Mathematical Entropy

In mathematics, particularly in geometry and differential equations, several phenomenon appear which behave similarly to entropy in physics.

Perelman Entropy[edit]

For more on this subject, see Perelman's proof of the Poincaré conjecture.

Ricci flow, used by Perelman for his solution of one of the greatest unsolved problems of the 20th century, the Poincare conjecture, behaves similarly to heat dispersion, which is a form of entropy.

Noticing this (and other similarities between the effect of Ricci flow on a manifolds metric and the effect of time on heat dispersion), Perelman formulated the concept of "Perelman entropy." By focusing on entropy, Perelman unlocked and solved one of the greatest mathematical mysteries of the 20th century.

Perelman's insight was to develop and use an estimate for entropy such that "a special eigenvalue of the manifold undergoing the Ricci flow and cut/paste process is increasing."^[1]

Nash Entropy[edit]

Nash entropy is a concept in game theory which behaves similarly to entropy in statistical mechanics.

Entropy in Engineering[edit]

Mathematical entropy is useful in solving shallow water equations.

References[edit]

1. ↑ http://comet.lehman.cuny.edu/sormani/others/perelman/introperelman.html