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The world in the head
The World in the Head collects the best of Robert Cummins' papers on mental representation and psychological explanation. Running through these papers are a pair of themes: that explaining the mind requires functional analysis, not subsumption under \"psychological laws\", and that the propositional attitudes--belief, desire, intention--and their interactions, while real, are not the key to understanding the mind at a fundamental level. Taking these ideas seriously puts considerable strain on standard conceptions of rationality and reasoning, on truth-conditional semantics, and on our interpretation of experimental evidence concerning cognitive development, learning and the evolution of mental traits and processes. The temptation to read the structure of mental states and their interactions off the structure of human language is powerful and seductive, but has created a widening gap between what most philosophers and social scientists take for granted about the mind, and the framework we need to make sense what an accelerating biology and neuroscience are telling us about brains. The challenge for the philosophy of mind is to devise a framework that accommodates these developments. This is the underlying motivation for the papers in this collection.
Book
Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements,
interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the
monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left
regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such
structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left
regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order
complexes of posets naturally associated to the left regular band.
The purpose of the present monograph is to further develop and
deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all
simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left
regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the
examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure
on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional
oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A
fairly complete picture of the representation theory for CW left regular bands is obtained.
eBook
Representation Theory of Geigle-Lenzing Complete Intersections
Weighted projective lines, introduced by Geigle and Lenzing in 1987, are important objects in representation theory. They have
tilting bundles, whose endomorphism algebras are the canonical algebras introduced by Ringel. The aim of this paper is to study their
higher dimensional analogs. First, we introduce a certain class of commutative Gorenstein rings
eBook
The Representation Theory of the Increasing Monoid
We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation
category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish
properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous
connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle
algebras.
eBook
The Irreducible Subgroups of Exceptional Algebraic Groups
This paper is a contribution to the study of the subgroup structure of exceptional algebraic groups over algebraically closed fields
of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group
A result of Liebeck and Testerman shows that each irreducible connected subgroup
eBook