There are very significant changes taking place in the university sector and
in related higher education institutes in many parts of the world. In this work
we look at financial data from 2010 and 2011 from the UK higher education
sector. Situating ourselves to begin with in the context of teaching versus
research in universities, we look at the data in order to explore the new
divergence between the broad agendas of teaching and research in universities.
The innovation agenda has become at least equal to the research and teaching
objectives of universities.

We describe many vantage points on the Baire metric and its use in clustering
data, or its use in preprocessing and structuring data in order to support
search and retrieval operations. In some cases, we proceed directly to clusters
and do not directly determine the distances. We show how a hierarchical
clustering can be read directly from one pass through the data. We offer
insights also on practical implications of precision of data measurement. As a
mechanism for treating multidimensional data, including very high dimensional
data, we use random projections.

We survey agglomerative hierarchical clustering algorithms and discuss
efficient implementations that are available in R and other software
environments. We look at hierarchical self-organizing maps, and mixture models.
We review grid-based clustering, focusing on hierarchical density-based
approaches. Finally we describe a recently developed very efficient (linear
time) hierarchical clustering algorithm, which can also be viewed as a
hierarchical grid-based algorithm.

Following a review of metric, ultrametric and generalized ultrametric, we
review their application in data analysis. We show how they allow us to explore
both geometry and topology of information, starting with measured data. Some
themes are then developed based on the use of metric, ultrametric and
generalized ultrametric in logic. In particular we study approximation chains
in an ultrametric or generalized ultrametric context.

Data analysis and data mining are concerned with unsupervised pattern finding
and structure determination in data sets. "Structure" can be understood as
symmetry and a range of symmetries are expressed by hierarchy. Such symmetries
directly point to invariants, that pinpoint intrinsic properties of the data
and of the background empirical domain of interest. We review many aspects of
hierarchy here, including ultrametric topology, generalized ultrametric,
linkages with lattices and other discrete algebraic structures and with p-adic
number representations.