Research and Publications
Michael Bush
mbush at smith.edu
Summary of Research Interests
I'm interested in various questions concerning the structure of Galois groups of maximal p-extensions with restricted ramification (in particular the situtation where p itself does not ramify) and the associated towers of fields. These pro-p groups are examples of arithmetic fundamental groups and are central objects that appear throughout number theory and arithmetic geometry.
As a graduate student my work centered around maximal unramified
p-extensions with p = 2 or 3 and quadratic base field. Recently I have carried some of the associated
group theoretical investigations further and have discovered several new infinite families of Schur
σ-groups. I'm also interested in mild pro-p groups (a notion due to John Labute). These groups have
many nice properties and can be shown to arise as the Galois groups of certain maximal tamely ramified
p-extensions.
Other things that I'm pursuing (or plan to pursue) in the near future
include:
- Computing the cohomological dimension (or finding bounds) when an extension is not mild.
- Classifying the 2-generated Schur σ-groups of small prime power order.
- Surveying the p-class towers of imaginary quadratic fields with small discriminant.
- Developing new techniques for establishing whether or not a field has finite or infinite tower and resolving whether or not there is a bound on the possible lengths of finite towers.
- Various questions related to group actions and certain types of orbit computations.
Publications
In Preparation:
- Schur σ-groups of small prime power order.
- Different Partners, Different Places: Mathematics applied to the construction of four-couple folk dances, joint with Gary Roodman.
Submitted:
- Heuristics for p-class towers of imaginary quadratic
fields, (pdf)
joint with Nigel Boston and Farshid Hajir.
Data files: README, data_partial.txt.gz, data_complete.txt.gz.
Published:
- Mild 2-relator pro-p-groups, (pdf)
M. R. Bush, J. Gärtner, J. Labute, D. Vogel,
New York J. Math., 17 (2011), 281--294.
Published version is freely available online from the journal.
- An irreducibility lemma, (pdf)
M. R. Bush, F. Hajir,
J. Ramanujan Math. Soc., 23, No. 1 (2008), 1--9.
- Mild pro-p Galois groups with 4 generators, (pdf)
M. R. Bush, J. Labute,
J. Algebra, 308 (2007), 828--839.
- Maximal unramified 3-extensions
of imaginary quadratic fields and SL_2(Z_3), (pdf)
L. Bartholdi, M. R. Bush,
J. Number Theory, 124 (2007), 159--166.
- Maximal unramified 3-extensions
of imaginary quadratic fields and SL_2(Z_3), extended abstract of talk (pdf)
M. R. Bush,
Oberwolfach Reports for 2006. - Computation of the Galois groups associated to the
2-class towers of some quadratic fields, (pdf)
M. R. Bush,
J. Number Theory, 100 (2003), 313--325.
- Computing left Kan extensions,
M. R. Bush, M. Leeming, R. F. C. Walters,
J. Symbolic Comput. 35 (2003), 107--126.
Available online from JSC. - Integral equation approximation for inhomogeneous fluids:
functional optimization,
M. R. Bush, M. Booth, A. D. J. Haymet, A. G. Schlijper,
Molecular Physics, Vol. 95, No. 3 (1998), 601--619.
Theses
- p-class towers of imaginary quadratic fields,
Ph.D. Dissertation, University of Illinois at Urbana-Champaign (May 2004). - The Todd-Coxeter procedure and its generalisations,
Honours Thesis, University of Sydney (November 1996).
Recent Talks
- Heuristics for p-class towers of imaginary quadratic fields, (slides)
talk given at the Maine-Quebec Number Theory Conference, October 2011.
- Symmetry : A mathematician's perspective, (slides)
10 minute talk for a general audience given in November 2010 as part of Smith's "Science at the Center" lunchtime talk series.
- De Bruijn sequences and linear recurrences, (slides)
talk geared towards high school students given at HCSSiM in July 2009.
Other notes and materials
-
MathSciNet reviews that I've written for various papers.
