Interdisciplinary Problem Session hosted by Mathematics Department: Advice

From my experience (mostly on the interface of biology): unless you are thinking about something like theoretical physicists, you need to consider that most researchers will not have a good understanding of what mathematicians actually do, and will not be able to recognize an interesting mathematical problem.

This means you essentially bring the mathematicians to a state in which they will be able to identify good problems (with help of the other researchers). This is not easy, but I think it is a more sensible strategy than trying to bring the non-mathematicians to a state in which they can identify good mathematical problems (though you can nudge them in this direction).

Thus, you need to set reasonable expectations with realistic goals. I suggest the following approach:

1. At the first stage, I would focus on introducing the mathematicians to the field, so they can start understanding the terms and what tools are being used. This will probably get the mathematicians thinking about how to introduce formality and so on. This step is also very important in order to establish a common language for discussion.
2. Next, I would let the researchers introduce important open problems in the field. These problems don't need to be mathematically interesting - because the researchers will probably not be able to tell what is mathematically interesting anyway. This helps avoid a common issue of theorists focusing on problems that are useless for practitioners.
3. After these are done, the participants should have the minimal set of tools to explore problems that are interesting for both sides. This should be mainly based on open discussion.

Realistically, if you are able to achieve the first 2 steps and get the people to start thinking about point 3 (as it can take some time to process all this new information) and possible establish some pre-collaborations, I would call it a success. While I strongly believe that many fields can benefit from mathematical insights, it can be very challenging to find a problem that is interesting for both sides.

BTW it may also help to invite some people which are doing active research at the interface of the two communities.

You could set up something like a help desk or a clinic. Have a website to advertise the services to the other departments and give some examples of things you could help with. An uncomplicated way to meet is important to reduce the barrier for others to contact you. A short meeting should be enough for you to asses if this is any thing the maths department could help with.

The University of Bristol has a statistics clinic working along this way. They have a drop in sessions every other week, where researchers form other departments can present a short summary of their problem. This is not sessions where you solve problems in front of others but it is a way to facilitate collaborations.