**UBC Mathematics Department**

*http://www.math.ubc.ca*

## Colloquium Abstract: Dr. Thomas Leness,
Michigan State University

*The Role of Compactness in Gauge Theoretic Invariants
*

Until 1994, the only known invariants of smooth four-manifolds were the
Donaldson polynomials. The definition of these invariants relied
upon a construction called the Uhlenbeck compactification and it was
thought by some that the invariant's sensitivity to the differential
structure of the manifold depended upon this compactification.
However, work of Kronheimer-Mrowka on the structure of Donaldson
polynomials and the new Seiberg-Witten invariant have shown that
the Donaldson polynomial is really only sensitive to certain cohomology
classes, similar to the canonical class of an algebraic surface.
The moduli space of PU(2) monopoles gives a method to
prove a relation between the Donaldson and Seiberg-Witten invariant
as well as some understanding of the role the Uhlenbeck compactification
plays in Donaldson theory.

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