Name: Tom Hoffman, Associate Professor
Department: Mathematics & Statistics
Date: February 1, 2013

Tom Hoffman is a mathematician.  And he is much like the wizard behind the curtain, as in, his research deals with numbers and patterns behind the scenes, which he gives to other researchers who can apply those components and make sense of their own research.

Take for instance when you talk on your cell phone.  Your conversation is broken down into words and those words are broken down into 0's and 1's and transmitted to another person's cell phone which translates those 0's and 1's back into words and your conversation.  That's actually a mathematical science called Coding Theory.  And someone broke down the elements of the words we speak into numbers, and recreated them out of numbers, in order for that technology to be explored and created!

Tom Hoffman works with Dr. Klaus Lux from the University of Arizona in a project he calls "Constructing Basic Algebras for Nearly Simple Groups."  Professor Hoffman says, "Groups are mathematicians' way of talking about symmetry.  The more complicated the symmetries, the larger the group."  He studies groups, explores them, and finds their, as of now, unknown properties.

Take for instance a square.  If you play with a square you find that there are eight rigid symmetries.  You basically took the group of symmetries of the square apart to find out what made it up.  Professor Hoffman is dealing with much more complex objects, breaking the objects down, trying to build them up again into a similar structure, in order to measure their symmetries, and find properties of their symmetry groups.

A common application for Professor Hoffman's research is in the biochemistry field.  For instance, he can take a virus and flip it, looking at its symmetry from different angles: Does it look the same? How can it be deconstructed, and then how can it be put together again? This is used to learn about the original virus, which can then lead to how a virus will affect the environment and those in it.

Computer programs are generally used to help explore symmetries, one of which is GAP (Groups, Algorithms, and Programming).  Tom Hoffman has also written a number of programs that can help take these objects apart for further study, which go by the name of Basic Package.

Once he has collected this make-up of the object, he relays those data to other researchers who then use that information in their research.  Groups are commonly applied in String Theory, Coding Theory, and Frame Theory.

If it has to do with breaking down anything into its basic form to look for patterns and symmetry, Tom Hoffman is your mathematician.

The principal block of the Higman-Sims group in characteristic 5

The Higman-Sims group is a rare connection between Tom Hoffman's different areas of research.  Tom Hoffman has worked with Jim Solazzo, focusing on objects called 2-graphs.  Higman-Sims is the symmetry group of a 2-graph on 176 points.  This two graph has 44,352,000 symmetries, so this is a large goup.‌