The smartest mathematican I know personally is Susan J. Sierra, who was a fellow math major with me at Oberlin College. After earning her PhD from the University of Michigan (Go Blue!) and some post-doc positions (ever heard of a place called "Princeton" where folks like a guy called Einstein used to be on the faculty), she is now on the faculty at the University of Edinburgh where she is working in the fields of noncommunitive algebric geometry, noncommunitive algebra, algebric geometry.

Hopefully, local fashion will not cause her to develop an affinity for tartan, however. The Carnegie Mellon Tartans are one of Oberlin's athletic rivals (and yes, they do generally crush us, as a quick Google search would make clear).

In other words, she's doing work in the areas of math that are the underpinnings of Yang-Mills theory (e.g. in the fundamental physics of the Standard Model that underlies nuclear physics which has been confirmed by every experiment in the last 40 years is a non-communitive algebra, in loop quantum gravity, in string theory, in higher dimensional physics (not just fundamental physics but also fields like condensed matter physics where higher dimensional formulations of problems can be easier to solve when transformed into algebric transformation of problems too hard to solve in their natural form), and a surprising number of practical problems that are more mundane.

A mundane example of noncommunitive geometry is the kind of math you need if you are a program like mapquest trying to determine the optimal route to get from point A to point B in a city with one way streets and rush hours. The time it takes to get from point A to point B in one direction and the shortest path between them may be different from the time it takes to get from point B to point A due to traffic loads, stop lights, one way streets and so on. While basic Newtonian mechanics assumes a communitive geometry, in general, any system with an arrow of time induced by CP violation, friction, or the second law of thermodynamics implies a noncommunitive geometry.

Noncommunitive algebra also has mundane as well as ordinary applications. For example, the mathematics behind tax planning is noncommunitive, because the tax code treats losses (i.e. negative numbers) very differently from profits (i.e. positive numbers) and is also asymmetric in time, for example, with different treatments of carryforwards of losses and carrybacks of losses. Any set of functions that operate differently forward and backward is noncommunitive. Noncommunitive algebras are also sometime called "non-Abelian", as Mr. Abel's name has become synonymous with communitive algebras (some odd associations and footnotes related to him can be found here).

At its most basic, algebric geometry is the study of equations that can translate into shapes, like the analytic geometry that you studied in pre-algebra or trig in high school, which have been widely known since Alexander the Great studied them as part of his education (although the conic sections weren't as neatly tied to equations then as they are now, something we owe to Descartes and his peers). But, geometries in more than our ordinary four dimensions are much harder to express in any way other than equations, and since fields like string theory call for more than four dimensions, one really can't make sense of any of it without a firm command of algebric geometry. It is also critical to fundamental issues in relativity and gravity such as the still unresolved question of whether it is conceptually possible for the geometric formulation of gravity in general relativity to have an equivalent formulation in the nature of a force exchanged by particles such a graviton, or whether there exists some proveable "no go theorem" that could establish that it is impossible to have a formulation of general relativity in a Minkowski space-time.

Also, if you had the impression that there are no unsolved problems remaining in mathematics, you are incorrect. Susan Colley, one of my math professors at Oberlin College, explains in a recent article just how many open questions remain in mathematics and provides some current examples such as the outstanding Millenium Prize questions (some of which, like the question in Yang-Mills theory, readers of this blog in academia are actively working in their own research to solve).

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