Georgia Institute of Technology College of Sciences

The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity"). We show that this is true if the original measure is log-concave, via two very different strategies - a dynamic variant of Bakry-Émery's $\Gamma$-calculus, and a mass-transportation argument. Moreover, we show that the dynamic $\Gamma$-calculus argument can also be extended to the discrete setting of measures on $\mathbb Z$, and that spectral monotonicity holds in this setting as well. Some results joint with B. Klartag.