familiar example of a mean field game arises in the traffic flow of a large city; commuters may select their travel routes and schedules in anticipation of what the traffic may be at a given time of day, but the choices they make in this regard may make that traffic different from what they initially expected. As a consequence, the actual distribution of traffic over time can behave in highly unintuitive ways; it is not always the obvious "rush hours" which exhibit the most congestion. (2/7)
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It is a highly nontrivial problem to determine if a mean field game even has any self-consistent solutions (Nash equilibria), let alone how to compute them numerically. And even if solutions exist, they could be highly non-unique; the initial conditions could permit multiple consistent solutions, not all of which are equally desirable.
Our mathematical understanding of these equations lags significantly behind that of nonlinear dynamical systems, which in turn lags behind that of linear dynamical systems. External shocks to such equations can have extremely unpredictable and poorly understood consequences, due to their complex effects on the agents' perception of future conditions; and their impact often cannot be cancelled out simply by applying further shocks in the opposite direction. (3/7)
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A real-time version of this phenomenon is playing out right now in my own field of mathematics in the United States (it is impacting countless other fields as well, but mathematics is the area where I have the most information). It is hiring season currently, with postdoctoral job offers being sent out as we speak, and candidates for tenure-track and tenured positions interviewing. However, the NSF postdoctoral fellowships, which were due to be announced some time this week, have been unexpectedly delayed due a recent executive order affecting almost all federal government funding.
Other major NSF funding sources for mathematics, such as the grants to mathematics institutions or the multiyear CAREER grants that have played a pivotal role in the trajectory of many talented mathematicians in the US, are also in a state of high uncertainty, in which it is not currently known whether grants which had been recommended for approval or renewal by the relevant NSF panels will actually get funded. (4/7)
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One can hope that some clarification of the executive order will come soon, and that funding that had already been approved can be disbursed as planned. However, even if this is the case, there are already other impacts that cannot be as easily reversed. In the last 24 hours I have recieved multiple emails from job applicants nervously asking whether the funding for their position is as secure as they had assumed; and at least one math institution is advising visitors to their programs to delay booking travel or accommodation until the funding situation is clarified. It is likely that there will be cancellations and reduced participation at multiple workshops and other research events because of this. (5/7)
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But even if the system state returns to something resembling what was the status quo 48 hours ago, the solution to this mean field game may now be quite different. Many potential job candidates will factor in increased uncertainty in the funding and regulatory environment for mathematics in the US going forward, and may therefore accept competing offers in other countries. The number of participants in major US-based mathematical events, such as the 2026 International Congress of Mathematicians in Philadelphia, may also be impacted. One could argue that any "brain drain" from the US would simply result in an equal and opposite "brain gain" in other countries, but this thinking again assumes an oversimplified linear model: in practice, the rest of the world would not be able to absorb all of the lost opportunities in the US in a single job cycle, and some mathematicians may end up leaving the field entirely, or not obtain as enriching a career as they would otherwise have been able to achieve. (6/7)
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The last comparable shock to my field was the 2008 global financial crisis. Due to that crisis, university budgets and endowments were suddenly upended. My colleagues and I received salary cuts; but more importantly, many open positions evaporated, and an entire generation of promising mathematicians was at risk of having their careers cut short. However, in the US a stimulus package was passed which, among other things, allowed the NSF to greatly expand its postdoctoral program on an emergency basis, and also maintain or temporarily expand other sources of NSF funding. This allowed hiring for that season to proceed at something close to normal levels, and in the end there was no major disruption to the expectations for what the career track of an academic mathematician in the US would be. To continue the mathematical formalism at the start of this post, the "mean field" of the mean field game stayed within the "effective regime" in which it could be understood by simpler models.
A similar stabilization of expectations for the future would be quite welcome and helpful at this point. But the current administration has thus far not demonstrated much ability to solve mean field games effectively. (7/7)
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The executive order has just been rescinded: https://www.cnbc.com/2025/01/29/white-house-rescinds-federal-funds-freeze-memo.html . However, as discussed above, due to the alterations in future expectations of funding stability, the current situation is not close to where it was before the order was first introduced.
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Even less clarity now, as the White House press secretary asserts that the original executive order is actually still in effect: https://thehill.com/homenews/administration/5113776-white-house-press-secretary-spending-freeze/
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@tao
Schroedinger’s orders…
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