Topology, Algebra and Categories in Logic

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Basic information

The link between logic and algebra, which goes back to the pioneering work of George Boole in the early nineteenth century, justifies the modern algebraic point of view on logical formalisms. Since the pioneering work of Marshall Stone on the topological duality of Boolean algebras in the 1930s, topological methods have acquired an increasing importance in the semantic study of logic, whether classical or not. The importance of the triangle formed by algebra, logic, and topology has been established in several logical frameworks: from modal logic to substructural logic, through intuitionist logic and topos theory. The theory of categories, which appeared in the middle of the twentieth century, provides effective tools for understanding this triangle, and for generalizing certain results to broader frameworks, for example when moving from the propositional calculus to the predicate calculus, or from the logic of provability to logic of evidence.

Given the success of these ideas and their applications in computer science, there has been a rapid development in recent decades, mainly driven by computer science communities. Applications range from the semantics of programming languages to complexity theory and system verifications. These recent developments have made it possible to build a new field of research in logic that is difficult to summarize with a single sentence or to characterize with a single paradigm. The researchers adapt and combine results and tools from various branches of mathematics and theoretical computer science, for example: universal algebra, general topology, category theory, order theory, and model theory. This field of research, which has by now acquired a profile of its own, is represented by the conference series TACL: Topology, Algebra and Categories in Logic.

A brief history of TACL

The series was founded in 2003. The impetus came from a project, centered around topological and algebraic methods in logic and supported by a bilateral US-Republic of Georgia grant, with the purpose of forging collaborations between the group of Leo Esakia in Tbilisi and the group in Las Cruces NM initiated by Mai Gehrke. The participants in Georgia were Leo Esakia (PI), Revaz Grigolia, Mamuka Jibladze, Dito Pataraia (participants), and Nick Bezhanishvili and David Gabelaia (students). The participants in the US were Mai Gehrke (PI), Guram Bezhanishvili, John Harding, and Patrick Morandi (participants), with Guram Bezhanishvili, who had been recruited a few years prior in NM, as major contributor to this project linking his group of origine and his new home.

The first meeting was held in Tbilisi, under the name International Conference on Algebraic and Topological Methods in Non-Classical Logics. In 2005, a second installment expanded the theme, paying particular attention to multivalued logics and residuated structures. This meeting took place in Barcelona, organized by the research group around Josep Maria Font and Ramon Jansana. In 2007, the third meeting was held in Oxford, organized by Mai Gehrke and Hilary Priestley. There were also satellite workshops organized by Bob Coecke, Alexander Kurz and Michael Zacharyaschev. By 2007, a cohesive community, much bigger than the original founding members, had emerged. It involved researchers from many different fields including lattices and Stone duality, modal and intuitionistic logic, universal algebra, and categorical methods in logic and computer science. And the conference series was successfully fostering meaningful interactions between these fields that had previously evolved separately. Although category theory had been a topic of the conferences since the beginning, it was agreed, at the Oxford meeting in 2007, that it should be made clearer by changing the name to Topology, Algebra, and Categories in Logic.

Also in 2007, a steering committee was formed. The purpose was to ensure the long term goals of the series while allowing the individual program committees and meetings to have a stronger emphasis on one or two of the fields feeding the TACL community. The founding steering committee members were

• Guram Bezhanishvili, New Mexico State University
• Mai Gehrke, CNRS and Université Côte d’Azur
• Rob Goldblatt, Victoria University, Wellington
• Ramon Jansana, University of Barcelona
• Hiroakira Ono, Japan Advanced Institute of Science and Technology
• Yde Venema, University of Amsterdam
• Michael Zakharyaschev, Birkbeck College, University of London

In 2011,

• Achim Jung, University of Birmingham

joined the steering committee and in 2018 Achim Jung, Hiroakira Ono, and Michael Zakharyaschev decided to step down. They have been replaced by three new members joining in the spring of 2018:

• Silvio Ghilardi, University of Milano
• Hilary Priestley, University of Oxford
• James Raftery, University of Pretoria

The biannual series continues to make strong links between fields, mainly general or pointfree topology, universal algebra and lattice theory, and category theory by focussing on shared applications in logic. It also brings researchers together from many parts of the world: Eastern and Western Europe, Africa, the Americas and the Far East, including Japan, Australia, and New Zealand. As the majority of participants come from Eastern and Western Europe, most of the conferences of the series have been held in Europe. As a consequence, a US-based sister series under the name Boolean algebras; Lattices, algebraic logic and quantum logic; universal Algebra, Set theory; and set-theoretic and point-free Topology (BLAST), was formed in 2008 and is held annually in the mountain/western/midwest region of the USA.

Based on a request from the participants of the TACL series, an associated summer school has been added since 2013. The purpose of the school is to prepare and initiate young researchers and newcomers to the interdisciplinary research featured at TACL and to foster and strengthen the sense of community among the young members of the community. The program is usually focused around 4 courses with at least one from each of topology, algebra, and categories as applied in logic. Lecturers are chosen for their leadership in research but also for their excellence in communication. An effort is made to have adequate funding available in order for potential participants with few financial means to be able to participate.

Previous instalments

• 2017: Prague (20-24 and 26-30 June 2017, 119 speakers, 150 participants), organised by Marta Bílková and Petr Cintula, homepage, CfP.

• 2015: Ischia (15-19 and 21-26 June 2015, 90 speakers, 170 participants), organised by Antonio Di Nola, Giacomo Lenzi, and Luca Spada, homepage, CfP.

• 2013: Nashville (24-27 July and 28 July-5 August 2013, 63 speakers, 89 participants), organised by Warren McGovern, Francesco Paoli, and Constantine Tsinakis, homepage, CfP.

• 2011: Marseille (26-30 July 2011, 82 speakers, 130 participants), organised by Luigi Santocanale, Yves Lafont, and Nicola Olivetti, homepage, CfP.

• 2009: Amsterdam (7-11 July 2009, 74 speakers, 120 participants), organised by Yde Venema and Alessandra Palmigiano, homepage.

• 2007: Oxford (5-9 August 2007, 70 speakers, 150 participants), organised by Mai Gehrke and Hilary Priestley, homepage, CfP.

• 2005: Barcelona (15-18 June 2005, 90 speakers and nearly 200 participants), organised by Josep Maria Font, homepage, abstracts, CfP.

• 2003: Tbilisi (7-11 July 2003, 27 speakers and about 50 participants), homepage.