It is possible to write any program (i.e. Turing complete) with just one single expression? So I'm currently taking discrete math II at my university and I came across a somewhat philosophical question (so I apologize if this is question is hard to answer precisely) of whether mathematicians and computer scientists view computation in the same way.

We computer scientists define computation with the Turing machine and/or lambda calculus. I was simply wondering if there is some language/notation (lambda calculus??) that allows a mathematician to write a computable (i.e. Turing computable) program using only one mathematical expression.

Again, I apologize if that's a bit of an abstract question (I'm new to these concepts and my terminology isn't great) but let me try to clarify: in math (or at least when I did math in school), there isn't the notion of control flow over time with different statements--there is only a single expression that evaluates to a value. Evaluating the expression may involve partially evaluating nested expressions in a certain order but the end result and the expression itself represented computation in some sort.

So now my question is this:

(Through the use of lambda calculus) is it possible to write any program (i.e. Turing complete) using only one expression with no more than one statement? Is lambda calculus set up in a way that removes or hides the notion of control flow?
I think that your general question is pretty broad, because you'd need to specify what you consider a program (e.g. is the identity function that outputs the input it receives a program?).

It is possible to have some sort of control flow in lambda expressions (there are only expressions in lambda calculus); consider e.g. Church Booleans that enable conditionals.

If you would like to see examples of lambda calculus expressions put to work like regular programs, I can recommend John Tromp's binary lambda calculus. I have been able to evaluate some of his programs using plain β-reductions (examples), so I can say it's definitely possible to write programs even with pure lambda calculus.