View source for User:DPatrick

This is the home page of [[David Patrick]].

== Places ==

I have lived in:
* Castleton, NY
* Batavia, NY
* Pittsburgh, PA
* Cambridge, MA
* Seattle, WA
* Gales Ferry, CT
* San Diego, CA

== My thesis ==

Since several people have asked about this...

My thesis was titled ''Noncommutative Ruled Surfaces'', which is a bridge between [[noncommutative]] [[algebra]] and [[geometry]].  My thesis research was supervised by [[Michael Artin]] at [[Massachusetts Institute of Technology|MIT]].

Basically, [[algebraic geometry]] gives us tools to go between algebra and geometry.  For example, we can think of <math>R[x,y]</math>, the [[ring]] of polynomials in two variables (x and y), as representing the Euclidean plane, and then [[ideal]]s of that ring represent subsets (called ''varieties'').  For examples, the ideal <math>(x-y)</math> would represent the line <math>x=y</math>, and the ideal <math>(x-2,y-3)</math> would represent the point <math>(2,3)</math>.

The ring <math>R[x,y]</math> is [[commutative]], meaning that <math>xy</math> and <math>\displaystyle yx</math> represent the same polynomial.  If we remove that assumption, we get noncommutative polynomials.  Geometry gets much harder.  In fact, we do have to place some restriction on the noncommutativity.  One common example is the so-called quantum polynomial ring <math>R\langle x,y\rangle/(xy-qyx)</math>, where <math>q</math> is a non-zero constant.  (<math>q=1</math> gives the usual commutative polynomial ring.)

It gets more complicated from here... 

If you want to see details, you can [http://www.msri.org/publications/ln/msri/2000/interact/patrick/1/index.html watch a lecture] I gave at MSRI in 2000, and read the accompanying lecture notes.  

You can also view a [http://citeseer.ifi.unizh.ch/cache/papers/cs/8682/http:zSzzSzwww.math.washington.eduzSz~patrickzSzwebprop.pdf/patrick97proposed.pdf research proposal] that I wrote in 1997 that more or less summarizes my thesis.

Finally, you can also see Paul Smith's [http://www.math.washington.edu/~smith/Research/research.html links] to people and sites about noncommutative algebra and geometry.