##### Department of Mathematics,

University of California San Diego

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### Food for Thought Seminar

## Dan Budreau

#### UCSD, Graduate Student

## The geometry of schemes (and other tall tales!)

##### Abstract:

This talk will motivate the use of schemes in understanding varieties. Schemes are notorious for their lack of transparency. In order to motivate the practicality of schemes, I will discuss the basics of varieties: specifically coordinate rings (the ring you get from a variety) and the various properties that they have. After that, schemes will be (roughly) defined and I will come to the heart of the talk---examples of what happens when we relax various conditions on coordinate rings. Very strange phenomena come up: we can get extremely local information, we can get points on varieties with tangent vectors, we can describe some strange varieties over the reals, etc. If time permits, I will talk about schemes as they pertain to number theory and give an example or two. This talk is meant to introduce schemes in a geometric sense. Most importantly, I want to brush aside the technical baggage that comes with high-end commutative algebra, sheaves, etc. in order to get at the heart of the geometry. Those with a penchant for pictures will be none too displeased.

### October 19, 2006

### 1:00 PM

### AP&M 7321

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