I am studying geometric analysis on possibly singular spaces.

For example, let us consider a sphere which is regarded as the earth.

Then the sum of angles of any triangle on the sphere is not equal to pi. More precisely it is greater than pi.

This result can be understood by using ``curvature'' which is a central notion in geometry. Moreover, the difference between the sum and pi can be written by using the area of the triangle.

Next let us consider a cube which is a typical example of singular spaces. The singular points consist of vertices. Then it is natural to ask whether the sum of angles of any triangle on the cube is equal to pi or not.

I am interested in such geometric/analytic problems and studying them by using heat equations on possibly singular spaces.