While many specialties are fully filled, we need pediatricians, family medicine, and internal medicine docs. They're generalists and where the largest shortage is. There were 147 unfilled slots for pediatricians, 805 for family medicine, and 357 for internal medicine. They don't have the applicants; it's not the slots.
> There were 147 unfilled slots for pediatricians, 805 for family medicine, and 357 for internal medicine. They don't have the applicants; it's not the slots.
Aren't a lot of these shortages scattered around rural areas where young doctors really don't want to move to? I understand from a buddy who is currently in med school that there are all sorts of incentive carrots being deployed to attract doctors to these underserved communities.
Basically, you interview at a bunch of programs and then rank them. The programs (hospitals) rank applicants and then the algorithm does its magic to "match" applicants to programs. Now, if one doesn't match with any of them, there's something called the scramble where a med student works with their program to match into a program somewhere in some specialty that has room. This is non-ideal, but can work out.
Generally speaking, the match algorithm is setup to guarantee all U.S. medical school graduates a match somewhere in something. In may not be what you want, but you will have a job. Then, preference is given to things like the island schools (affiliated medical schools in the Carribean, which are very expensive, but somewhat easier to get into), and then to other international medical schools. Somewhere in there are also foreign physicians who want to work in the U.S., but are forced to redo residency.
I don't know everything about how it works, but that's the general idea. To that end, I don't fully understand the stats you pulled from the reference. That doesn't mean they're not valid, but I don't know.
And, yes, often times, there are open slots at some program in the middle of nowhere. As much as there can be incentives such some debt relief by working in rural hospitals, the jobs are not a good fit for a lot (most) people. I mean, someone just worked extremely hard for 10 years or more and you want them to go live in a town of 10k people. It's not that it's not important, but you can't force people to do it and it takes a particular personality to be happy there. A lot of highly educated people want to live in urban centers with amenities. Not all, but probably most.
Places like Canada use their foreign docs to fill this rural gap. A not small number of the rural docs are foreign born and trained and they essentially work this crappy jobs until they have permanent residency and then they move to more desirable markets. It's a trade, I guess, but there's not a small amount of resentment about it.
> A not small number of the rural docs are foreign born and trained and they essentially work this crappy jobs until they have permanent residency and then they move to more desirable markets.
Not sure that I follow how "rural" necessarily begets "crappy" though. Is the working quality of life somehow that much worse, or is it the relative social isolation and/or lack of recreational options while off duty, or is it really just a case of urbanite out of their accustomed habitat?
It's a combination of factors. Rural hospitals and clinics tend to be under-resourced with lack of equipment in buildings that aren't particularly nice. As far as small town, if you like it, great. However, people who are highly educated tend to like to be around others who are similarly educated and that's difficult to find in a rural town unless it's also a university town. There tends to be a lack of school options for their children and given how much they spent on their own education, they tend to prioritize this highly. There tends to be a lack of town infrastructure like good grocery stores, or theater, or museums, or other amenities. Docs also have their own medical needs and understand that those can't be met at small clinics, so they like to have access to good hospitals. Imagine intimately knowing all the ways something like childbirth can kill you and also knowing that there's not an appropriately trained surgeon in town. By the time one finishes their training, they're probably in their 30s and may want to find a partner. Options tend to be limited in small towns. On the darker side of things, foreign people are often not particularly welcomed in rural towns and this can be a particularly bitter experience for the foreign docs that are essentially forced to work there.
So, no, it's not just an urbanite out of their comfort zone. There's a whole host of issues. And, to be clear, we need people to work these jobs, but it's not particularly pleasant for a lot of them.
My statement above was correct. There are students who graduate from accredited medical schools with MD/DO degrees but don't get matched. Part of that is because some of them simply don't apply to programs that have extra openings. Medicare / Medicaid pay primary care physicians below market rates so students are naturally reluctant to pursue those specialties.
If they don't match, they're allowed to scramble and move into one of those programs with open positions. If they don't choose to, that's on them, but it's still not a problem with number of residency slots.
