Speaker
Ruy Exel Tight representations of inverse semigroupsTitle Many of the most important
C*-algebras studied over the last thirty years contain a
generating set consisting of partial isometries. In the
vast majority of cases, the multiplicative semigroup
generated by such partial isometries is an inverse
semigroup. Cuntz-Krieger algebras, graph C*-algebras,
higher-rank graph C*-algebras, C*-algebras of ample etale
groupoids and certain C*-algebras associated with tilings
are all instances of this general phenomenon. It is
therefore sensible to ask to what extent these algebras
may be reconstructed exclusively from this inverse
semigroup. As it turns out, the classical notion of the
C*-algebra of an inverse semigroup, such as the one
presented in Paterson's book, is not suitable to achieve
this reconstruction. In this talk, we will describe an
alternative construction of a C*-algebra from an inverse
semigroup based on the notion ofAbstract tight representations
which may be used to recover the algebras listed above
from their generating inverse semigroup.Slides This was a blackboard talk. |

Speaker Robert GrayTitle Finite
Gröbner-Shirshov bases for plactic algebras and
biautomatic structures for plactic monoids The
plactic monoid, having origins in work of Schensted and
Knuth, is concerned with certain combinatorial problems
and operations on Young tableaux. It was later studied
in depth by Lascoux and Schützenberger and has
since become an important tool in several aspects of
representation theory and algebraic combinatorics.
Various aspects of the corresponding semigroup algebras,
the plactic algebras, have been investigated, for
instance by Cedo & Okniński, and Lascoux &
Schützenberger. In this talk, I will discuss a
result in which we show that every plactic algebra of
finite rank admits a finite Gröbner-Shirshov basis.
The result is proved by using the combinatorial
properties of Young tableaux to construct a finite
complete rewriting system for the corresponding plactic
monoid. Also, answering a question of Efim Zelmanov, I
will explain how this rewriting system and other
techniques may be applied to show that plactic monoids
of finite rank are biautomatic. This is joint work with
A. J. Cain and A. Malheiro.Abstract Slides |

Speaker
Peter Higgins
Burrows-Wheeler transformations and de Bruijn sequencesTitle Abstract The Burrows-Wheeler (BW) transform is a
very effective method of lossless data compression of a
string x: BW(x) is the final column of the array
consisting of all the cyclic conjugates of x arranged in
dictionary order. Surprisingly x can be recovered
from BW(x), which implicitly exploits the structure of
ordinary languages (human or machine) to generate
strings with long single-letter substrings, thereby
lending to easy data compression. Moreover the
calculation of the inverse transform can be carried out
in linear time. A de Bruijn sequence is a string of
length k to the power n that contains every substring of
length n as a cyclic factor (over an alphabet of size
k): for example 00011101 is a binary de Bruijn sequence
for strings of length 3. The class of de Bruijn
sequences is contained in the set of BW inverses of a
very simply described collection of strings. In
particular, the de Bruijn sequence that is first in the
lexicographic order among all such strings arises in a
particular way through taking the inverse
Burrows-Wheeler transform of one especially simple
string, and so it can be readily calculated in that
fashion. Slides |

Speaker
Zur IzhakianTitle Supertropical
algebra and representations
Tropical mathematics is carried out over idempotent
semirings, a weak algebraic structure that on the
one hand allows descriptions of objects having a
discrete nature, but on the other hand, its lack of
additive inverse prevents the access to basic
mathematical notions. To overcome these drawbacks,
we use a supertropical semiring - a ``cover''
semiring structure having a
distinguished ``ghost ideal'' that plays the role of
the zero element in many of the theorems. This
supertropical structure is rich enough to permit a
systematic development of tropical algebraic theory,
yielding direct analogues to many important results
and notions from classical commutative algebra.
These provide a suitable algebraic framework that
allows natural representations of matroids and
semigroups.Abstract Slides |

Speaker Johannes KellendonkTitle The local
structure of aperiodic mediaAbstract Tilings became fashionable in physics
because they can model aperiodic media such as
quasicrystals. When this became clear the question was
asked as to how the structure of the tiling determines
physical properties. Are there fundamental physical
differences between crystals and quasi-crystals? Are
there fundamental differences between quasi-crystals
based on Penrose tilings and those based on octagonal
tilings? For this to be analyzed, one has to have a
mathematical way to translate the structure of the
tiling into a structure of the Schroedinger equation. My
idea was to do that with the help of what is now called
the tiling semigroup. It is not the only way to do it,
and the consequences in terms of real physical
properties whose nature depends substantially on what
type of tiling is used are less spectacular than one had
hoped for, but it led to a nice mathematical theory. One
of my goals is to describe how one could in principal
measure the golden ratio, thereby distinguishing the
Penrose from the octagonal tilings. This uses, next to
tiling semigroups, a little bit of non-commutative
topology.Slides This was a
blackboard talk. |

Speaker Wolfgang KriegerTitle A
construction of subshifts and a class of semigroups
(joint work with Toshihiro Hamachi) The talk
is based on the paper by T. Hamachi and W. K., Abstract A
construction of subshifts and a class of semigroups,
arXiv 1303.4158. R-graphs and R-graph semigroups
are introduced. Subshifts with property (A) are
constructed from R-graphs. As special cases the
Markov-Dyck shifts are shown to have property (A). The
R-graph semigroups, that are associated to topologically
transitive subshifts with Property (A), are
characterized. The slides
are not available, but
the speaker recommends the arXiv
paper.Slides |

