 
Summary: Research Summary
Daniel Allcock
July 31, 2009
I have tried to make this summary readable by any mathematician, and
the first paragraph of each section mostly readable by any scientist.
First I discuss my past research, and then starting on page 10 my
current research projects. These include discussion of my published
papers relevant to them, namely [12][15][16][18][28].
My field is group theory (the study of symmetry) and its applica
tions outside group theory. For me this has mostly meant applications
in algebraic geometry, some by myself and some jointly with several
different coauthors. Most notable is a long series of papers concerning
the spaces of deformations of various classical geometric objects like
curves in the plane, or cubic surfaces. I have also contributed to a
number of problems in topology and pure group theory, for example
hyperbolic reflection groups and the topology of their associated hy
perplane arrangements. My work in algebraic geometry and reflection
groups led to my current research project concerning the monster finite
simple group, described in section 6.
In 2006 I began moving into a new field: representation theory. In
