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--- seminars:alge [2026/01/30 08:13]
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+++ seminars:alge [2026/02/26 14:24] (current)
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    * **February 3**\\  <html> <span style="color:blue;font-size:120%"> Tim Riley (Cornell University) </span></html> \\      **//Conjugator length//** \\   \\  <WRAP center box 90%> **//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. </WRAP>
-    * **February 10**\\  <html> <span style="color:blue;font-size:120%"> Ryan McCulloch (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
+    * **February 10**\\  <html> <span style="color:blue;font-size:120%"> Ryan McCulloch (Binghamton University) </span></html> \\      **//A p-group Classification Related to Density of Centralizer Subgroups//** \\   \\  <WRAP center box 90%> **//Abstract//**: If $\mathfrak{P}$ is a property pertaining to subgroups of a $p$-group $G$, and if each subgroup with property $\mathfrak{P}$ contains $Z(G)$, then a group $G$ whose subgroups are dense with respect to property $\mathfrak{P}$ must satisfy the following criteria:
-   * **February 17**\\  <html> <span style="color:blue;font-size:120%"> Tae Young Lee (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
+$|Z(G)|= p$ and every subgroup $H$ of order at least $p^2$ contains $Z(G)$.
-   * **February 24**\\  <html> <span style="color:blue;font-size:120%"> Lei Chen (Bielefeld University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
+I will discuss our progress in obtaining a classification of all such $p$-groups. This is joint work with Mark Lewis and Tae Young Lee. </WRAP>
-   * **March 3**\\  <html> <span style="color:blue;font-size:120%">  (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
+   * **February 17**\\  <html> <span style="color:blue;font-size:120%"> Tae Young Lee (Binghamton University) </span></html> \\      **//Title: Finite groups with many elements of the same order//** \\   \\  <WRAP center box 90%> **//Abstract//**: It is a well-known fact that if more than 3/4 of the elements of a finite group are involutions then the group is abelian. Berkovich proved that if more than 4/15 are involutions then the group must be solvable. Motivated by these results, Deaconescu asked the following question: If at least half of the elements are of the same order, $k$, does the group have to be solvable? In this talk, we prove this when $k = p^a$ for primes $p$ except when $p = 2,3$ and $a > 1$, and give counterexamples for larger powers of 2 and 3 except $k = 4$, and also for several other types of composite numbers. We also show that when $k > 4$, it is always possible to find a non-solvable group such that at least 3/19 of its elements have order $k$. This is a joint work with Ryan McCulloch. </WRAP>
+   * **February 24**\\  <html> <span style="color:blue;font-size:120%"> Lei Chen (Bielefeld University, by Zoom) </span></html> \\      **//Covering a finite group by the conjugates of a coset//** \\   \\  <WRAP center box 90%> **//Abstract//**: It is well known that for a finite group G and a proper subgroup A of G, it is impossible to cover G with the conjugates of A. Thus, instead of the conjugates of A, we take the conjugates of the coset Ax in G and check if the union of $(Ax)^g$ covers G-{1} for g in G. Moreover, if $(Ax)^g$ covers G for all Ax in Cos(G:A), we say that (G,A) is CCI. We are aiming to classify all such pairs. It has been proven by Baumeister-Kaplan-Levy that this can be reduced to the case where A is maximal in G, and so that the action of G on Cos(G:A) is primitive, here Cos(G:A) stands for the set of right cosets of A in G. And they showed that (G,A) is CCI if G is 2-transitive. By O'Nan-Scott Theorem and CFSG (classification of finite simple groups), we see that G is either an affine group or almost simple. In the paper by Baumeister-Kaplan-Levy, it is shown that affine CCI groups are 2-transitive. Thus, it remains to consider the almost simple groups. By employing the knowledge of buildings, representation theory, and Aschbacher-Dynkin theorem, we prove that, apart from finitely many small cases, the CCI almost simple groups are 2-transitive.  </WRAP>
+   * **March 3**\\  <html> <span style="color:blue;font-size:120%"> Chaitanya Joglekar (Binghamton University) </span></html> \\      **//Lattice basis reduction and the LLL algorithm//** \\   \\  <WRAP center box 90%> **//Abstract//**: A lattice L is a subgroup of $\mathbb{R}^n$ isomorphic to $\mathbb{Z}^n$. Finding a vector in L of the shortest length has many applications in number theory, cryptography and optimisation. While finding a vector with the shortest length is an NP hard problem, the LLL algorithm finds a “short enough” vector in Polynomial time.
+In this talk, we will go over the LLL algorithm and demonstrate one of its applications, finding a Diophantine approximation for a finite set of rational numbers. </WRAP>
    * **March 10**\\  <html> <span style="color:blue;font-size:120%"> Hanlim Jang (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
-   * **March 17**\\  <html> <span style="color:blue;font-size:120%">  (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
+   * **March 17**\\  <html> <span style="color:blue;font-size:120%"> William Cocke (Carnegie Mellon University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
    * **March 24**\\  <html> <span style="color:blue;font-size:120%">  (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
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    * **April 7**\\  <html> <span style="color:blue;font-size:120%"> No Meeting (Monday Classes Meet) </span></html>\\
-   * **April 14**\\  <html> <span style="color:blue;font-size:120%">  (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
+   * **April 14**\\  <html> <span style="color:blue;font-size:120%"> Luna Gal (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
    * **April 21**\\  <html> <span style="color:blue;font-size:120%">  (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>

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