Lecture 1 (Monday, 5/14)
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Title:  The mathematics of doodling
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Abstract:  Doodling has many mathematical aspects: patterns, shapes,  numbers, and more.  Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles.  I'll begin by doodling, and see where it takes us.  It looks like play, but it reflects what mathematics is really about:  finding patterns in nature, explaining them, and extending them.  By the end, we'll have seen some important notions in geometry, topology, physics, and elsewhere; some fundamental ideas guiding the development of mathematics over the course of the last century;  and ongoing work continuing today.
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Lecture 2 (Tuesday, 5/15)
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Title:  Cutting and pasting in algebraic geometry
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Given some class of "geometric spaces", we can make a ring as follows.  
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(i) (additive structure)  When U is an open subset of such a space X,  
[X] = [U] + [X \ U];
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(ii) (multiplicative structure)  [X x Y] = [X] [Y].
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In the algebraic setting, this ring contains
surprising structure, connecting geometry to arithmetic and topology.  I will discuss some remarkable
statements about this ring (both known and conjectural), and present new statements
(again, both known and conjectural).  A motivating example will be polynomials in one variable.  (This talk is intended for a broad audience.)  This is joint work with Melanie Wood.