Speaker
Ganna Kudravtseva Etale
groupoids and their morphisms Title
Etale groupoids are topological objects that are
closely related to inverse semigroups and quantales
looked at as algebraic and ordered structures.
Recent results by Mark Lawson, Daniel Lenz and Pedro
Resende establish several one-to-one correspondences
between these objects and their morphisms. In
my talk, I discuss these results and put special
attention to the notion of a morphism between etale
groupoids. These can be defined in several ways
depending on the requirements to the corresponding
morphisms at the algebraic side, e.g., meet
preservation, preservation of the top element etc. I
will also discuss parallel results when etale
groupoids are replaces by sheaves and normal bands
arise as algebraic structures.Abstract Slides |

Speaker
Daniel Lenz Order
based constructions of groupoids from inverse
semigroupsTitle We discuss how the various groupoids
associated to inverse semigroups can be obtained by
simple order based constructions. This applies in
particular to the universal groupoid of an inverse
semigroup introduced by Paterson. Along the way one
obtains canonically a reduction of this
groupoid. In the case of inverse semigroups
arising from graphs and tilings respectively
this reduction is the graph groupoid introduced by
Kumjian et al. and the tiling groupoid of
Kellendonk. We discuss some topological features of
this reduction as well as the structure of its
open invariant sets. Abstract This can be used to investigate the ideal structure of an associated reduced C*-algebra. The results might also be useful for classification questions. Slides This was a
blackboard talk. |

Speaker Stuart Margolis Representation
theory of finite semigroups and combinatorial
applicationsTitle Whereas
the representation theory of finite groups has
played a central role in that theory and its
applications for more than a century, the same
cannot be said for the representation theory of
finite semigroups. While the basic parts of the
representation theory of finite semigroups wereAbstract developed in the 1950s by Clifford, Munn and Ponizovksy, there were no ready-made applications of the theory either internal to semigroup theory or in applications of that theory to other parts of mathematics and science. Thus, in a paper in 1971, the only application of the theory mentioned was Rhodes's use of it to compute the Krohn-Rhodes complexity of a completely regular semigroup. Over the past few years, this situation has changed completely arising from at least three sources. The first is the theory of monoids of Lie type developed by Putcha, Renner and others The second is the theory of quasihereditary and stratified algebras; finite regular semigroups have quasihereditary algebras whereas an arbitrary finite semigroup has a stratified algebra, both naturally so, via the J-order. The third is applications to algebraic combinatorics and probability theory via monoid structures on objects such as real and complex hyperplane arrangements, ordered matroids, interval greedoids and more. The purpose of this talk is to survey these latter combinatorial applications where finite semigroups and their linear representations play a central role. Slides |

Speaker
Walter MazorchukTitle Linear
representations of semigroups from 2-categoriesAbstract In this talk I will try to describe
how one can use 2-categories to construct linear
representations of semigroups. The primary examples
will be representations of the Hecke-Kiselman and
Catalan monoids, following a recent joint paper with
A.-L. Grensing.Slides |

Speaker
Jan OkninskiTitle Faithful linear representations of
algebras, semigroups and semigroup algebras Abstract Given an (infinite) semigroup S and
its semigroup algebra K[S] over a field K can we
embed S or K[S] in a matrix algebra M_{n}(L) over a
potentially very big field extension L of K for some
n? What are the advantages of having such an
embedding? We shall describe some illustrative
examples and maybe some problems. This is related to
the so called representability problem for algebras,
which is often stated in the context of noetherian
algebras and algebras satisfying a polynomial
identity. Slides |

Speaker
Pedro Resende Inverse
semigroups and groupoids via quantales Title Abstract
Several relations between inverse semigroups and
topological groupoids are known and well documented
in the literature. Many of them can be placed within
a general equivalence between inverse semigroups, or
inverse semigroup actions, and their étale
groupoids of germs, where the germs can be
described, for instance, in terms of certain kinds
of filters of the inverse semigroups. In this talk I
describe how these constructions actually have two
facets that can be dealt with in two separate
stages, provided one is ready to replace topological
spaces by locales --- of whose theory I will give an
overview. The first stage is concerned with
creating, in a point-free way, the algebraic
structure of a groupoid from that of an inverse
semigroup, whereas the second one deals with the
groupoid topology itself. It is only at the second
stage that germs appear. What this means is that,
instead of working directly with topological
groupoids, the first stage relates inverse
semigroups to localic groupoids, that is groupoids
in the category of locales, whereas the second stage
is essentially a consequence of obtaining the
spectrum of a locale according to general locale
theory: the germs are the points. This separation of
concerns yields an axiom-of-choice-free theory at
the first stage, and it allows for a clearer
understanding of the purely algebraic aspects,
namely as regards a third actor that comes into
play: the relation between inverse semigroups and
localic étale groupoids is mediated by a
class of quantales known as inverse quantal frames,
which arise as completions with respect to arbitrary
joins of the inverse semigroups and which can also
be thought of as being groupoid ``topologies''
equipped with multiplication and involution obtained
pointwise from those of the groupoids.
Alternatively, they are convolution algebras of the
groupoids, constructed with locale-valued functions
instead of field-valued ones. Moreover, this theory
can be extended to more general groupoids by
similarly generalizing the class of quantales.Slides |

Speaker Pedro
Silva Title Boolean representations of
simplicial complexes Abstract Zur Izhakian and John
Rhodes introduced a theory of representations by
boolean matrices over the superboolean ring which
can be applied to various algebraic and
combinatorial structures. In joint work with John
Rhodes, we have been exploring the immense potential
of boolean representations of abstract simplicial
complexes, providing in particular interesting
connections with lattice theory, graph theory and
finite geometries.Slides |