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--- seminars:arit [2026/03/09 18:01] – borisov
+++ seminars:arit [2026/07/13 14:10] (current) – aborisov
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    * **February 3 (2:45-3:45 pm, cross-listed from Algebra Seminar) ** \\    **//Speaker//**: Tim Riley (Cornell University) \\      **//Title//**: Conjugator length \\ **//Abstract//**: The conjugacy problem for a finitely generated group $G$ asks for an algorithm which, on input a pair of words u and v, declares whether or not they represent conjugate elements of $G$. The conjugator length function $CL$ is its most direct quantification: $CL(n)$ is the minimal $N$ such that if $u$ and $v$ represent conjugate elements of $G$ and the sum of their lengths is at most $n$, then there is a word $w$ of length at most $N$ such that $uw=wv$ in $G$.  I will talk about why this function is interesting and how it can behave, and I will highlight some open questions.  En route I will talk about results variously with Martin Bridson, Conan Gillis, and Andrew Sale, as well as recent advances by Conan Gillis and Francis Wagner.\\
-   * **February 10**  \\    **//Speaker//**: Alexander Borisov (Binghamton)  \\      **//Title//**: A structure sheaf for Kirch topology, an update \\ **//Abstract//**:  Kirch topology on $\mathbb N$ goes back to a 1969 paper of Kirch. It can be defined by a basis of open sets that consists of all infinite arithmetic progressions $a+d\mathbb N_0$, such that $gcd(a,d)=1$ and $d$ is square-free. It is Hausdorff, connected, and locally connected. I will give an update on my current work on a natural presheaf of functions on this topological space: locally integer polynomial functions. In particular, I will discuss when the sheafification is equal to the presheaf, and when it is bigger. I will also discuss (Cech) cohomology. In particular, I will give examples with trivial and nontrivial H^1. No prior knowledge of the topic is assumed. This talk will also serve as an introduction to Mithun's talk next week. \\
+   * **February 10**  \\    **//Speaker//**: Alexander Borisov (Binghamton) \\      **//Title//**: A structure sheaf for Kirch topology, an update \\ **//Abstract//**:  Kirch topology on $\mathbb N$ goes back to a 1969 paper of Kirch. It can be defined by a basis of open sets that consists of all infinite arithmetic progressions $a+d\mathbb N_0$, such that $gcd(a,d)=1$ and $d$ is square-free. It is Hausdorff, connected, and locally connected. I will give an update on my current work on a natural presheaf of functions on this topological space: locally integer polynomial functions. In particular, I will discuss when the sheafification is equal to the presheaf, and when it is bigger. I will also discuss (Cech) cohomology. In particular, I will give examples with trivial and nontrivial H^1. No prior knowledge of the topic is assumed. This talk will also serve as an introduction to Mithun's talk next week. \\
    * **February 17**  \\    **//Speaker//**: Mithun Veettil (Binghamton) \\      **//Title//**: Some results on the  Locally LIP functions \\ **//Abstract//**: Locally LIP functions are obtained as a result of sheafification of the presheaf LIP on some infinite subset $X$ of $N={1,2,3,...}$, with a prescribed topology. Often we work with Kirch topology on $N$ that makes $N$ a connected, locally connected, and Hausdorff space. \\ If the set $X$ is a union of non-connected open sets, then we can easily define a locally LIP function on $X$ that is not a LIP function globally. In fact, even if the space $X$ is connected, a locally LIP function on $X$  need not be a LIP function on $X$. In this talk, we will look at $X=N$\ $6N$, which is connected, and construct a locally LIP function that is not LIP on $X$. Also, we will show that this is not the case if one works with $\mathbb{Z}[1/2]$ instead of $\mathbb{Z}$ for the above set $X$.\\