I have three answers. The first two involve mathematics with an applied flavor, with strong connections to mathematics of a purer flavor. The last one is purer in origin, but full of potential for applications.

**$\color{red}{\large\text{I)}}$ $\color{blue}{\large\text{Image Analysis}}$**

This is clearly an applied area. But it has strong connections to purer areas like harmonic analysis, PDE, geometric measure theory, and variational analysis.

The mathematical branch of image analysis heated up a great deal in the 1990's as a cumulative result of S. Geman and D. Geman (1984). *Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images* (Google citations = 13672), Mumford and Shah (1989) *Optimal approximations by piecewise smooth functions and associated variational problems* (Google Citations = 3140), and Rudin, Osher and Fatemi (1992) *Nonlinear total variation based noise removal algorithms* (Google citations = 5079). Innovations in applied harmonic analysis - wavelets! - also had a large impact.

Interesting mathematics could be applied to problems that could be seen, literally! This inspired many applied and (pure) mathematicians to explore and contribute. This was strengthened by the formation of the SIAM activity group in about 2001, as well as by the fact that it had strong, influential participants like Guillermo Saprio, Andrea Bertozzi, Don Geman, Stan Osher, Tony Chan, David Mumford ... to name only a very few. Another reason image analysis became interesting to mathematically serious folk (as well as many dabblers) was that the ideas went both ways -- cool mathematics could be applied, but also, applications generated exciting, new mathematical problems.

$\color{blue}{\text{Examples}}$

*The Mumford-Shah functional*: This variational functional introduced by David Mumford and Jayant Shah to solve segmentation problems became an object of study attracting lots of intense scrutiny from the likes of E. De Giorgi, L. Ambrosio, G. David and others. And as fas as I know, the structure theory is still not complete.

*ROF functional -- TV Denoising*: This functional and it's variants generated a huge amount of interest. In fact that interest has not died off, especially if one looks at the endless variations that have been generated. Interesting algorithms as well as purer investigations using the tools of geometric measure theory have generated new ideas, even in geometric measure theory. Example: Allard's 2007 paper, *Total variation regularization for image denoising, I. Geometric theory*, uses geometric measure theory tools to definitively expose the nature of TV regularized image functional minimizers.

*Geman and Geman*: as is clear from the Google citations, it has had an enormous influnece in applications. I know the least about this subject, so I am not aware of the details of its impact on mathematics.

The area is stronger than ever and is characterized by a constant influx of fresh ideas, some of which generate very interesting and rich innovations in mathematics. For example, the CS topic brought up by Yuichiro has a big intersection with mathematical image analysis.

**$\color{blue}{\text{Dicussion:}}$** Is this a grand challenge? I would argue that it is, but it is much more of a grass roots effort, not dominated by one personality but rather driven by a large number of ingenious people and real world problems. So it is different than the Grothendieck or Lurie or Thurston programs. It is more chaotic, more accessible, yet rich with motivations and inspirations that lead very deeply as well. It feels to me like something at the intersection of mathematics and physics.

**$\color{red}{\large\text{II)}}$ $\color{blue}{\large\text{Mathematics for and from the Data Deluge}}$**

It is not news that massive overloads of data are being generated, nor is it a new idea that old analysis tools are not enough. Those who know something of both the current data challenges and available mathematical technology realize that:

- Those mathematical tools are largely unexplored for their potential to data, and
- data problems are powerful sources of new ideas in those (purer) mathematical areas.

This is definitely another grand challenge which in fact subsumes the previous grand challenge of mathematical image analysis. It is of course driven by real world applications, but this in no way lessens the mathematical challenges. But it does broaden them tremendously.

$\color{blue}{\text{What is the nature of the mathematics involved in this challenge?}}$ It is very wide ranging, from geometric measure theory, harmonic analysis and PDE to graph theory, probability and statistics. Real problems are agnostic as to where insights might come from!

$\color{blue}{\text{What are the big questions?}}$ How do we extract information from very high dimensional data? How do we characterize streaming data on the fly? How do we find the proverbial needle in the haystack? etc. etc. etc.

How does this translate into mathematical programs of research? In tremendously varied ways. One has to look at research in mathematics, electrical engineering and computer science (at least) to get a grasp on the large scale of the intellectual energy devoted to these problems.

**$\color{red}{\large\text{III)}}$ $\color{blue}{\large\text{Analysis in Metric Spaces}}$**

I am a neophyte here, but this area is both rather hot and very intriguing. *Currents in metric spaces* by L Ambrosio, B Kirchheim (2000), *Differentiability of Lipschitz functions on metric measure spaces* by J Cheeger (1999), and monographs like Heinonen's *Lectures on Analysis in Metric Spaces* (2001) as well as various papers on analysis in sub-Riemannian spaces are examples and starting points for exploration. (The Helsinki school in analysis seems to me one major driving force here.)

There appears to me to be huge opportunities for progress here. Lots of exciting questions!

I also believe that the potential of this area for use in understanding and modeling data in metric spaces is just beginning to be realized. Data often comes with some notion of distance, but no natural embedding in a vector space. As the numbers of mathematicians working simultaneously in both pure and applied modes grow, I believe areas like analysis in metric spaces will become exploited for their power to illuminate applied problems.

Metathread created tea.mathoverflow.net/discussion/1505 Please post further contributions on the appropraiteness of the question their; for those not yet meta active: there is an extra sign-up for meta (do not try to use your MO login) but it is simple and instant (despite the wording 'apply for membership') $\endgroup$ – user9072 Dec 31 '12 at 14:1610more comments