Mathematical Research Interests
My research interests include complex analytic geometry, the singularity theory of holomorphic maps, and connections to representation theory. I am especially interested in logarithmic vector fields, the vector fields that are tangent to a hypersurface or analytic germ; and free divisors, hypersurfaces for which the module of logarithmic vector fields is a free module. I am also interested in the teaching of mathematics, the use of computers for mathematics, and mathematical modeling.
For more information, please see my research statement.
Publications and Preprints
Below are my publications, in (roughly) reverse chronological order. Please contact me if you need access to any of these papers and cannot obtain them by other means. Similar lists are available on my CV, at the arXiv, at Google Scholar, and at ResearchGate.
- Ragnar-Olaf Buchweitz and Brian Pike.
Lifting free divisors.
Published in Proc. London Math. Soc..
(Preprint).
Suppose that \(\varphi:X\to S\) is a map and \(f=0\) defines a free divisor in \(S\). When does \(f\circ \varphi=0\) define a free divisor on \(X\)? - Brian Pike.
Additive relative invariants and
the components of a linear free divisor.
(Preprint).
The first half of this paper studies prehomogeneous vector spaces (PVS), linear representations of linear algebraic groups that have an open orbit. The relative invariants of a PVS are functions defining the hypersurface components of the complement of the open orbit, and are related to multiplicative characters of a quotient \(H\) of the group, that is, homomorphisms \(H\to (\mathbb{C}\setminus \{0\},\cdot)\). I define the notion of additive relative invariants, and prove that they are related to additive functions of \(H\), that is, homomorphisms \(H\to (\mathbb{C},+)\). In the second half of the paper, I study linear free divisors, a type of free divisor that comes from a PVS. I describe the relationship between the group's structure and the corresponding linear free divisor; for instance, I can compute the number of components of the linear free divisor from the group's structure. - Brian Pike.
On Fitting ideals of logarithmic
vector fields and Saito's criterion.
(Preprint).
This studies the ideals generated by minors of a matrix containing the coefficients of a set of logarithmic vector fields. (These are Fitting ideals of an appropriate module.) I found an upper bound for these ideals, and give a criterion for having a generating set of the logarithmic vector fields of a hypersurface, generalizing a criterion of Saito for free divisors. - James Damon and Brian Pike.
Solvable groups, free divisors and
nonisolated matrix singularities II: Vanishing topology.
Published in Geometry & Topology,
18 (2014) 911-962.
DOI 10.2140/gt.2014.18.911.
(Preprint)
This studies the topology of complex analytic sets, in particular a generalization of the classical Milnor number of isolated hypersurface and isolated complete intersection singularities. Although this singular Milnor number is defined quite generally, even for many non-isolated singularities, formulas to compute it are known in only a few cases. Here we describe an approach to find computable algebraic formulas for certain classes of singular Milnor numbers, and apply this technique to certain matrix singularities. We also explore how this technique can be used to calculate the vanishing Euler characteristic for many types of singularities, for example, Cohen-Macaulay singularities defined as \(2\times 3\) matrix singularities. The software used for some of the calculations in this paper may be found here. - James Damon and Brian Pike.
Solvable groups, free divisors and
nonisolated matrix singularities I: Towers of free divisors.
Published
in Annales de l'institut Fourier,
65 no. 3 (2015), p. 1251-1300.
DOI 10.5802/aif.2956.
(Preprint)
This work investigates certain prehomogeneous vector spaces. We show that when a PVS has the additional structure of a block representation, then various operations can be performed (e.g., quotient) that will produce new block representations, in particular, new PVSs. When a PVS is associated with a linear free divisor, then these operations often produce a PVS associated with a linear free divisor. In particular, we can build infinite towers of linear free divisors, and we give many examples of this. - Brian Pike.
Singular Milnor Numbers of Non-Isolated Matrix Singularities.
Ph.D. Thesis, University of North Carolina at Chapel Hill, 2010.
MR2782347.
My goal was to find computable formulas for the singular Milnor number. My approach was a 2-step process: find a number of (mostly linear) free divisors, and then use a relatively simple Euler characteristic argument to express the singular Milnor number in terms of previously-known formulas. Most of this work was published in the two "Solvable groups, free divisors, and nonisolated matrix singularities" papers, listed above. - James Damon and Brian Pike.
Solvable group representations and
free divisors whose complements are \(K(\pi,1)\)'s.
Published in
Topology and its Applications, 159 (2012), no. 2, 437-449.
DOI 10.1016/j.topol.2011.09.018.
(Preprint)
MR2868903.
This describes a number of families of (linear) free divisors whose complements are Eilenberg-MacLane \(K(\pi,1)\) topological spaces. - Pike, David A., Lígia Pizzatto, Brian A. Pike, and Richard Shine.
Estimating survival rates of uncatchable animals: the myth of high juvenile mortality in reptiles.
Published in
Ecology, 89 (2008), p. 607-611.
DOI 10.1890/06-2162.1.
Many juvenile animals are hard to catch and study. The prevailing belief is that reptiles have high juvenile mortality. Here we develop a mathematical model that uses properties of the adult population to estimate the juvenile mortality rate in a stable population. We apply the model to many species of reptiles.
The LaTeX source files for many of my papers include proofs, calculations, or computer programs, all carefully written up and then placed in comment environments that do not impact the resulting PDF file. If this is useful to you, then you can download the LaTeX source from the arXiv.
Talks
You probably want the Notes, not the Slides.
- The number of irreducible components of a linear free divisor, Joint Mathematics Meeting 2014, Baltimore, MD, January 15, 2014: Slides, Notes.
- Block representations and their properties, Workshop on Free Divisors, University of Warwick, U.K., May 31, 2011: Slides.
- Linear free divisors arising from representations of solvable groups, 11th International Workshop on Real and Complex Singularities, Sao Carlos, SP, Brazil, July 27, 2010: Slides, Notes.