Tynan Kelly

Access to advanced education in the mathematical sciences has become much easier and more general over the last century. Today's graduates hoping to pursue a career in research have vastly more practical options than did their counterparts of a century ago.

Institutional support is also vast compared to a century ago. In the US alone, major research universities offering tenure to professors in the mathematical sciences have more than doubled in just the past 25 years. Conferences, summer schools, and prestigious prizes abound.

Over the decades, the number of PhDs granted in the mathematical sciences has steadily grown. In the US alone, nearly three times more PhDs in mathematics were granted between 1995-9 than between 1960-4. (In physics, the corresponding growth factor was just over two.)

The global community of researchers in the mathematical sciences has grown over the last century by more than an order of magnitude, and technologies and circumstances obviously favorable to its success have improved and spread spectacularly.

Yet it is impossible to argue that all these advantages enjoyed today by researchers in the mathematical sciences have led to equally spectacular improvements in the overall quality of their achievements. At best, it may be possible to argue that no steep decline has occurred.

In particular, looking just at mathematics, we find the following names associated with profound innovations made in the first two decades of the last century: Frobenius, Burnside, Poincaré, Hilbert, Minkowski, Hadamard, Cartan, Takagi, Ramanujan, Weyl, Hecke, Noether, Banach.

It is impossible to argue that the first two decades of mathematical research in this century have produced any innovations as profound as group representation theory, functional analysis, dynamical systems theory, the geometry of fiber bundles, or class field theory.

Great researchers in mathematics are certainly not ten times more numerous today than they were a century ago; indeed, it takes some audacity to argue that we have as many. (It's far from clear, for example, whether anyone alive today can bear close comparison with Poincaré.)