Frank Morgan

When friends hear that I’m on a math visit to Pakistan, they often say something about safety. Actually, despite heightened security, I’ve felt completely safe and at home here. The culminating experience was setting up my computer well before talks at QAU, NUST, and GCU* and finding I could leave it unattended, rarely the case in the US or anywhere else.

Incidentally, as of 2008-09, 46% of undergraduates in Pakistan are women, up from 37% in 2001-02. During this period total enrollment has more than doubled. Similarly 31% of PhD students are women, up from 20%.

*See the article by Loring Tu on the GCU “Abdus Salam School of Mathematical Sciences in Pakistan” in the August 2011 Notices of the American Mathematical Society.

The most important lemma of regularity for an m-dimensional area-minimizing surface S in Rⁿ is the monotonicity of the mass ratio about a point p of S, i.e., the ratio of the area inside an n-ball about p of radius r to the volume of an m-ball of radius r [FF, Thm. 9.13 and proof], [M, 9.3]. Here in Lahore, Pakistan, at the Abdus Salam School of Mathematical Sciences, Rein Zeinstra told me that such monotonicity generalizes a similar result by Lelong [L] for complex analytic varieties (which are area minimizing). Similarly, the compactness theorem of geometric measure theory generalizes a later such result for complex analytic varieties. See a Math Review by Stoll.

[FF] Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. 72 (1960), 458-520.

[L] Pierre Lelong, Propriétés métriques des variétés analytiques complexes définies par une équation, Ann. Sci. École Norm. Sup. 67 (1950), 393–419.

[M] Frank Morgan, Geometric Measure Theory: a Beginner’s Guide, Academic Press, 4th ed., 2009.