I am wondering how a mathematician can find a sabbatical in the US. More specifically is it necessary to have people interested in a collaboration with them in the department they wants to visit? Is it possible to find a department for sabbatical in order to start working in a new field?
The main thing to keep in mind is that from the host department’s point of view they aren’t inviting you to come “do a sabbatical” - that’s not an official status at any US department I know. They are inviting you to come and do a certain kind of work (usually with a title such as Visiting Professor/Researcher/Scholar, which is usually but not always unpaid), whether it’s teaching or research or some combination of the two. The fact that you’re on sabbatical from your home institution (as opposed to some other status like being on leave) is immaterial.
is it necessary to have people interested in a collaboration with them in the department they wants to visit?
I think it’s fair to say that it’s necessary to give the department a reason to invite you. They need to have some concrete expectation that your presence will be beneficial to them. The most common such reason (in my very anecdotal experience) is that you will collaborate with some of the department’s faculty, but if you’re offering to teach, run a seminar, mentor graduate students etc, those could all be useful things that might make the department receptive to inviting you. And a collaboration might not necessarily be very concrete - it may be that you are simply an expert in some field that’s of interest to someone and they feel that having you around would be nice for them and their grad students, postdocs etc, regardless of whether there is an official intent to collaborate.
It could also be that you are a famous researcher at the top of your field. In that case I doubt you will need to offer anything concrete. Just say “I was wondering if it would interest your department to have me visit next year”... Your reputation will do the convincing for you.
Is it possible to find a department for sabbatical in order to start working in a new field?
I don’t know, but I don’t see why not. Again, the key is that you have to give people a reason to want to have you around. If you want to expand from field X to field Y and the department has people who are experts on Y but are interested in learning more about X (or about techniques from X that might help them attack the problems of Y) and think you could help them, I think that could be a win-win for everyone and a perfectly good reason to invite you for a visit.
One common option for sabbatical leaves in mathematics in the US is to spend a semester or year at one of the NSF funded mathematical sciences institutes. There are currently 9 such institutes, see
For semester-long visits, these institutes will typically provide housing and some funding for travel expenses.
I've done semester-long sabbatical leaves at IPAM and ICERM. I've also visited for shorter periods at IMA.
Let me complement the answer of Dan Romik with some information from the other side, for those who may not have it: the home institution. And, this may apply only in the US.
A Sabbatical originated in Jewish Agricultural Law. The land should lay fallow for one year in seven and the farmer should take the year off. Wikipedia will give some background on the origin, but not much on the current practice.
Academic sabbaticals aren't universal, but they are pretty common. When available at an institution they may be a contractural right or not. They may be for a full year or a shorter period, say one Semester. They may be fully paid or not. There may be expectations or not. A lot of variation is possible. Some institutions won't offer them at all, though an occasional year without pay, funded elsewhere, is usually a possibility, though that isn't really a sabbatical.
The ideal is that you are more or less entitled to one every seven years, but need not take it. Fully paid with all benefits paid. Few if any expectations about what you do for the year. Earning additional income permitted.
A year in the south of France is possible under the ideal, tasting fine wine, even if (especially if?) you are a mathematician. Starting up a company based on your research is fine. Moving to a new place and working with new people is fine and is hoped for by the home institution.
You may, however, need to arrange it so that continuing work gets carried on in your absence. The lab may not be able to be shut down. Doctoral students still need to be served.
But that ideal isn't open to everyone. To a large extent it depends on the financial position generally of the home institution and how solid their planning is. Having 1/7 of the faculty not "producing" anything is pretty expensive to maintain.
So, more realistically, you may have to negotiate a bit and you may not get a sabbatical every seven years. You may have to make an application. You may have to explain how you will be productive during that period. You may be required to file a report at the end detailing your productivity. There may be some restrictions about teaching at another institution. There may be an expectation that your research will continue. Changing sub-fields is unlikely to be restricted, but changing fields entirely might be an issue.
And you probably have to agree to return to the home institution for at least one year after a sabbatical, even for the ideal model. The institution hopes to get something back from you in productivity or renewed vigor.