I very much agree that pay is a barrier to entering specialties like family medicine. Though it depends on the market, I normally see family medicine at around $200k/year and that's not great if one needs to take something like $750k debt to get there along with eight years of training after a bachelors. If we want to fix that, then we need to make the value proposition better and reduce the medical school debt, improve working conditions, and/or increase pay.
So, yes, if one wants to maximize their earning potential, then they need to enter one of the specialty residencies and fellowships. Those are currently filled. However, that's not where the biggest need is and I contend that's not why there's a physician shortage.
No, they are not paid too much. There's a lot of incorrect assertions here, so it'll take a lot to work through them.
Physician pay depends on specialty, but it can range from the low $100kish mark for pediatricians to $500-750k for certain kinds of surgeons. Family medicine tends to be around $200k. However, this amount ranges vastly by market and top pay often goes to those willing to work in more rural hospitals because no one wants to. For example, pay in NYC for physicians is appalling low compared to the rest of the U.S. market. In addition, certain systems have hard caps. For example, the VA hospitals cap physician pay inclusive of bonus at $400k. This is documented and you can in fact just look up a random doc at the VA with one of the many federal pay search tools.
While some doctors can make more, it typically because they own a practice and that increased pay comes from good old fashioned capitalism. Meaning, they tax the amount their nurses, NPs, medical assistants, etc. make just like all businesses make money per head on their employees. Whether you believe this is right or wrong is up to you. However, this is not any different that someone who runs, for example, a yard care business. More accurate pay can be found by those who work directly for large hospitals.
Next, the cost of medical education in the United States is vastly higher than other countries. Right now, medical school will cost you somewhere from $400-600k. This is in addition to whatever debt accrued during undergraduate. Further, medical school applications are highly competitive, so students often accrue additional debt by completing a masters in something like public health prior to entry to medical school. This means that someone may have upwards of $750k of debt when they finish medical school, but they still have somewhere between 3-10 years of residency and fellowship before they make attending money. During this time, the debt accrues interest and balloons.
Now, once you become an attending, you're still not good and expenses are vast. Shift work can vary from something like 7 12-hour shifts in a row for intensivists to 14 shifts in a row for hospitalists. Note, just because it says its a 12-hour shift doesn't mean you work 12 hours. They still need to chart and bill and if it's busy, that may be another few hours after the shift is over. In some remote clinics, an ER physican may work 7 24-hour shifts in a row. That may sound absurd and unsafe and it likely is, but it's the reality of the work. If someone is working that schedule, they have increased expenses to just, frankly, live. On the low end, it's very difficult to cook in that environment, so you have to buy a lot of premade food. On a more expensive end, having children on this schedule is extremely difficult. You either require a spouse that doesn't work or you need something like a night nanny. If you're working 12 hour shifts, you must sleep at night and you can't be up to take care of a baby otherwise you run the risk of killing someone the next day. Unless you're paying someone under the table, current nanny rates in large markets are about $20-25/hour. Insurance rates are also high. I don't mean malpractice either. Generally speaking, one needs to carry disability insurance because if one gets into a car accident and breaks their magic hands, there's no way to pay back that debt otherwise. These policies are thousands a year. That's just the start. They pay a large amount of money to buy their time back because they don't have it.
Next, there's a myth about limiting residency slots in order to increase pay, at least recently. I will not defend the AMA and some of they took, especially in the 1990s. Here's the 2025 residency match data:
The number of offered and filled slots is on page 2 (or 13 depending on how you count). Some specialties filled all of their slots. Where the U.S. vastly lacks is pediatricians, family medicine, internal medicine (who can work like family medicine if need be.) Pediatricians had 147 unfilled slots. Family medicine had 805 unfilled spots. Internal medicine had 357 unfilled spots. These spots can be filled by people who graduated from U.S. medical schools, island medical schools, Mexican medical schools, or a vast array of other foreign medical schools. However, they're not filled because they don't have the applicants. That's not medical school collusion. That's the hard reality that medical school is extremely expensive and the training is extremely long.
Now, how do other countries handle things? One, their medical school is not as crushingly expensive. Two, places like Europe cap the number of hours a physican can work. If you want to pay American physicians less, you'd need to blow out their medical school debt, reduce their hours, and offer better benefits. Until then, no, really, they're not overpaid.
