Difference between revisions of "User:DPatrick"

Revision as of 12:19, 10 July 2006

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 MIT.

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

The ring $R[x,y]$ is commutative, meaning that $xy$ and $\displaystyle yx$ 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 $R\langle x,y\rangle/(xy-qyx)$ , where $q$ is a non-zero constant. ( $q=1$ gives the usual commutative polynomial ring.)

It gets more complicated from here, but you get the idea. If you want to see details, you can watch a lecture I gave at MSRI in 2000, and read the accompanying lecture notes.

You can also see Paul Smith's links to people and sites about noncommutative algebra and geometry.

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 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<x,y>/(xy-qyx)</math>, where <math>q</math> is a non-zero constant.  (<math>q=1</math> gives the usual commutative polynomial ring.)
+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, but you get the idea.  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 see Paul Smith's [http://www.math.washington.edu/~smith/Research/research.html links] to people and sites about noncommutative algebra and geometry.

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Difference between revisions of "User:DPatrick"

Revision as of 12:19, 10 July 2006

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My thesis