Difference between revisions of "Applied Algebra Seminar/Abstracts F13"
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+ | == September 26 == | ||
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+ | |- valign="center" | ||
+ | | [[Image:Aasf13 jimdemmel.jpg|200px]] [http://www.cs.berkeley.edu/~demmel/ Jim Demmel], UC-Berkeley (Math/CS) | ||
+ | |- valign="top" | ||
+ | |''TBD'' | ||
+ | |- valign="top" | ||
+ | |TBD | ||
+ | |} | ||
+ | |||
== October 31 == | == October 31 == | ||
{| cellpadding="5" width="80%" | {| cellpadding="5" width="80%" | ||
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− | | [[Image:Aasf13 andrewbridy.jpg|200px]] | + | | [[Image:Aasf13 andrewbridy.jpg|200px]] [http://www.math.wisc.edu/~bridy/ Andrew Bridy], UW-Madison (Math) |
|- valign="top" | |- valign="top" | ||
− | |Functional Graphs of Affine-Linear Transformations over Finite Fields | + | |''Functional Graphs of Affine-Linear Transformations over Finite Fields'' |
|- valign="top" | |- valign="top" | ||
− | |A linear transformation <math>A: (\mathbb{F}_q)^n \to (\mathbb{F}_q)^n</math> gives rise to a directed graph by regarding the elements of <math>(\mathbb{F}_q)^n</math> as vertices and drawing an edge from <math>v</math> to <math>w</math> if <math>Av = w</math>. In 1959, Elspas determined the "functional graphs" on <math>q^n</math> vertices that are realized in this way. In doing so he showed that there are many non-similar linear transformations which have isomorphic functional graphs (and so are conjugate by a non-linear permutation of <math>(\mathbb{F}_q)^n | + | |A linear transformation <math>A: (\mathbb{F}_q)^n \to (\mathbb{F}_q)^n</math> gives rise to a directed graph by regarding the elements of <math>(\mathbb{F}_q)^n</math> as vertices and drawing an edge from <math>v</math> to <math>w</math> if <math>Av = w</math>. In 1959, Elspas determined the "functional graphs" on <math>q^n</math> vertices that are realized in this way. In doing so he showed that there are many non-similar linear transformations which have isomorphic functional graphs (and so are conjugate by a non-linear permutation of <math>(\mathbb{F}_q)^n</math>). I review some of this work and prove an new upper bound on the number of equivalence classes of affine-linear transformations of <math>(F_q)^n</math> under the equivalence relation of isomorphism of functional graphs. This bound is significantly smaller than the number of conjugacy classes of <math>\operatorname{GL}_n(q)</math>. This is joint work with Eric Bach. |
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Latest revision as of 10:15, 23 August 2013
September 26
Jim Demmel, UC-Berkeley (Math/CS) |
TBD |
TBD |
October 31
Andrew Bridy, UW-Madison (Math) |
Functional Graphs of Affine-Linear Transformations over Finite Fields |
A linear transformation [math]A: (\mathbb{F}_q)^n \to (\mathbb{F}_q)^n[/math] gives rise to a directed graph by regarding the elements of [math](\mathbb{F}_q)^n[/math] as vertices and drawing an edge from [math]v[/math] to [math]w[/math] if [math]Av = w[/math]. In 1959, Elspas determined the "functional graphs" on [math]q^n[/math] vertices that are realized in this way. In doing so he showed that there are many non-similar linear transformations which have isomorphic functional graphs (and so are conjugate by a non-linear permutation of [math](\mathbb{F}_q)^n[/math]). I review some of this work and prove an new upper bound on the number of equivalence classes of affine-linear transformations of [math](F_q)^n[/math] under the equivalence relation of isomorphism of functional graphs. This bound is significantly smaller than the number of conjugacy classes of [math]\operatorname{GL}_n(q)[/math]. This is joint work with Eric Bach. |