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Ruy Exel Title Tight representations of inverse semigroups Abstract 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 of 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 Gray Title Finite Gröbner-Shirshov bases for plactic algebras and biautomatic structures for plactic monoids Abstract 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. Slides |
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Peter Higgins Title Burrows-Wheeler transformations and de Bruijn sequences 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 |
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Speaker
Zur Izhakian
Title Supertropical algebra and representations Abstract 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. Slides |
| Speaker Johannes Kellendonk Title The local structure of aperiodic media Abstract 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 Krieger Title A construction of subshifts and a class of semigroups (joint work with Toshihiro Hamachi) Abstract The talk is based on the paper by T. Hamachi and W. K., 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. Slides The slides are not available, but the speaker recommends the arXiv paper. |
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Ganna Kudravtseva Title Etale groupoids and their morphisms Abstract 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. Slides |
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Daniel Lenz Title Order based constructions of groupoids from inverse semigroups Abstract 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. 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 Title Representation theory of finite semigroups and combinatorial applications Abstract Whereas the representation theory of fi nite 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 fi nite semigroups. While the basic parts of the representation theory of finite semigroups were 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 |
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Walter Mazorchuk Abstract 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 |
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Jan Okninski Title 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 |
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Pedro Resende Title Inverse semigroups and groupoids via quantales 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 |