Thematic Sessions
Algebraic aspects of symbolic dynamics
Organizers: Jorge Almeida, Alfredo Costa and Herman Goulet-Ouellet
Abstract: Symbolic dynamics, which could be described as the "dynamical study of infinite words", dates back to the seminal work of Hedlund and Morse in the late 1930s. While the initial motivations came from classical dynamical systems, modern symbolic dynamics has evolved to interact with many other fields, such as combinatorics on words, automata theory, logic, numeration systems, fractal geometry, tilings and, as hinted by the title of this session, algebra.
The interplay of symbolic dynamics with algebra unfolds in a variety of ways, reflecting the overall richness of the fields. This includes the study of dynamical invariants like dimension groups, associated with the space of invariant measures, or Schützenberger groups, associated with the structure of free profinite semigroups; structural work on automorphism groups of symbolic systems, which also relate to semigroup theory via the construction of Ellis semigroups; relationships with subgroups of the free group, for instance via the theory of codes; cohomological methods linked with cocycles and coboundary, often applied to the study of continuous eigenvalues of symbolic systems; various results exploring the relationship between automatic sequences and power series over finite fields; algebraic methods used in the study of specific systems, like interval exchanges or hypercubic billiards; or the study of symbolic systems, particularly of finite type, over general classes of groups.
The aim of the thematic session is to gather researchers interested in various algebraic aspects of symbolic dynamics to share their point of view and present their latest results.
Language theory in (semi)group theory
Organizer: Carl-Fredrik Nyberg Brodda
Abstract: Language theory and semigroup theory go hand in hand, and have done so for decades. The same is true for group theory and language theory, but this interaction often has a subtly different flavour. To highlight these differences and similarities, we will go through the ways in which these interactions take place by giving one talk focussed on the first, and one talk focussed on the second. On the group side, we will focus on decision problems and subsets defined by languages inside groups, including recent work on Fatou properties. On the semigroup side, we will focus on links with the word problem, and structural properties of semigroups with strong language-theoretic properties.
Numerical Semigroups: From Algebraic Geometry to Error Correction and Beyond
Organizer: Mahir Can
Abstract: Numerical semigroups play an important role in deciphering the local properties of algebraic curves. Recent decades have witnessed significant advancements in numerous areas within the theory of algebraic curves and numerical semigroups. Geometric advancements include the development of valuation-based techniques, particularly those pertaining to tropical geometry. The algebraic realm has seen the exploration of generalized numerical semigroups and the identification of related theories, such as the submonoids of groups of integer points of algebraic groups. This thematic session aims to facilitate discussions on select recent discoveries to spark further advancements at the intersection of the aforementioned theories.
Solving Equations in Algebraic Structures
Organizers: Volker Diekert and Alexei Miasnikov
Abstract: Solving equations is a fundamental mathematical challenge. In the thematic session we deal with solving equations in groups and monoids or, more general, in rings or other structures. The focus of the workshop is on decidability, complexity, and (efficient) algorithms.
Frequently we are not only interested in solvability, but also in a "nice" description of all solutions. This has been particular successful for describing the set of all solutions of an equation over a free group using Makanin-Razborov diagrams, which led to the positive solution of the Tarski problems about the elementary theory in free groups. However, Makanin-Razborov diagrams are quite difficult to understand and their construction is extremely involved. Another possibility to describe all solution of an an equation over a free group is to use concepts of formal language theory and to describe the solution set using a finite nondeterministic automaton where the transitions are labeled by endomorphisms over a free monoid with involution. This approach has a deep connection to Lindenmayer systems, which were introduced to describe the behavior of plant cells and to model their growth processes.
The thematic session includes a lecture by Alexei Miasnikov on "Diophantine problems in groups" and by Armin Weiß on "On the epimorphism problem (and what it has to do with equations)".
Participants of the thematic session who wish to present their own research are invited to contact the organizer Volker Diekert.
Symmetries in discrete objects
Organizer: Maria Elisa Fernandes
In this session, six speakers will present recent results on the following topics, all involving the study of their symmetries: incidence geometries and configurations, Cayley maps, abstract polytopes and hypertopes, and graphs.
Combinatorial and geometric inverse monoid theory
Organizer: Robert Gray
Abstract: Combinatorial inverse monoid theory studies inverse monoids defined by presentations in generators and relations. There is a powerful range of geometric methods for studying such monoids such as the Scheiblich/Munn description of free inverse monoids and Stephen's procedure for constructing Schutzenberger graphs. These methods can be used to prove both algebraic structural results about inverse monoids, and also results about algorithmic properties such as the word problem. Of particular interest is the class of special inverse monoids which have presentations where every relation is of the form w=1. Stephen's procedure specialises to give a particularly beautiful geometric theory for this class. Further motivation for studying these inverse monoids comes from a celebrated theorem of Ivanov, Margolis and Meakin (2001) which shows that that the (still open) one-relator word problem for general monoids reduces to the word problem for certain one-relator special inverse monoids. The thematic session aims to review recent advances in this area and highlight current open problems and directions for possible future research.
Stallings graphs and the lattice of subgroups of a free group
Organizer: Enric Ventura
Abstract: The aim of this thematic session is to give an introduction to the study of the lattice of subgroups of free groups under the modern point of view of Stallings graphs. These modern graphical techniques are more friendly, intuitive and powerful than the purely combinatorial classical ones in order to understand the basic properties of the lattice of subgroups of a free group, and provide easy, fast and intuitive algorithms to solve many computational problems about it: membership to a subgroup, computation of rank and basis, deciding finite index, computing intersections, etc.
During the first hour I'll present a quick introduction to these techniques, avoiding proofs and technical details, but including the basic definitions, concepts, and properties of Stallings graphs and foldings, and their applications to solving the membership problem and the intersection problem in free groups, among other issues (Hall’s theorem, residual finiteness, Howson’s property, Hanna Neuman inequality, malnormality, etc).
During the second hour, I'll concentrate on two more advanced applications of these techniques: (1) the study of algebraic and free extensions between subgroups of a free group, and Takahasi’s theorem; and (2) the study of the relative order of elements and the spectrum of subgroups of a free group.
Identities and quasi-identities: finite axiomatization and complexity issues
Organizer: Mikhail Volkov
Abstract: Several crucial properties of algebraic structures can be expressed in the language of identities and quasi-identities. Studies of this language focus on two fundamental questions: Are (quasi-)identities of an algebraic structure $S$ finitely axiomatizable (the Finite Basis Problem)? How to check whether a given (quasi-)identity holds in $S$ (the (Quasi-)Identity Checking Problem)? The thematic session aims to review recent advances in both these problems and suggest avenues for further progress.
Augmented Intelligence Theorem Proving (AgITP)
Organizers: João Araújo and Michael Kinyon
Abstract: This thematic session has several aims: a) to provide a general overview of the power and limitations of the AgITP tools; b) to train participants on how to use them; c) to demonstrate some packages in action; d) to present real applications, results, and papers obtained with these tools.
Exercises for the session can be found here