We’ve discussed a few modern areas of research in mathematics: the search for a combinatorial proof of the density Hales-Jewett theorem lead by Timothy Gowers and some of the open questions about the number 
A few days ago President Obama gave a speech at the National Academy of Science in which he gave a good answer to this question:
As Vannevar Bush, who served as scientific advisor to President Franklin Roosevelt, famously said: “Basic scientific research is scientific capital.”
The fact is, an investigation into a particular physical, chemical, or biological process might not pay off for a year, or a decade, or at all. And when it does, the rewards are often broadly shared, enjoyed by those who bore its costs but also by those who did not.
That’s why the private sector under-invests in basic science – and why the public sector must invest in this kind of research. Because while the risks may be large, so are the rewards for our economy and our society.
No one can predict what new applications will be born of basic research: new treatments in our hospitals; new sources of efficient energy; new building materials; new kinds of crops more resistant to heat and drought.
It was basic research in the photoelectric effect that would one day lead to solar panels. It was basic research in physics that would eventually produce the CAT scan. The calculations of today’s GPS satellites are based on the equations that Einstein put to paper more than a century ago.
– President Obama
We should point out that President Obama did make one mistake. He forgot to mention that Einstein’s theory of relativity (on which GPS technology depends) itself depends on the work of mathematicians which came even earlier. In fact, in a recent article by Alicia Dickenstein in the Bulletin of the American Mathematical Society talks about how in the first page of Einstein’s handwritten notes for his paper on general relativity he writes about how importantly his work depends on the mathematics done by pure mathematicians (who had no idea it was useful in the “real world”!). Dr. Dickenstein tells the story like this:
These are Albert Einstein’s words on the first page of his most important paper on the theory of relativity:
“The theory which is presented in the following pages conceivably constitutes the farthest-reaching generalization of a theory which, today, is generally called the “theory of relativity”; I will call the latter one—in order to distinguish it from the first named—the “special theory of relativity,” which I assume to be known. The generalization of the theory of relativity has been facilitated considerably by Minkowski, a mathematician who was the first one to recognize the formal equivalence of space coordinates and the time coordinate, and utililzed this in the construction of the theory. The mathematical tools that are necessary for general relativity were readily available in the “absolute differential calculus,” which is based upon the research on non-Euclidean manifolds by Gauss, Riemann, and Christoffel, and which has been systematized by Ricci and Levi-Civita and has already been applied to problems of theoretical physics. In section B of the present paper I developed all the necessary mathematical tools—which cannot be assumed to be known to every physicist—and I tried to do it in as simple and transparent a manner as possible, so that a special study of the mathematical literature is not required for the understanding of the present paper. Finally, I want to acknowledge gratefully my friend, the mathematician Grossmann, whose help not only saved me the effort of studying the pertinent mathematical literature, but who also helped me in my search for the field equations of gravitation.”
So, indeed, he was not only paying homage to the work of the differential geometers who had built the geometry theories he used as the basic material for his general physical theory, but he also acknowledged H. Minkowski’s idea of a four dimensional “world”, with space and time coordinates. In fact, Einstein is even more clear in his recognition of the work of Gauss, Riemann, Levi-Civita and Christoffel in [7], where one could, for instance, read, “Thus it is that mathematicians long ago solved the formal problems to which we are led by the general postulate of relativity.”
