Every now and again, academic departments are threatened with closure by university administrations.
One current example: Pure Mathematics in Leicester is under threat of closure (making academic staff redundant).
What successful strategies have departments been able to use to see off threats like this?
These decisions are all about money, and there's no easy way to get money.
A more detailed answer: https://ep3guide.org/toolkit
This is, in fact, the second time that Leicester's maths department has faced these kind of threats. The previous time (in 2016), they backed down following a petition and other objections organised by a variety of mathematicians.
Check with whatever accreditation outfit your school uses. I was once at an engineering school and the engineering faculty really thought that they could teach the "math their students needed" and would really liked to have gotten rid of the math department. But whatever accreditation body they used to have an accredited engineering program insisted that they have a real math department staffed by real mathematicians.
So the answer might be "if you close your math department, you'll lose your accreditation." And who wants to send their kid to an unaccredited school?
First of all, let me say that I find what is happening at Leicester utterly horrible.
However, the truth is that pure mathematicians do, in general, tend to have a disdain for applied work. In not a few places "mathematics departments", which originally contained within them areas such as statistics or computer science, tended to eject any type of applied field and let them create their own departments, while leaving only the purest of the purest within the administrative division labelled "mathematics". This attitude is now coming back to bite them.
As a long-term strategy, it might serve mathematics departments well to maintain more diversity in their research focus, and not segregate "pure" and "applied" research. After all, there isn't really a very clear line between "pure" and "applied": a single researcher may do some of both, and some types of research touch on both. If the University of Leicester did not have two clearly separated administrative divisions labelled "pure math" and "applied math", then they could not pull this off.
This is not meant as criticism, but as a pragmatic suggestion.
If the academic department offers courses that directly contribute to the university's mission and vision, then this can be used to justify its continued existence. For example, a religiously-affiliated university would not close a theology department even if the department has dwindling enrollment.
At least in the US, the outlook for many universities is utterly dire. Enrollments have plummeted due to the virus, but at least in the US education has been headed towards a major correction for some time. Many will go bankrupt, or be gobbled in an acquisition or merger. Mergers and acquisitions almost always mean redundancies, either by thinning a department or eliminating it entirely.
Thus, the answer to the OP is that it greatly depends on why exactly a department might be eliminated. In some cases there may be strategies that can be employed to save a program if it's a marginal case (such as the petition mentioned in another answer). However, in many cases the best thing you can do if the writing is on the wall is to polish the resume and get out ahead of everyone else.
Making the studied subject useful at something would be a good start.
In the specific case, the definition of "pure mathematics" deserves an historical introduction, to make clear it has nothing to do with the problem-solving mathematics. It is a subject that was founded only in the thirties by the Bourbaki group in France. The group was chasing the goal of finding a complete and coherent foundation of mathematical analysis, since few members of the groups had found some counterexamples in the Gourstat's book used back then.
The complete detachment from physics, and consequent creation of theories descending only from axioms, where no single problem is ever solved, is what characterises the "pure" approach established back then.
Fortunately or not, Goedel incompleteness theorem wiped out the possibility of the coherent and complete theory sought by Bourbaki for good. Despite this event, the branch of "pure mathematics", as it fits very well with the academic environment and the need of a hierarchy not based on merit (if no problem is solved, who will tell who deserves a promotion and how does not), is still a subject per se, and it had become as well as a pedagogical methodology (and a detrimental one, according to many).
Interesting to notice that there have never been such a thing as "pure mathematics" before Bourbaki, having maths always something applied to solve real problems for the entire history of humanity: finding algorithms, and generalisations only when it was possible to apply a solution found in one instance to solve a wider range of problems. Archimedes, Gauss or Euler would have not understood the question: "are you a pure mathematician?".
To the commenters distressed at the idea of a university without a math department, they can be reassure that "pure mathematics" has nothing to do with the mathematics they have studied for the A-level, or the mathematics they know of as a tool to solve problems.
Besides the theoretical groundlesness of pure mathematics, as mathematically proven by Goedel, the "pure" mathematics studies certainly have a detrimental effects on their students. I recently interviewed a candidate with a pure mathematics background, graduated with full marks with a thesis on a subject he was not able to explain during the interview, who turned out to have never heard of graph theory. Also he heard of the existence of differential equations, but as something for another department, not for him, so he had never solved one.
There is the curious situation of an increasing amount of industries thirsty for mathematical talents, as problem solvers and algorithms developer, and the "pure" mathematics department is producing graduates with the title of mathematicians who are crippled by too much useless theory, lost in a machinery of details, and unable to solve anything or compute anything. Leaving the abstract manipulation of symbols, and teaching Algorithms and Differential Equations ("with their applications", as purists like to put it) instead of axiomatic theories closed in boxes, would be the best strategy to keep the maths departments alive, and provide students with capabilities other than knowledge.
---- Edit:
I must have touched a point, since all my posts have been downvoted shortly after writing this one. Bringing down the reputation of the author instead of talking about ideas, now, that's a strategy never used before! .-)
