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 ==== Upcoming talks ====  ==== Upcoming talks ==== 
  
 +**September 2 2025: Gandhar Joshi**
  
-**April 8 2025: [https://trios.tsukuba.ac.jp/en/researcher/0000003624|Hajime Kaneko]** //Analogs of Markoff and Lagrange spectra on+**September 16 2025: Kaisei Kishi** 
 + 
 +**October 28 2025: Idrissa Kaboré** 
 + 
 +**November 11 2025: Aleksi Vanhatalo** 
 + 
 +**November 25 2025: Ignacio Mollo** 
 + 
 +** December 9 2025: Florin Manea** 
 + 
 +**December 23 2025: Savinien Kreczman** 
 + 
 + 
 +==== Past talks 2025 ==== 
 + 
 +**July 15 2025: [[https://apacz.matinf.uj.edu.pl/users/1719-elzbieta-krawczyk|Elżbieta Krawczyk]]**  //Quasi-fixed points of substitutions and substitutive systems// 
 + 
 +{{ seminar2025:20250715krawczyk.pdf |slides}} 
 + 
 +{{ seminar2025:20250715krawczyk.mp4 |video of the talk}} 
 + 
 + 
 +We study automatic sequences and automatic systems (symbolic dynamical systems) generated by general constant length (nonprimitive) substitutions. While an automatic system is typically uncountable, the set of automatic sequences is countable, implying that most sequences within an automatic system are not themselves automatic. We provide a complete classification of automatic sequences that lie in a given automatic system in terms of the so-called quasi-fixed points of the substitution defining the system. Quasi-fixed points have already appeared implicitly in a few places (e.g. in the  study of substitutivity of lexicographically minimal points in substitutive systems or in the study of subsystems of substitutive systems) and they have been described in detail by Shallit and Wang and, more recently, Béal, Perrin, and Restivo . We conjecture that a similar statement holds for general nonconstant length substitutions. 
 + 
 + 
 + 
 +**June 17 2025: [[https://sites.google.com/view/noysofferaranov/bio|Noy Soffer Aranov]]** //Escape of Mass of Sequences// 
 + 
 +{{ seminar2025:20250617aranov.pdf |slides}} 
 + 
 +{{ seminar2025:20250617aranov.mp4 |video of the talk}} 
 + 
 + 
 +One way to study the distribution of nested quadratic number fields satisfying fixed arithmetic relationships is through the evolution of continued fraction expansions. In the function field setting, it was shown by de Mathan and Teullie that given a quadratic irrational $\Theta$, the degrees of the periodic part of the continued fraction of $t^n\Theta$ are unbounded. Paulin and Shapira improved this by proving that quadratic irrationals exhibit partial escape of mass. Moreover, they conjectured that they must exhibit full escape of mass. We construct counterexamples to their conjecture in every characteristic. In this talk we shall discuss the technique of proof as well as the connection between escape of mass in continued fractions, Hecke trees, and number walls. This is part of joint works with Erez Nesharim and Uri Shapira and with Steven Robertson. 
 + 
 + 
 +**June 3 2025: [[https://www.curtisbright.com/|Curtis Bright]]** //Mathematical Problems with SATisfying Solutions// 
 + 
 +Automated reasoning tools have been effectively used to solve a variety of problems in discrete mathematics.  In this talk, I will introduce satisfiability (SAT) solvers and highlight a variety of problems in discrete mathematics that have been tackled with a SAT solver.  As a case study, I will demonstrate how a SAT solver can be used to make progress on a question arising in combinatorics on words involving North-East lattice paths. 
 + 
 +{{ seminar2025:20250603bright.pdf |slides}} 
 + 
 +{{ seminar2025:20250603bright.mp4 |video of the talk}} 
 + 
 + 
 +**May 20 2025: [[https://www.researchgate.net/profile/Pranjal-Jain-10|Pranjal Jain]]** //Partial Sums of Binary Subword-Counting Sequences// 
 + 
 +{{ seminar2025:20250520jain.pdf |slides}} 
 + 
 +{{ seminar2025:20250520jain.mp4 |video of the talk}} 
 + 
 + 
 +Let $w$ be a finite word over the alphabet $\{0,1\}$. For any natural number $n$, let $s_w(n)$ denote the number of occurrences of $w$ in the binary expansion of $n$ as a scattered subsequence. The talk aims to provide a systematic way of studying the growth of the partial sum $\sum_{n=0}^N (-1)^{s_w(n)}$ as $N \to \infty$. In particular, these techniques yield several classes of words $w$ with $\sum_{n=0}^N (-1)^{s_w(n)} = O(N^{1-\epsilon})$ for some $\epsilon >0$. We begin by motivating the ideas using the case of $w=01,011$. The talk is based on joint work with Shuo Li. 
 + 
 +**May 6 2025: Jarkko Peltomäki** //The repetition threshold for ternary rich words// 
 + 
 +{{ seminar2025:20250506peltomaki.pdf |slides}} 
 + 
 +{{ seminar2025:20250506peltomaki.mp4 |video of the talk}} 
 + 
 + 
 +In the recent years, it has been popular to determine the least critical exponent in specific families of infinite words. In this talk, I will explain what is known about the least critical exponents for infinite words that are rich in palindromes. In particular, I will outline the proof of our result that the least critical exponent for ternary rich infinite words equals $1 + 1/(3 - \mu) \approx 2.25876324$, where $\mu$ is the unique real root of the polynomial $x^3 - 2x^2 - 1$. This result is based on proving a structure theorem for ternary rich infinite words that avoid $16/7$-powers. In addition, I discuss some recent progress on determining the least asymptotic critical exponents for rich infinite words. 
 + 
 +Joint work with L. Mol and J. D. Currie. 
 + 
 + 
 + 
 +**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// 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 The Lagrange spectrum is related to the rational approximations of badly
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 substitutions. This is joint work with Wolfgang Steiner. substitutions. This is joint work with Wolfgang Steiner.
  
