The geometry referred to is not the geometry of lines and angles but
the geometry of topology, shape, and knots. For example, galaxies are not
distributed randomly in the universe, but they tend to form clusters, or
sometimes strings, or even sheets of high galaxy density. How can this be
handled statistically? The Euler characteristic (EC) of the set of high-
density regions has been used to measure the topology of such shapes; it
counts the number of connected components of the set, minus the number of
"holes," plus the number of "hollows." Despite its complex
definition, the exact expectation of the EC can be found for some simple models,
so that observed EC can be compared with expected EC to check the model. A
similar problem arises in functional magnetic resonance imaging (fMRI), where
the EC is used to detect local increases in brain activity caused by an external
stimulus. Research work extended these ideas to manifolds so that changes
in brain shape can be detected via structure masking, surface extraction,
and 3D deformation fields. Worsley also noted some curious random fields,
the excursion sets of which are strings; using the Siefert representation,
he showed that these strings can be knotted.

### Detecting changes in brain shape, scale, and connectivity
via the geometry of random fields

Three types of data are available to test for changes
in brain shape: 3D binary masks, 2D triangulated surfaces, and trivariate 3D
vector displacement data from the nonlinear deformations required
to align the structure with an atlas standard. The Euler characteristic
of the excursion set of a random field is used as a tool to test for localized
shape changes. These ideas are extended to scale space, where the scale
of the smoothing kernel is added as an extra dimension to the random
field. Extending this further still, the fields of correlations
between all pairs of voxels are examined, which can be used to assess brain
connectivity. Shape data is highly nonisotropic, that is, the effective smoothness
is not constant across the image, so the usual random field theory
does not apply. Worsley and colleagues proposed a solution that warps the
data to isotropy using local multidimensional scaling and then showed that
the subsequent corrections to the random field theory can be done without actually
doing the warping—a result guaranteed in part by the famous Nash
Embedding Theorem. This was formalized by Jonathan Taylor,
who extended Robert Adler's random field theory to arbitrary manifolds.

### Advances in random field theory

After Robert Adler's 1981 book on the geometry of random
fields, many successful applications to astrophysics
and brain mapping provoked a
flurry of new theoretical work. Worsley and colleagues traced the history
of this development over the subsequent 20 years, touching on
Robert Adler's early work on the expected EC of excursion
sets; David Siegmund's and Jiayang Sun's approach to finding
the *P*-value of the maximum using Weyl's tube formula;
Kuriki's and Takemura's link between the two; Naiman's and
Wynn's improved Bonferroni inequalities; Robert Adler's
proof that the expected EC really does approximate the
*P*-value of the maximum; Jonathan Taylor's extensions to
manifolds; the role of the Nash Embedding Theorem; and
Jonathan Taylor's remarkable and unexpected Gaussian Kinematic
Fundamental Formula for finding EC densities.

### The statistical analysis of
fMRI data

Worsley proposed a method for the
statistical analysis of fMRI data
that sought a compromise between validity,
generality, simplicity, and execution
speed. The method is based on
linear models with local AR(p)
errors. The AR(p) model is fitted
via the Yule-Walker equations
with a simple bias correction
that is similar to the first step
in the Fisher scoring algorithm
for finding ReML estimates. The
resulting effects are then combined
across runs in the same session,
across sessions in the same subject,
and across subjects within a population
by a simple mixed effects model.
The model is fitted by ReML using
the EM algorithm after reparameterization
to reduce bias, at the expense
of negative variance components.
The residual degrees of freedom
are boosted using a form of pooling
by spatial smoothing. Activation
is detected using Bonferroni,
False Discovery Rate, and nonisotropic
random field methods for local
maxima and spatial extent. Worsley also examined an alternative
method based on conjunctions and used a simple method
to estimate and make inference
about the delay of the hemodynamic
response function at every voxel.
This work contributed toward the advance of the optimal design of fMRI experiments.