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: Christmas cracker pulling: How to send everyone home a winner

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

Christmas cracker pulling: How to send everyone home a winner

Dec 15, 2014

According to experts' statistical analyses, if you're expecting 10 guests for dinner on Christmas day, 15 crackers—those festive cardboard tubes filled with a one-size-fits-no-one paper hat, a small toy, and a groan-inducing ...

Mathematicians prove the Umbral Moonshine Conjecture

Dec 15, 2014

Monstrous moonshine, a quirky pattern of the monster group in theoretical math, has a shadow - umbral moonshine. Mathematicians have now proved this insight, known as the Umbral Moonshine Conjecture, offering ...

Uncovering complex network structures in nature

Dec 10, 2014

The global spread of Ebola is due to the complex interactions between individuals, societies, and transportation and trade networks. Understanding and building appropriate statistical and mathematical models ...

Shifting boundaries and changing surfaces

Dec 10, 2014

New research published in the Proceedings of The Royal Society A by members of the Mathematical Soft Matter Unit at the Okinawa Institute of Science and Technology Graduate University examines the en ...

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.