Is any of today's applied mathematics research in academia still done primarily on pencil and paper? Does there exist applied mathematics research that involves primarily pencil-to-paper work, without the use of computers?  Or, are computers quickly replacing everything that an applied mathematician can once do by hand / traditionally does by hand? 

Anything in academia in applied math research that computers can't do?  

For instance, do PDE theorists use computers or strictly pencil and paper / chalk and chalkboard -- and leave the numerics of PDEs to numerical analysts instead?  Is there such a distinction, or is there instead always some sort of collaboration between the two types of applied mathematicians?

Another example that I can think of is the work of probability theorists.

Any insight to my very naive question would be greatly appreciated :)

Thanks,
I am an applied mathematician and I can't work without paper and pencil and in most cases I can't get things done without a computer. 

My computer can't tell me in advance if my algorithm will converge, at what rate and what the error will be. My computer can't figure out how to discretize some new problem (unless I tell him how). My computer (up to now) never had a great idea. 

On the other hand, I can't invert matrices with pen and paper and I even can multiply a large matrix like that. In fact I can't even store most data I use on paper... (and also access rates for data on paper are pretty slow). 

tl;dr I basically always use paper, pen and computer.