Research with Students
Recently a senior seminar student and I studied a mathematical connection to music. Specifically, we generalized rhythm patterns where there are a finite number of beats and "hits" (the left picture above) to allow for hits anywhere in a stretch of time (right picture above). Then we studied a measure of the "evenness" of those rhythm patterns. Our work was published in The Pentagon! (Also, here's a paper that introduces and studies the "discrete rhythm patterns" where there are a finite number of beats.)
My Grad School Research
Being interested in form and structure led me to the land of algebraic geometry and category theory. My work centered on Bridgeland stability conditions (which give labels to things living inside derived categories) and the moduli spaces involved.
Here's a brief description of the picture above, which shows different aspects of Bridgeland stability conditions. (Read them from right to left)
If you pick some invariants then you get a moduli space parameterizing the semistable objects with those invariants. As you move your stability condition around, the stability of objects can change, and thus so can the moduli space.
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This represents one of the defining components of a BSC. It is a map sending certain algebraic objects related to S to the complex numbers. With this map, each object is labeled as semistable or unstable.
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This is Stab(S), the set of all BSCs for S. It is a manifold that you can move around on by slightly changing your BSC.
The blue and red curves are "walls" where a given object's stability can change. |
This is a surface S for which we can find Bridgeland stability conditions (BSCs).
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Check out this poster for some pretty pictures related to my research (see, ellipses and hyperbolas!). I presented the poster at the Fall '13 WAGS (Western Algebraic Geometry Symposium).
This paper is about when objects called "line bundles" (which you can think of as various hair-do's for your surface) get the labels semistable and unstable:
-"Bridgeland Stability of Line Bundles on Surfaces" (joint with Daniele Arcara)
This paper is about when Bridgeland stability conditions are related to dots and arrows (and the nice things you can then say about the moduli spaces):
-"Projectivity of Bridgeland Moduli Spaces on Del Pezzo Surfaces of Picard Rank 2" (joint with Daniele Arcara)
This paper is about when objects called "line bundles" (which you can think of as various hair-do's for your surface) get the labels semistable and unstable:
-"Bridgeland Stability of Line Bundles on Surfaces" (joint with Daniele Arcara)
This paper is about when Bridgeland stability conditions are related to dots and arrows (and the nice things you can then say about the moduli spaces):
-"Projectivity of Bridgeland Moduli Spaces on Del Pezzo Surfaces of Picard Rank 2" (joint with Daniele Arcara)