There's a well-known way of arranging fields of study according to "purity". This xkcd cartoon shows how it's done:
It's not surprising that those who are fondest of this simplified view are physicists and mathematicians, but that's not the point I want to make. The "purity order" can be seen in several different ways. For one, it's a hierarchy of rigorousness. Mathematicians can write down unassailable proofs. Physicists have to make experiments, but hey, at least they're experimenting with the basic building blocks of everything else! Chemists don't have to deal with the vagaries of living beings; biologists are at least spared most of the wilfulness and lack of cooperation of conscious subjects, and so on.
What those lower down in this particular hierarchy can (and do) argue is that, in an important sense, they are closer to human experience, to "real life": the society we live in and interact with, our own minds, our bodies and the way they function.
But there's an interesting twist, and it's one of the reasons I'm looking forward to the HLF with its particular mix of attendees. Mathematics and computer science close the loop between the most abstract reaches of science and some of the most direct application.
Number theory, the study of integers and their properties, would seem to be one of the most basic and pure fields of mathematics, and has been called the "Queen of Mathematics". Yet it also is the basis of public key encryption which, as ever-new revelations about PRISM and similar systematic invasions of privacy show, is directly relevant to our daily lives indeed. HLF attendee Shafi Goldwasser, who will be attending HLF is a good example. She makes the connection between rigorous mathematical analysis and everyday concepts of secrecy and the security of encrypted messages.
Many more connections can be made - computers themselves are this kind of pure-applied hybrid, with the Turing machine as both an abstract tool for defining certain sorts of mathematical proof and, at the same time, close kin to the ways the processors in real computers work.
There will be plenty of mathematical purity pervading the HLF - but there will be direct, and sometimes unexpected connection to applications, as well.