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start [2025/03/12 07:35] – 139.124.146.3start [2025/04/22 14:24] (current) – 139.124.146.3
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 ==== Upcoming talks ====  ==== Upcoming talks ==== 
  
-**March 25 2025: Vuong Bui** //An explicit condition for boundedly supermultiplicative subshifts// 
- 
-We study some properties of the growth rate of  $\L(\A,\F)$, that is, the language of words over the alphabet $\A$ avoiding the set of forbidden factors $\F$. We first provide a sufficient condition on $\F$ and $\A$ for the growth of $\L(\A,\F)$ to be boundedly supermultiplicative.  That is, there exist constants $C>0$ and $\alpha\ge0$, such that for all $n$, the number of words of length $n$ in $\L(\A,\F)$ is between $\alpha^n$ and $C\alpha^n$. In some settings, our condition provides a way to compute $C$, which implies that $\alpha$, the growth rate of the language, is also computable whenever our condition holds. 
-We also apply our technique to the specific setting of power-free words where the argument can be slightly refined to provide better bounds. Finally, we apply a similar idea to $\F$-free circular words and in particular we make progress toward a conjecture of Shur about the number of square-free circular words. 
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- 
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-**April 8 2025: Hajime Kaneko** 
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-**April 22 2025: Be'eri Greenfeld** 
  
 **May 6 2025: Jarkko Peltomäki** **May 6 2025: Jarkko Peltomäki**
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 ==== Past talks 2025 ==== ==== Past talks 2025 ====
 +
 +**April 22 2025: [[https://sites.google.com/view/beeri-greenfeld|Be'eri Greenfeld]]** //On the Complexity of Infinite Words//
 +
 +
 +{{ seminar2025:20250422greenfeld.mp4 |video of the talk}}
 +
 +Given an infinite word $w$, its complexity function $p_w(n)$ counts the number of distinct subwords of length $n$ it contains. A longstanding open problem in the combinatorics of infinite words is the {\it inverse problem}: describe which functions $f: \mathbb N \to \mathbb N$ arise as complexity functions of infinite words. Such functions must be non-decreasing and, unless eventually constant, strictly increasing; they must also be submultiplicative, i.e., $f(n+m)≤f(n)f(m)$. Many interesting results, both positive and negative, have been obtained in this direction.
 +
 +We resolve this problem up to asymptotic equivalence in the sense of large-scale geometry. Specifically, given any increasing, submultiplicative function $f$, we construct an infinite recurrent word $w$ such that $c f(cn) ≤ p_w(n) ≤ d f(dn)$ for some constants $c,d>0$. For uniformly recurrent words, we obtain a weaker version allowing a linear error factor. Time permitting, we will discuss connections and applications of these results to asymptotic questions in algebra.
 +
 +Joint work with C. G. Moreira and E. Zelmanov.
 +
 +
 +**April 8 2025: [[https://trios.tsukuba.ac.jp/en/researcher/0000003624|Hajime Kaneko]]** //Analogs of Markoff and Lagrange spectra on
 +one-sided shift spaces//
 +
 +{{ seminar2025:20250408kaneko.pdf |slides}}
 +
 +{{ seminar2025:20250408kaneko.mp4 |video of the talk}}
 +
 +
 +The Lagrange spectrum is related to the rational approximations of badly
 +approximable numbers. The discrete part of the spectrum is denoted in terms
 +of Christoffel words. A multiplicative analog of the Lagrange spectrum was
 +recently investigated, which is defined by Diophantine approximations of geometric sequences and more general linear recurrences. Dubickas investigated
 +the minimal limit points of certain multiplicative Markoff-Lagrange spectra in
 +terms of symbolic dynamical systems and certain substitutions.
 +
 +In this talk, we study an analog of the Markoff-Lagrange spectrum for general
 +one-sided shift spaces. As our main results, we determine the discrete parts and
 +minimal limit points in terms of $S$-adic sequences, where $S$ is an infinite set of
 +substitutions. This is joint work with Wolfgang Steiner.
 +
 +
 +**March 25 2025: [[https://page.mi.fu-berlin.de/vbui/|Vuong Bui]]** //An explicit condition for boundedly supermultiplicative subshifts//
 +
 +
 +{{ seminar2025:20250325bui.pdf |slides}}
 +
 +{{ seminar2025:20250325bui.mp4 |video of the talk}}
 +
 +
 +
 +We study some properties of the growth rate of  $L(A,F)$, that is, the language of words over the alphabet $A$ avoiding the set of forbidden factors $F$. We first provide a sufficient condition on $F$ and $A$ for the growth of $L(A,F)$ to be boundedly supermultiplicative.  That is, there exist constants $C>0$ and $\alpha\ge 0$, such that for all $n$, the number of words of length $n$ in $L(A,F)$ is between $\alpha^n$ and $C\alpha^n$. In some settings, our condition provides a way to compute $C$, which implies that $\alpha$, the growth rate of the language, is also computable whenever our condition holds.
 +We also apply our technique to the specific setting of power-free words where the argument can be slightly refined to provide better bounds. Finally, we apply a similar idea to $F$-free circular words and in particular we make progress toward a conjecture of Shur about the number of square-free circular words.
 + 
 +
 +
 **March 11 2025:[[https://www.irif.fr/users/letouzey/index|Pierre Letouzey]]** //Generalizing some Hofstadter functions: G, H and beyond// **March 11 2025:[[https://www.irif.fr/users/letouzey/index|Pierre Letouzey]]** //Generalizing some Hofstadter functions: G, H and beyond//