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start [2026/02/28 08:14] – anna.fridstart [2026/05/13 19:31] (current) – anna.frid
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 ==== Upcoming talks ====  ==== Upcoming talks ==== 
  
 +**POSTPONED <del>May 26 2026: Reem Yassawi</del>**
  
-**March 03 2026: [[https://orcid.org/0000-0002-2499-3401|Ingrid Vukusic]]** //Balanced rectangles over Sturmian words//+**June 9 2026:  [[https://kmlinux.fjfi.cvut.cz/~balkolub/| Ľubomíra Dvořáková]]** //Attractors of sequences coding beta-integers//
  
-One of the many properties of the famous infinite Fibonacci word $01001010\cdots$ is that it is \textit{balanced}. This means that any two blocks of the same length have either the same weight or their weights are off by $1$. Results by Berth\'e and Tijdeman show that a 2-dimensional variant of the Fibonacci word cannot be balanced. However, for certain pairs $(m,n)$, the $m\times n$ rectangles are balanced, and recently all such $(m,n)$ were fully characterised. We extend this characterisation to completely general Sturmian words. The proof is based on Diophantine approximation, and I will explain why Ostrowski representations are ``everything one could wish for''.+String attractor is an intensively studied object in Combinatorics on Words. 
 +In our talk, we will recall known results and also some previously used techniques. 
 +We will then describe minimal string attractors of prefixes of simple Parry sequences. 
 +These sequences form a coding of distances between consecutive beta-integers in numeration systems with a real base beta. 
 +Simple Parry sequences have been recently studied from this point of view and (not necessarily minimal) attractors of their prefixes have been described and a conjecture that attractors of alphabet size should be sufficient was stated. We prove this conjecture. Moreover, we provide attractors of prefixes of some particular form of binary non-simple Parry sequences.
  
 +**June 23 2026: Benoit Cloitre**
 +
 +**July 7 2026: Delaram Moradi**
 +
 +==== Past talks 2026 ====
 +
 +
 +**May 12 2026: [[https://www-lmpa.univ-littoral.fr/~lvivion/|Léo Vivion]]** //New examples of words for which binomial complexities and subword complexity coincide//
 +
 +{{ seminar2026:20260512vivion.pdf |slides}}
 +
 +(The talk was not recorded at the speaker's request.)
 +
 +In 2015, Rigo and Salimov introduced a family of complexities that forms a scale between abelian complexity and factor complexity: the $k$-binomial complexities. In particular, they showed that Sturmian words satisfy the following remarkable combinatorial property: their $2$-binomial complexity is equal to their factor complexity. Since then, the only other known example satisfying this property is the Tribonacci word.
 +
 +In this talk, I will present some stability results for words whose $k$-binomial complexities coincide with their factor complexity, notably under letter deletion and a coloring operation. These stability properties allow to show that several well-known families of words also have their $2$-binomial complexity equal to their factor complexity.
 +
 +
 +**April 28 2026:  [[https://sites.google.com/ug.uchile.cl/espinoza|Bastiàn Espinoza]]** //The Thue-Morse word in base 3/2//
 +
 +{{ seminar2026:20260428espinoza.pdf |slides}}
 +
 +{{ seminar2026:20260428espinoza.mp4 |video of the talk}}
 +
 +
 +A rich family of symbolic dynamical systems of low complexity is given by automatic sequences. These sequences are obtained by feeding the base-$k$ expansions of integers, for a fixed integer base $k$, into a finite automaton.
 +In this talk, we turn to automatic sequences in rational bases, using as a guiding example a rational-base analogue of one of the most classical integer-base automatic words.
 +Namely, we consider the Thue--Morse word in base $3/2$, whose $n$-th term is given by the sum modulo 2 of the digits in the base-$3/2$ representation of $n$.
 +Our results show that, although this base-$3/2$ variant is substantially more complex than classical automatic words (for instance, it is not generated by iterating a single substitution, and its factor complexity grows superlinearly) it nevertheless retains several characteristic properties of the integer-automatic world.
 +More precisely, we prove uniform recurrence, establish the existence of letter frequencies, and show several combinatorial symmetries of its language analogous to those of the classical base-2 word.
 +Our approach relies on describing the word via the periodic iteration of two substitutions, studying the induced action of these substitutions on the $2$-adic integers, and applying Pontryagin duality on this group.
 +
 +This is joint work with Julien Cassaigne, Michel Rigo, and Manon Stipulanti.
  
