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: Using Twitter to probe political polarization

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

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

Volunteers not safe from workplace bullying

52 minutes ago

Despite the charitable nature of volunteering, new research from Murdoch University and Edith Cowan University has found one in three volunteers can experience workplace bullying.

The problem with solitary confinement

52 minutes ago

It's a practice that has been in the news since the Ashley Smith case first made headlines and, last week, the Ontario government announced it had launched a review of its solitary confinement policies.

100-million-year-old scale insect practiced brood care

1 hour ago

Scientists at the University of Bonn, together with colleagues from China, UK and Poland, have described the oldest evidence of brood care in insects: it is in a female scale insect with her young that is encased in amber ...

The humorous path to academic success

4 hours ago

Academics and universities are in a race: to produce high impact publications, to gain citations, bring in grant income and climb university rankings. In this rat race, perhaps the true path to academic success ...

Scientists ask, peer review on fast track at what price?

5 hours ago

A fast-track peer-review trial is in the news. A Nature Publishing Group (NPG) -owned journal's editorial board member has resigned in protest over a pilot project where researchers pay for faster peer review. ...

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.