Posts tagged with Robert Ghrist










I view a mathematics library the same way an archaeologist views a prime digging site. There are all these wonderful treasures that are buried there and hidden from the rest of the world.

If you pick up a typical book on sheaf theory, for example, it’s unreadable. But it’s full of stuff that is very, very important to solving really difficult problems.

And I have this vision of digging through the obscure text and finding these gems and exporting them over to the engineering college and other domains where these tools can find utility.




More drawings by Robert Ghrist. Point-set topology, differential topology, geometric topology, symplectic topology, algebraic topology illustrated. From his (free) notes on applied homology.

GLOSSARY

  • Topology = connections between things.
  • Manifolds come in a wide variety of shapes, but they’re all tame.
  • Vector fields = arrows on a manifold.
  • Differential = linear approximation.
  • Symplectic = isotropic + antisymmetric + bilinear. (erm, not as complicated as it sounds)
  • Phase Space means these things can be conceptual rather than literal.










A beautiful depiction of a 1-form by Robert Ghrist. You never thought understanding a 1→1-dimensional ODE (or a 1-D vector field) would be so easy!

image

What his drawing makes obvious, is that images of Phase Space wear a totally different meaning than “up”, “down”, “left”, “right”. In this case up = more; down = less; left = before and right = after. So it’s unhelpful to think about derivative = slope.

image

BTW, the reason that ƒ must have an odd number of fixed points, follows from the “dissipative” assumption (“infinity repels”). If ƒ (−∞)→+, then the red line enters from the top-left. And if ƒ (+∞)→−∞, then the red line exits toward the bottom-right. So no matter how many wiggles, it must cross an odd number of times. (Rolle’s Thm / intermediate value theorem from undergrad calculus / analysis)

Found this via John D Cook.

(Source: math.upenn.edu)