In what fields would you like to see applications of mathematics? There are very few disciplines which mathematics has not penetrated. As a pupil finds such gem in the calculus problem of theory of rumors, he wonders if such field has application in vaudevillian arts such as tarot reading or theory of juggling. As one digs further one may finds terms as oracle, Napier's bone, Polya urn model, Witch of Agnesi, Faddeev-Popov ghost and Soul theorem which of course has nothing to do with occult in general. Although the field has been applied to understanding phenomenon which at first seemed improbable to understand gambling or betting on race horses, one is equally vexed to find there is no mathematics of vexillology. One fondly remembers the scene in A Beautiful Mind where a young John Nash was studying flight of pigeons for game theory. 
Since a mathematician is prone to see patterns between seemingly disconnected fields it brings me to question what further areas can be bridged by cross-pollination?
Some fuel for thought of fields where the subject can be applied:


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*Mathematics of Vexillology and Heraldry 
How can a formal theory be constructed that will aid to generate random,  new geometrical patterns (designs) from the constraint of 63 representing symbols?

*Mathematics of Choreography
Can one apply braid theory to choreograph  puppetry, fencing, kali or dancing patterns from software?

*Surveillance Theory 
How can we generalize the concept of Panopticon or sousveillance?

*Mathematics of Locksmithing
Although theory of locksmithing exist, can one apply algebraic topology to generalize the concept of locks using knot theory? 

*Mathematics of Smuggling or "Hustling" 
In youtube one can find videos of the show Hustle which conceives elaborate means of schemes and hustle. How can the process of transporting a concealed object from an agent to agent around obstacles be modeled or formalized using graph theory?

*"Theory of Plots and Sub-Plots" 
Aristotle's Three Act structure is still used in screenwriting. However, can one create templates for story? Do patterns exist between story plots or heist? Can it be modeled mathematically? or even  plots based on data-mining of musical scores.

*Formalizing punchlines
In pursuit of Loebner prize now bots are eavesdropping on twitters; computers can even detect sarcasm. Similarly is it possible to create a program generating witty, one-liner advertising slogans from data-mining from twitters, proverbs, foreign expressions, existing ad slogans, etcetera?

*Mathematics of Culinary Theory & Recipes

*Can swarm intelligence be used to model NFL or NHL offense patterns?
Of course, a general question would be to ask: Can creativity be automated? But, one can, break down the notion of creativity to further sub-divisions.
What fields would you like to see applications of mathematics?
I sincerely believe this endeavor will encourage non-mathematicians chancing upon this site shed their inputs for a collaboration in a marriage of disparate fields.
 A: Mathematics of Shamanism? Many cultures still use shamans and witch doctors for healing. I would love to understand how efficient the interventions of these folks are compared to placebo effects. If their interventions are better than placebo (especially I suspect for mental condtions, which in many societies are seen as a paranormal visitation), what can modern medicine learn from them on the art of putting up a show and leveraging a belief system for healing?
(Sorry for my spelling, english is not native to me)
A: Disclaimer: Although OP relates to applications of mathematics that one would like to see, the answer provided below discusses about interdisciplinary studies in mathematics that are already being researched.
The key terms here are "interdisciplinary studies in mathematics" and by no means this is a new concept and enough research has been devoted in this field.
In addition to Journal of Interdisciplinary Mathematics published in India (for instance, Volume 1, 1998), there do exist an area of research called Digital humanities (whose companion volume can be accessed here) and an example of further subdivision of the field would be usage in stylometry. Some examples include: (i) A Naive Bayes classifier for Shakespeare's second-person pronoun, (ii) The Potosi principle: religious prosociality fosters self-organization of larger communities under extreme natural and economic conditions or (iii) The liberty of invention: alchemical discourse and information technology standardization [here].
In 2007, eight symposia were held in San Francisco where some of the topics included (AMS Conference notification):

• The Science and Modeling of Hurricanes (organized by Clint Dawson)
   • New Vistas in the Mathematics of Ecology and Evolution
  (organized by Simon Levin)  •  Controversies in Forest Fire
  Suppression and Management (organized by John Braun)  •
  Blockbuster Science: Math & Science Behind Movies & Entertainment
  (organized by Tony Chan) 

Additionally, Arizona State University held 3rd Annual Intermountain/Southwest Conference on Industrial and Interdisciplinary Mathematics topics of which are here. 
Although this borders more on artificial intelligence and robotics, Swarm intelligence is another umbrella term that takes muse from nature such as river formation or brooding behavior of cuckoo species, or flashing behavior of fireflies. (also stigmergy) Searching under mathematical optimization problems or differential evolution yields eagle strategy analyzing foraging behavior or jeep problem as example of applied mathematics. For another paper on interdisciplinary field of mathematical ecology this AMS paper by Hayward and Henson can be consulted.
Furthermore, Mathematical Connections explore "the interplay between mathematics and the humanities" founded in 1992 by Steve Whittle and Keith Luoma of Augusta State University and this link contains some interesting titles.
And finally for popular account: Consilence mentions Condorcet's application of mathematics in social sciences, Diaconis and Graham also has this book on mathematics of magic that talks about probability and I Ching and about fractals in African settlements.
The website Numb3rs by Topic has wealth of materials that discusses the math behind many interesting applications such as:


*

*counterfeiting

*horse race betting

*accidents

*Poisson distribution observed for soldiers disabled by horse-kicks in the Prussian Cavalry
etc

A: I would like to see application of math using Minkowski geometry and projective geometry in camps and after school programs for high school students.
A: Psychology. We finally need good mathematical constructions to connect thinking to actual neural nets.
A: Neither looking for nor expecting the bounty, but have a look at DARPA 23 math challenges. 
Overall they are broader in scope than the 7 Clay Millennium Prize Problems... for example, DARPA challenge #1 relates to modeling the brain, as per Xnyyrznaa's comment. 
My favorite is #7: Occam's razor in many dimensions, regarding which I'll post Q's soon.
A: I started my college education wanting to be a theoretical mathematician.  My interests were abstract algebra, complex analysis and to some extent number theory.  Number theory interested me because my father had a pure math bent and contributed to that field.  But he went into nuclear engineering and wound up becoming famous for his work on reactors.  Mathematics served him well because of the discipline it teaches you in solving problems.  But the aspects of mathematics that was important in his work was nonlinear differential equations.  My path to pure math got short curcuited by Vietnam which prevented me from going to graduate school to study math.  I wound up as a civilian working for the army and discovered the importance of operations research and statistics in every aspect of my work.  After the war I went on to get advanced degrees in those fields.  Some may think of them as area in applied mathematics.  I won't argue with that.  But I do think that the discipline of mathematics could pay more attention to these applied areas as well as computer science.  Although I love it that technology has allowed the four color problem and Fermat's Last Theorem to be solved in my lifetime I feel that the core of applied mathematics should interact more with statistics, operations research and computer science.
When great minds from different disciplines interact great new discoveries are made.  as a side note my father did work on magic squares, the Tarry-Escott problem and Fermat's little theorem.  He did not live long enough to see the four color problem and Fermat's Last Theorem solved but I think he would have really enjoyed learning about it even more than I did.  I am sure in his school days studying number theory under Leonard Dickson at the University of Chicago he must have given a lot of thought to Fermat's Last Theorem as I think all numbrer theorist must have.
