Mathematicians...

Mathematicians applied to industry don't work to produce theorems: they work to build formal models of real phenomena, then derive computational tools usable in production from those models. They observe the physical or behavioral phenomenon, translate it into a system of equations (differential, stochastic, constrained optimization), identify the parameters from the experimental data, build a simulation or control algorithm, and validate it against real data.

In the industrial setting, a consolidated practice is bilateral work between academic mathematicians and industrial engineers: a documented example is the control problem in blast furnaces, where mathematics was actually used in factory production. 

in a design project.

Possible tools an applied mathematician brings to a design project are: 

- differential equations for dynamic systems, 

- control theory for systems with feedback, 

- variational and optimization methods to find optimal solutions within constraints, 

- differential geometry and topology for configuration spaces) 

- statistics/Bayesian inference for uncertain systems.

Designers... in a computational learning project.

Designers translate mathematical variables and thresholds into perceptual language. They design how the algorithm communicates with the user: e.g. in an automotive HMI project, progressively increasing haptic resistance on the steering wheel, changing timbre of audible feedback, varying brightness on the dashboard. 

They take the mathematician's parameters and convert them into design variables, meaning that response time might become perceived latency, feedback gain might become heaviness of the control.

Together!

Mathematicians govern the formal space of the system, the equations and their parameters; designers govern the perceptual space of interaction, i.e. how those parameters become experience.