### Two Problems in Mathematical Physics: Villani's Conjecture and a Trace Inequality for the Fractional Laplacian

- Series
- Dissertation Defense
- Time
- Monday, August 29, 2011 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Amit Einav – School of Mathematics, Georgia Tech

The presented work deals with two distinct problems in the field
of Mathematical Physics, and as such will have two parts addressing each
problem.
The first part is dedicated to an 'almost' solution of Villani's conjecture,
a known conjecture related to a Statistical Mechanics model invented by Kac
in 1956, giving a rigorous explanation of some simple cases of the Boltzman
equation. In 2003 Villani conjectured that the time it will take the system
of particles in Kac's model to equalibriate is proportional to the number of
particles in the system. Our main result in this part is an 'almost proof'
of that conjecture, showing that for all practical purposes we can consider
it to be true.
The second part of the presentation is dedicated to a newly developed trace
inequality for the fractional Laplacian, connecting between the fractional
Laplacian of a function and its restriction to the intersection of the
hyperplanes x_n =...= x_n-j+1 = 0 , where 1 <= j < n.
The newly found inequality is
sharp and the functions that attain inequality in it are completely
classified.