If you want to start pointing fingers, try the vertical integration of insurance companies, pharmacy benefit managers, and hospitals. I don't have the numbers readily available, so I'll stop here. But, really, it's not the docs.
I'm an applied mathematician and this is the most common layout for dense matrices due to BLAS and LAPACK. Note, many of these routines have a flag to denote when working with a transpose, which can be used to cheat a different memory layout in a pinch. There are also parameters for increments in memory, which can help when computing across a row as opposed to down a column, which can also be co-opted. Unless there's a reason not to, I personally default to column major ordering for all matrices and tensors and use explicit indexing functions, which tends to avoid headaches since my codes are consistent with most others.
Abstractly, there's no such thing as memory layout, so it doesn't matter for things like proofs, normally.
Automatic differentiation was actively and continuously used in some communities for the last 40 years. Louis Rall has an entire book about it published in 1981. One of the more popular books on AD written by Griewank was published in 2000. I learned about it in university in the early 2000s. I do agree that the technology was not as well used as it should have been until more recently, but the technology was well known within numerical math world and used continuously over the years.
Greek War of Independence (1821-1832)
French invasion of Spain (1823)
Russo-Persian War (1826-1828)
Russo-Turkish War (1828-1829)
Hungarian Revolution and War of Independence (1848-1849)
First Schleswig War (1848-1851)
Wars of Italian Independence (1848–1866)
Crimean War (1854–1856)
Second Schleswig War (1864)
Austro-Prussian War (1866)
Franco-Prussian War (1870-1871)
Russo–Turkish War (1877–1878)
Serbo-Bulgarian War (1885)
Greco–Turkish War (1897)
Together, that adds up to multiple decades of war.
I think https://ourworldindata.org/grapher/deaths-in-wars-project-ma... puts this in perspective. The period from 1815 to 1915 was a much more peaceful period measured by deaths in war than 1915 to 2015, though 1975 forward seems like a return to that level (but world population is so much larger now that it's even better than it seems).
Counting the years when there was a war anywhere is Europe, you'll end up with a large number.
I'm counting how often each country was at war. Several countries had no wars, and even the most war torn country didn't fight for more than 10-15 years.
That's really not true if you look at the European neighbors and European territories of Russia and the Ottoman Empire.
Also not true of Spain, which spent a lot of time in internal warfare (with occasional outside interventions.)
But, yes, excluding those, most of the countries in Europe were too busy fighting endless wars throughout their (or their allies’ or enemies’) colonial empires (whether to expand them, defend them, or put down or assist rebellions in them) to bother fighting other powers in Europe in that period.
True. That said, I'll also mention that tomography is a very rich, interesting field that's still open to new innovations. I work in the area and unfortunately needed to pass on a muon tomography contract some years ago. By the way, you may know this, but the following is for the broader audience.
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If anyone is interested, the book Parameter Estimation and Inverse Problems by Aster, Borchers, and Thurber give an easy introduction to simple tomography problems in their book. Example 1.12 in their second edition has a very basic setup. More broadly, tomography intersects with an area of study called PDE constrained optimization. Commonly, tomography problems are setup as a large optimization problem where the difference between experimental data and the output of a simulation are minimized. Generally, the simulation is parameterized on the material properties of whatever is under study and are the optimization variables. The idea is that whatever material property that produces a simulation that matches the experimental data is probably what's there. This material property could be something simple like density or something more complicated like a full elasticity tensor.
What makes this difficult, is that most good simulations come from a system of differential equations, which are infinite dimensional and not suitable for running directly in an optimization algorithm. As such, care must be taken into discretizing the system carefully, so that the optimization tool produces something reasonable and physical. Words you'll see are things like discretize-then-optimize or optimize-then-discretize. Generally speaking, the whole system works very, very poorly if one just takes an existing simulator and slaps an optimizer on it. Care must be taken to do it right.
