seminars:alge
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| * **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> | * **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> \\ **//Title//** \\ \\ <WRAP center box 90%> **//Abstract//**: Text of Abstract </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 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> | ||
seminars/alge.1771442933.txt · Last modified: 2026/02/18 14:28 by tongviet