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Can a professor (let's say a math professor) do something like an internship at a tech company during the summer as a way to keep their industry skills sharp / relevant? Does anyone ever do this?
Perhaps the word "internship" is not quite right, but the idea would be to show up, be useful for 6 weeks or so, then go back to the university and teach students relevant real-world skills they'll need in industry.
Yes, but they are usually called "visiting researchers" or, in some cases, "residents", rather than "interns". See for example Facebook, Microsoft, or Google.
Yes, absolutely. I am not able to answer for every discipline and country, but this is common in many places. There are company schemes, government schemes, charitable schemes and university funded schemes to support such arrangements.
For a specific example, in the UK the Royal Academy of Engineering provides Industrial Fellowships to support such activities.
Mathematics professor Robert Talbert, in a 2018 article titled "What I Learned On My 'Secret Sabbatical' As a Scholar-in-Residence at a Private Company", described how he was an intern for a year at the furniture company Steelcase:
Last fall I started my first day on the job as an embedded faculty member with a corporation—as a scholar-in-residence at Steelcase Education. But don’t be too impressed by the title; according to the employee system, I was just an intern.
Actually intern is probably the best lens through which to look at what I’ve been doing at Steelcase for the last eight months. I was nobody special: Just a guy, at the bottom of the org chart, working with and around a lot of people smarter and more talented than I am in any number of ways, and tasked with making their work and the collective work of the organization better. It’s kept me a little bit humbler than I would have been otherwise.
Math perspective: Of course they can (in the sense of there is no law against profs doing internships). In practise, I doubt many profs do this. Universities and professors do generally value research higher than real world skills. If a university cares about teaching real world skills, they might hire anyway a professional from the real world to teach a course.
However, there are some (mostly applied) math profs who work closely with industry. This might influence their teaching, i.e. some of them surely tell the students a little bit about the world outside academia.
(For a newcomer to industry, it would be also doubtful if they could be useful within 6 weeks.)
The short answer is of course. But it isn't necessarily going to be easy to get your foot in the door for a short time. You would almost be better off applying for jobs and then giving your notice before returning to teach.
There are two issues at play here.
The company the professor would intern at would not want to sink cost into temporary training.
The company would not want to expose private data. Especially to a mathematician that would usually mean working with customer data or insights, or company revenue etc. Companies usually don't want to expose themselves or their clients data to what would be a temporary hire that isn't getting vital work done.
Basically, a company tends not to want to hire or give space to someone who isn't going to produce some results or profit for them. If a mathematics professor genuinely expressed interest in helping the company in exchange for learning about the industry along the way (to benefit students as well) I do believe that it would just be a matter of communicating with enough companies.
Heck, it would probably benefit the professor to say that they will recommend future graduates to that company if they allow them to learn about the industry.
Just take a Sabbatical, which lets you take time off to work in-industry. Some universities require you to take a sabbatical of a certain length at fixed times. It's odd that you haven't run into this, unless you are relatively new.
The problem for you is that it won't be 6 weeks, it will be closer to 6 months. Nobody can be useful in 6 weeks, so you can't get cutesy over-the-summer work, especially not unpaid.