As far as the optimizer, the scale is pretty huge. It's common to see hundreds of millions of variables if not more. In addition, the models normally need to be bounded, so there are inequalities that must be respected. For example, if something like a density isn't bounded to be positive (which is physical), then the simulator itself may diverge (a simulator here may be something like a Runge-Kutta method.)
Anyway, it's a big combination of numerical PDEs, optimization, HPC, and other tools just to get a chance to run something. Something like the detector in the article is very cool because it may be a realistic way to get data to test against for super cheap.
In the U.S., there is typically a separation between calculus and real analysis. Though, the amount of difference between the two depends on the university.
In calculus, there is more emphasis on learning how to mechanically manipulate derivatives and integrals and their use in science and engineering. While this includes some instruction on proving results necessary for formally defining derivatives and integrals, it is generally not the primary focus. Meaning, things like limits will be explained and then used to construct derivatives and integrals, but the construction of the reals is less common in this course. Commonly, calculus 1 focuses more on derivatives, 2 on integrals, and 3 on multivariable. However, to be clear, there is a huge variety in what is taught in calculus and how proof based it is. It depends on the department.
Real analysis focuses purely on proving the results used in calculus classes and would include a discussion on the construction of the reals. A typical book for this would be something like Principles of Mathematical Analysis by Rudin.
I'm not writing this because I don't think you don't know what these topics are, but to help explain some of the differences between the U.S. and elsewhere. I've worked at universities both in the U.S. and in Europe and it's always a bit different. As to why or what's better, no idea. But, now you know.
Side note, the U.S. also has a separate degree for math education, which I've not seen elsewhere. No idea why, but it also surprised me when I found out.
There's a POV that learning math and learning how to teach math effectively are two orthogonal things.
If one only took the method of teaching that is most common in US university lecture halls, and applied it to a small class of pre-teens or teenagers, it probably wouldn't be very effective.
Honestly, the parent is pretty accurate. No one is claiming that P = NP. However, the technology to solved mixed integer programs has improved dramatically over the last 30 years and that improvement in performance has outpaced computational speed by multiple orders of magnitude. It's the algorithms.
I just went to pull up some numbers. The following comes from a talk that Bob Bixby gave at ISMP in 2012. He is the original author of CPLEX and one of the current owners of Gurobi. Between 1991 and 2007, CPLEX achieved a 29530x speedup in their solvers. Their 1997/98 breakthrough year attributes the following speedups, Cutting planes: 33.3x, presolve: 7.7x, variable selection: 2.7x, node presolve 1.3x, heuristics: 1.1x, dive probing 1.1x. I don't have a paper reference for these numbers and I don't think he has published them, but I was at the talk.
The point is that integer programming solvers perform unreasonably well. There is theory as to why. Yes, there is still a lot of searching. However, search in-and-of-itself is not sufficient to solve the problems that we regularly solve now. Further, that increase in performance is not just heuristics.
Laugh. Probably! I gave a talk at that conference titled, "Software Abstractions for Matrix-Free PDE Optimization with Cone Constraints." I still work in the field, so you want to talk algorithms sometime, feel free to send me an email. I keep my email off of HN to limit spam, but if you search for the lead author on that presentation, it should list my website.
I'll second this. Their methods are very powerful and very fast. For those out of the loop, the Chebyshev (and ultra-spherical) machinery allows a very accurate (machine precision) approximation to most functions to be computed very quickly. Then, this representation can be manipulated more easily. This enables a variety of methods such as finding the solution to differential algebraic equations to machine precision or finding the global min/max of a 1-D function.
I believe they use a different algorithm now, but the basic methodology that used to be used by Chebfun can be found in the book Spectral Methods in Matlab by Trefethen. Look at chapter 6. The newer methodology with ultraspherical functions can be found in a SIAM review paper titled, "A Fast and Well-Conditioned Spectral Method," by Olver and Townsend.
https://www.nrmp.org/match-data/2025/05/results-and-data-202...
While many specialties are fully filled, we need pediatricians, family medicine, and internal medicine docs. They're generalists and where the largest shortage is. There were 147 unfilled slots for pediatricians, 805 for family medicine, and 357 for internal medicine. They don't have the applicants; it's not the slots.