What things are mathematical research grants typically used on?

This post How do mathematicians conduct research? includes some explanation on things mathematicians do. I am now also a little curious about how much money/resources are used in this process.

For simplicity, let's suppose a pure maths professor gets a research grant of, say, 200 thousand, or 1 million USD. (Over a couple of years.) Note I have edited the amount according to the comments below.

Then, what is the typical list of things on which he or she spends the money, and how much roughly will be spent on each item? 1 million or 100 thousand is a pretty large amount, so the list of things on which we spend the money is probably a bit long, I guess. For example, roughly how much of the grants will be spent on PhD students? (I know the quantity is not fixed. I am just looking for some rough idea.)

I think an answer to this question would be very helpful for understanding how research in pure maths works. (After all, the spending of maths research is going to be very different from a scientist who works in a lab!)

Note that I am specifically asking about pure mathematics (analysis, geometry, number theory, etc, etc). I hope that this will narrow the scope of this question to a sufficient extend for an answer.

• Is that one million to spend over five years? If the time is really short that rules out hiring a postdoc. Aug 22, 2021 at 4:15
• A million dollar grant would be a very large award for an NSF DMS grant. See the awards at nsf.gov/mps/dms/awards.jsp Aug 22, 2021 at 4:53
• A million dollars over ten years is sort of possible. I have in mind the Simons Investigators awards. Even these do not guarantee funding past half a million. The question might be more reasonable if it used less fantastic amount of money. Aug 22, 2021 at 5:00
• I have edited the "amount of money". Hope it is now not "too large". Aug 22, 2021 at 5:59
• @BrianBorchers, but it would be fairly typical of an ERC grant. Aug 22, 2021 at 14:24

To supplement the other answer in regards to how much is spent on what, I think the only general statement that can be made about larger grants is that most of the money will be spent on salaries. (Where by larger I mean enough to pay someone a salary in the first place.) The university's cut is fixed anyway, the amount one can reasonably spend for office equipment and similar expenses is reached quickly and I don't think many people would manage to spend more than their own salary on travel (though the group's travel budget should increase with the number of people of course).

However, whose salaries are paid with the money will strongly depend on personal preferences. Some people prefer working with people who already know what they are doing and will thus hire mostly post-docs or invite colleagues for extended stays. Others might prefer having more PhD students, who require a bit more handholding, but can be set on longer projects and who aren't already stuck in their way of doing research.

Personally I have seen groups with half a dozen or more PhD students and only sometimes a single post-doc, and I have seen groups with 3-4 post-docs and only a single PhD student. Both worked well in their own way.

Finally one should also remark that sometimes it is more about people than about positions. If you have a really promising student in a class and some grant money coming in, you'll naturally try to recruit them as a PhD student, even if you'd normally would prefer a post-doc. Similarly, if one of your close collaborators in a different place recommends you one of their PhD students who is close to finishing, you might think about opening a post-doc position.

• I don't know about Math specifically, but in most fields some fraction of the Principle Investigator's salary would be paid by the grant to cover the time they spend doing or supporting the research proposed. Aug 22, 2021 at 13:11
• Thank you for the answer! Aug 22, 2021 at 23:14

Depends on the country and where the grant comes from. At some universities, for consultancy type grants, they will be a tax. This tax is used to support any overheads related to administering the grant. After that, you may use it to supplement your salary, hire a post-doc or research assistant, buy out your teaching, pay to visit collaborators (and vice-versa), attend conferences, pay publication cost (page charges), buy yourself a new computer and pay for software licenses.

• A good list, the only missing point that I've seen parts of larger grants used for is organizing workshops, summer schools and so on.
– mlk
Aug 22, 2021 at 8:08

Perhaps I should answer as I hold a ~£1M grant in pure mathematics. This grant is to be held over 4 years.

The most pricy thing that this grant yields is overheads. I would say a bit less than half of the grant is overheads.

The next most expensive things is researcher time. Despite what I thought when younger, the university does not just cost the salary, but also must cost pension contributions, insurance, raises, office costs (estate costs), etc. This tends to make my university request funds from funding bodies that are somewhere around double the amount any researcher actually gets paid. My particular grant is called a fellowship, where it not only pays for postdoctoral salary, but my time through teaching and administrative release. With this chunk of cash, I have my time for 4 years and 5 years of postdoctoral salary.

Then comes travel expenses for not only me, but all of the postdoctoral researchers which adds up quickly. Also for visiting researchers to spend time here. Next there is some amount of funding for conferences, public engagement and impact work in order to disseminate the research. The consumables are simply laptops for postdocs, which is negligible compared to the staff costs. This all comes to about 10% of the grant.

Let me give you a related perspective: theoretical physics.

First, most individual grants aren’t in the million dollar range but probably 20 to 40 times smaller. I think the average individual grant in theoretical physics in Canada is in the \$35k-\$40k range. Six-figure grants are rare, although there are “team grants” or “group grants” that will fund many researchers from the same pot.

In Europe some grants come with student funding earmarked.

Some researchers manage to concurrently hold several grants so their net grant intake is in the 6-figure yearly but this is not that common in physics, and probably even less so in math.

Most of the money goes to support students or postdocs. Some of the money goes towards conference or collaboration travel, possibly some visits by external guests and computer stuff - licences etc. In Canada there are equipment grants for larger items so basically the money is largely spent on humans or human interactions.

If you are part of a large group or institute that does run a larger grant, you may have access to parts of the funds to support students or postdocs, travel, promotion etc (it depends on how the institute is organized).

• As far as I know, at least in mathematics, there's an important difference between the Canadian NSERC and the U.S. NSF. An NSF grant can and usually does include summer salary for the PI; an NSERC grant does not. Aug 23, 2021 at 0:10
• @AndreasBlass correct. Yet grants in the millions? Aug 23, 2021 at 0:11
• I've never seen anything near a million-dollar NSF grant, but I wouldn't be surprised if some superstars have. Aug 23, 2021 at 0:14
• @AndreasBlass yes no doubt the Ed Wittens of this world have huge grants, but this is exceptional and likely not a good template for discussion. Aug 23, 2021 at 0:53

In my experience a lot of grants in pure maths don't directly award money but rather pay the salary of one person at a given level of seniority for a given amount of time. At the postdoc level one can apply for various grants that pay exactly the applicants salary for a time frame usually between 6 months and 3 years. Similarly a professor might apply for grants that pay exactly one PhD student or one postdoc. Sometimes the grant comes with an additional allowance for books or travel but this is usually tiny compared to the salary (less than 5%).

So in practise there are very few if any grants that just award 100.000 dollars, they directly award the salary for say one postdoc which might end up being 100.000 dollars depending on the duration.

• Note that the "employee cost" of any employee, including post docs is far more than the salary. Taxes, health care, office space... Aug 22, 2021 at 15:36
• @Buffy True. My point was more in saying that the premise of the question 'what does a mathematician do with a 100k \$ grant' is somewhat flawed because for most grants the mathematician doesn't actually get 100k \$ but rather one paid researcher. Aug 22, 2021 at 16:34

Further to earlier answers, note that it's not unheard of for a Mathematics department to have an experimental laboratory that needs equipment and consumables.

• Sorry, after I posted, I saw the aspect of OP's spec that asked to focus on pure mathematics. But nevertheless, I guess this answer may be useful to a future reader who comes here on the basis of the title. Aug 22, 2021 at 10:17
• You might modify the answer rather than just comment it, actually. But I'll also note that, today, even a pure mathematician might need substantial computing power. Aug 22, 2021 at 12:45