-**March 17 2026: Gwenael Richomme** 
  
-**March 31 2026: Paulina Cecchi Bernales** 
  
 **April 14 2026: Idrissa Kaboré** //On modulo-recurrence and window complexity in infinite words// **April 14 2026: Idrissa Kaboré** //On modulo-recurrence and window complexity in infinite words//
 +
 +
 +{{ seminar2026:20260414kabore.pdf |slides}}
 +
 +{{ seminar2026:20260414kabore.mp4 |video of the talk}}
  
 In this talk, first, I will recall the notions of modulo-recurrent words and of window complexity. These notions are introduced in 2007. Then, I present some properties of these notions. After that, I will present the notions of uniform modulo-recurrence and of strong modulo-recurrence. These notions are defined recently in a joint work with Julien Cassaigne. Sturmian words are uniformly (resp. strongly) modulo-recurrent words. Then, I will address the window complexity of the Thue-Morse. To finish, I will present a recurrent aperiodic word with bounded window complexity. In this talk, first, I will recall the notions of modulo-recurrent words and of window complexity. These notions are introduced in 2007. Then, I present some properties of these notions. After that, I will present the notions of uniform modulo-recurrence and of strong modulo-recurrence. These notions are defined recently in a joint work with Julien Cassaigne. Sturmian words are uniformly (resp. strongly) modulo-recurrent words. Then, I will address the window complexity of the Thue-Morse. To finish, I will present a recurrent aperiodic word with bounded window complexity.
  
-**April 28 2026: [[https://kmlinux.fjfi.cvut.cz/~balkolub/| Ľubomíra Dvořáková]]** //Attractors of sequences coding beta-integers// 
  
-**May 12 2026: Léo Vivion**+**March 31 2026: [[https://sites.google.com/view/paulinacb|Paulina Cecchi Bernales]]** //Interplay between combinatorics and dynamical properties in minimal symbolic systems. 
 +//
  
-**May 26 2026: Reem Yassawi**+{{ seminar2026:20260331cecchi-bernales.pdf |slides}} 
 + 
 +{{ seminar2026:20260331cecchi-bernales.mp4 |video of the talk}} 
 + 
 +Symbolic systems, also known as subshifts, are topological dynamical systems whose phase space consists of infinite sequences of symbols, evolving under the action of the (left) shift transformation. As infinite words, the elements of a symbolic system possess combinatorial properties associated with their language, and it is natural to ask how these combinatorial properties determine or interact with the properties of the system as a dynamical system, or vice versa. In this talk, I will briefly review the basic notions of symbolic systems and some classical results which connect their combinatorial and dynamical properties, particularly for subshifts generated by substitutions. I will then present more recent results illustrating the impact of combinatorics on the spectral properties of minimal subshifts. In these results, combinatorial objects such as extension graphs or coboundaries will play an important role. This is a part of a joint work with V. Berthé and B. Espinoza. 
 + 
 + 
 +**March 17 2026: [[https://www.lirmm.fr/~richomme/CV/responsabilites.html|Gwenaël Richomme]]** //On some 2-binomial coefficients of binary words: geometrical interpretation, partitions of integers, and fair words// 
 + 
 +{{ seminar2026:20260317richomme.pdf |slides}} 
 + 
 +{{ seminar2026:20260317richomme.mp4 |video of the talk}} 
 + 
 +The binomial notation $\binom{w}{u}$ represents the number of occurrences of the word 
 +$u$ as a (scattered) subword in $w$. We first introduce and study possible uses of 
 +a geometrical interpretation of $\binom{w}{ab}$ and \binom{w}{ba} when $a$ and $b$ are distinct 
 +letters. We then study the structure of the $2$-binomial equivalence class of a 
 +binary word $w$ (two words are $2$-binomially equivalent if they have the same 
 +binomial coefficients, that is, the same numbers of occurrences, for each word 
 +of length at most $2$). Especially we explain the existence of an isomorphism 
 +between the graph of the $2$-binomial equivalence class of $w$ with respect to a 
 +particular rewriting rule and the lattice of partitions of the integer \binom{w}{ab} 
 +with \binom{w}{a} parts and greatest part bounded by \binom{w}{b}. Finally we study binary 
 +fair words, the words over $\{a, b\}$ having the same numbers of occurrences of $ab$ 
 +and $ba$ as subwords $(\binom{w}{ab} = \binom{w}{ba})$. In particular, we sketch a proof of a recent 
 +conjecture related to a special case of the least square approximation. 
 + 
 + 
 + 
 +**March 03 2026: [[https://orcid.org/0000-0002-2499-3401|Ingrid Vukusic]]** //Balanced rectangles over Sturmian words// 
 + 
 + 
 +{{ seminar2026:20260303vukusic.pdf |slides}} 
 + 
 +{{ seminar2026:20260303vukusic.mp4 |video of the talk}} 
 + 
 + 
 +One of the many properties of the famous infinite Fibonacci word $01001010\cdots$ is that it is \textit{balanced}. This means that any two blocks of the same length have either the same weight or their weights are off by $1$. Results by Berth\'e and Tijdeman show that a 2-dimensional variant of the Fibonacci word cannot be balanced. However, for certain pairs $(m,n)$, the $m\times n$ rectangles are balanced, and recently all such $(m,n)$ were fully characterised. We extend this characterisation to completely general Sturmian words. The proof is based on Diophantine approximation, and I will explain why Ostrowski representations are ``everything one could wish for''.
  
-**June 9 2026: Bastiàn Espinoza** 
  
-**July 7 2026: Delaram Moradi** 
  
-==== Past talks 2026 ==== 
 **February 17 2026: [[https://sites.google.com/view/stevenrobertson|Steven Robertson]]** //Number walls of automatic sequences// **February 17 2026: [[https://sites.google.com/view/stevenrobertson|Steven Robertson]]** //Number walls of automatic sequences//