Title: Spectral monotonicity under Gaussian convolution


Abstract: 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"). This turns out to be true if the original measure is log-concave (i.e., has a density of the form $e^V$ where $V$ is a concave function). We will discuss a simple proof of this result, which is based on ideas from fluid mechanics (or, in mathematicians' terminology, measure transportation). If time permits, we will also discuss a different proof based on Markov semigroup ideas; this latter proof turns out to extend to the discrete setting of measures on $\mathbb Z$. Some results joint with B. Klartag.