Mathematics Colloquium
3:00 p.m.

Math Annex 1100

Alastair Craw

University of Utah

The McKay correspondence

By associating a graph to certain finite subgroups of SL(2,C), John McKay uncovered a beautiful link between representations of any such group and geometrical properties of a complex surface determined uniquely by the group. Years later, several string theorists conjectured that a similar statement should hold for finite subgroups of SL(3,C). However, the suggestion that a complex manifold of dimension three was uniquely determined by a finite subgroup of SL(3,C) flatly contradicted the ideology that had arisen throughout the 1980's that led to Mori's Fields Medal in 1990. Nevertheless, the string theorists were right!

I plan to review McKay's original observation before describing why the physicists' conjecture was a surprise to the algebro-geometric community. The main goal of the talk is to explain why the physicists were indeed correct and to mention very recent joint work with Akira Ishii which sheds more light on this question from the algebro-geometric point of view.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).

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