Columbia University has a dual-degree program with Juilliard, I can put you in touch with a few of my friends who did that (with a CS degree on the Columbia side)
Strictly mathematically speaking, the partition function is a sum of exponentials, which are each dimensionless.
From a probabilistic perspective, the partition function is the normalizing constant for the Boltzmann distribution; if the partition function takes value Z, then the probability of any state of your system is given by P_state = exp(-E_state/T) / Z. Since the numerator is dimensionless (it represents an unnormalized probability), and the left hand side represents a probability (which is dimensionless), the denominator Z must also be dimensionless.
I realize now that my initial post may have implied otherwise, but I'm mostly looking to start establishing relationships that may lead to entrepreneurial collaboration in the future. I'm definitely not looking to jump straight into anything at the moment – it's not like there's a particular idea I'm in love with, and I'm naturally a very cautious person so I'd like to take the time to get to know people before starting a project with them.
My concern is moreso that it takes a lot of time and effort to build trust, and so I'd like to start meeting such people sooner rather than later.