Department of Mathematics
Colloquium
by
Prof. Shreeram Abhyankar

During my Ph.D. work, my guru Zariski advised me to use Chevalley's local rings to algebracize Jung's surface desingularization of 1908 for carrying it over from the complex domain to the case of positive characteristic. In my 1955 paper [A01], I concluded that this cannot be done because in that case the algebraic local fundamental group above a normal crossing of the branch locus need not even be solvable. In my 1957 paper [A02], by taking a section of the unsolvable surface covering, I was led to a conjecture about the structure of the algegraic fundamental group of an affine curve. After some initial work by myself, Nori and Serre, this conjecture was settled affirmatively by Raynaud [Ray] and Harbater [Har] in 1994. A chatty discussion of the curve case can be found in my 1992 paper [A04]. In my 1997 paper [A05], this led me to explicitize the conjectures about higher dimensional algebraic fundamental groups which were implicit in my papers of 1955 [A01] and 1959-60 [A03].

Additional references:

[A01]	S. S. Abhyankar, On the ramification of algebraic functions, American Journal of Mathematics 77 (1955), 572-592.
[A02]	S. S. Abhyankar, Cooverings of algebraic curves, American Journal of Mathematics 79 (1957), 825-856.
[A03]	S. S. Abhyankar, Tame coverings and fundamental groups of algebraic varieties Parts I to VI, American Journal of Mathematics 81-82 (1959-60), 46-94, 120-190 + 341-388.
[A04]	S. S. Abhyankar, Galois theory on the line in nonzero characteristic, Bulletin of the American Mathematical Society 27 (1992), 68-133.
[A05]	S. S. Abhyankar, Local fundamental groups of algebraic varieties, Proceedings of the American Mathematical Society 125 (1997), 1635-1641.
[Har]	D. Harbater, Abhyankar's conjecture on Galois groups over curves, Inventiones Mathematicae 117 (1994), 1-25.
[Ray]	M. Raynaud, Revetment de la droit affine en characteristic p > 0 et conjecture d'Abhyankar, Inventiones Mathematicae 116 (1994), 425-462.