Frank Quinn (mathematician)

Frank Quinn
Born1946 (age 79–80)
EducationPrinceton University
Known forQuinn theorem
Scientific career
FieldsMathematics
WorkplacesVirginia Tech
William Browder
Michel Kervaire
Websitehttps://personal.math.vt.edu/fquinn/

Frank Stringfellow Quinn, III (born 1946) is an American mathematician and professor of mathematics at Virginia Polytechnic Institute and State University, specializing in geometric topology.

Contributions

He contributed to the mathematical field of 4-manifolds, including a proof of the 4-dimensional annulus theorem. In surgery theory, he made several important contributions: the invention of the assembly map, that enables a functorial description of surgery in the topological category, with his thesis advisor, William Browder, the development of an early surgery theory for stratified spaces, and perhaps most importantly, he pioneered the use of controlled methods in geometric topology and in algebra. Among his important applications of "control" are his aforementioned proof of the 4-dimensional annulus theorem, leading to the Quinn theorem, his development of a flexible category of stratified spaces, and, in combination with work of Robert D. Edwards, a useful characterization of high-dimensional manifolds among homology manifolds.

In addition to his work in mathematical research, he has written articles on the nature and history of mathematics and on issues of mathematical education.

Quinn theorem

In differential topology, the Quinn theorem is a result about topological 4-manifolds. It shows the importance of the topological property of compactness for smoothability, hence if there exists a compatible smooth structure. Quinn proved it in 1984.

Formulation

Every non-compact topological 4-manifold is smoothable. A counterexample is the E8 manifold, which is a compact and non-smoothable topological 4-manifold. A more general formulation is: A topological 4-manifold has a smooth structure in the complement of any closed set with at least one point in each compact component.

Awards and honors

In 2012 he became a fellow of the American Mathematical Society.

Selected publications

References

  1. ↑ Freed, Daniel; Uhlenbeck, Karen (1984). Instantons and Four-Manifolds. Mathematical Sciences Research Institute Publications. Vol. 1. Springer. Theorem 1.4 on p. 25. doi:10.1007/978-1-4613-9703-8. ISBN 978-1-4613-9705-2.
  2. ↑ Freedman, Michael; Quinn, Frank (1990). "8.2 Smoothing open 4-manifolds". Topology of 4-Manifolds. p. 116. ISBN 9780691632346.
  3. ↑ Scorpan, Alexandru (2005). The Wild World of 4-Manifolds. Mathematical Sciences Research Institute Publications. Vol. 1. American Mathematical Society. p. 219. ISBN 978-1-4704-6861-3.
  4. ↑ List of Fellows of the American Mathematical Society, retrieved 2013-05-27.