Heat distributions help researchers to understand curved space

Aug 26, 2014
Heat distributions help researchers to understand curved space

The heat equation is one of the most important partial differential equations. The behavior of the solution to the equation reflects the geometry of the underlying space very well. Therefore, this equation has been investigated very extensively in both analysis and geometry. The solution evolves over time so that the Dirichlet's energy functional decreases most efficiently. Recently, F. Otto introduced another characterization: the solution evolves so that the Boltzmann entropy increases most efficiently from the viewpoint of optimal transportation. Both of these characterizations enable us to study the heat equation on spaces admitting singularities, where usual differential calculus does not work. However, their identification in such spaces is unknown.

Now, Tokyo Tech mathematicians have provided a first identification result for this question by reducing the problem to the uniqueness of the second characterization. Their result is closely related to the theory of Ricci curvature in terms of optimal transportation by using the Boltzmann entropy, which appears in the second characterization.

Furthermore, Kuwada and his collaborators revealed a connection between this theory and another Ricci curvature theory present in the first characterization, known as Bakry-Émery . Their completely new approach will open the door to further extensive studies on geometric analysis of non-smooth spaces, and will help further understanding of the curvature of space.

Explore further: When vaccines are imperfect: What math can tell us about their effects on disease propagation

More information: Nicola Gigli, Kazumasa Kuwada and Shin-ichi Ohta. "Heat flow on Alexandrov spaces." Communications on Pure and Applied Mathematics, Vol. 66, no.3 (2013) 307-331. DOI: 10.1002/cpa.21431

add to favorites email to friend print save as pdf

Related Stories

Mathematicians Solve 140-Year-Old Boltzmann Equation

May 13, 2010

(PhysOrg.com) -- Two University of Pennsylvania mathematicians have found solutions to a 140-year-old, 7-dimensional equation that were not known to exist for more than a century despite its widespread use in modeling the ...

Quantum steps towards the Big Bang

Sep 03, 2013

(Phys.org) —Present-day physics cannot describe what happened in the Big Bang. Quantum theory and the theory of relativity fail in this almost infinitely dense and hot primal state of the universe. Only ...

Recommended for you

When shareholders exacerbate their own banks' crisis

Nov 21, 2014

Banks are increasingly issuing 'CoCo' bonds to boost the levels of equity they hold. In a crisis situation, bondholders are forced to convert these bonds into a bank's equity. To date, such bonds have been ...

Trouble with your boss? Own it

Nov 21, 2014

Don't get along with your boss? Your job performance may actually improve if the two of you can come to grips with the poor relationship.

User comments : 0

Please sign in to add a comment. Registration is free, and takes less than a minute. Read more

Click here to reset your password.
Sign in to get notified via email when new comments are made.