-**April 22 2025: Be'eri Greenfeld** 
  
-**May 6 2025: Jarkko Peltomäki**+**March 25 2025: [[https://page.mi.fu-berlin.de/vbui/|Vuong Bui]]** //An explicit condition for boundedly supermultiplicative subshifts//
  
-**May 20 2025: Pranjal Jain** 
  
-**June 3 2025: Curtis Bright**+{{ seminar2025:20250325bui.pdf |slides}}
  
-**June 17 2025: [[https://sites.google.com/view/noysofferaranov/bio|Noy Soffer Aranov]]**+{{ seminar2025:20250325bui.mp4 |video of the talk}}
  
-**July 15 2025: [[https://apacz.matinf.uj.edu.pl/users/1719-elzbieta-krawczyk|Elżbieta Krawczyk]]**  //Quasi-fixed points of substitutions and substitutive systems// 
  
-We study automatic sequences and automatic systems (symbolic dynamical systems) generated by general constant length (nonprimitive) substitutions. While an automatic system is typically uncountable, the set of automatic sequences is countable, implying that most sequences within an automatic system are not themselves automatic. We provide a complete classification of automatic sequences that lie in a given automatic system in terms of the so-called quasi-fixed points of the substitution defining the system. Quasi-fixed points have already appeared implicitly in a few places (e.g. in the  study of substitutivity of lexicographically minimal points in substitutive systems or in the study of subsystems of substitutive systems) and they have been described in detail by Shallit and Wang and, more recently, B\'eal, Perrin, and Restivo . We conjecture that a similar statement holds for general nonconstant length substitutions. 
- 
- 
- 
-==== Past talks 2025 ==== 
- 
-**March 25 2025: [[https://page.mi.fu-berlin.de/vbui/|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 $\alpha\ge 0$, 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 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.