Anisotropic Preiss density theorem and an advertisement for GMT

This talk will advertise a course I will offer in March (pending interest) and contain some new results. In the 1920s Besicovitch asked the question: What can one say about the structure of sets $E$ in the plane, with the property that $\lim_{r \downarrow 0} \frac{ \mathcal{H}^{1}(B(x,r) \cap E)}{2r}= 1$ for almost every $x \in E$? Here, $\mathcal{H}^{1}$ denotes the Hausdorff $1$-measure, which should be thought of as measuring the "length" of a set. Besicovitch's work can be viewed as the founding idea behind the field of Geometric Measure Theory (GMT). After many results by Besicovitch, Federer, Marstrand, and Mattila, the groundbreaking work of Preiss in 1987 provided a very satisfying answer to the "density question" in the setting of measures in all dimensions. Preiss' work relies on the new notion of tangent measures. Due to a lack of flexibility in Preiss' work, much of it has been described as "searching for a needle in a haystack, and finding only needles". But, the tools related to tangent measures themselves are very flexible. In recent work with Tatiana Toro and Bobby Wilson, we show that if $\mu$ is a locally finite measure on $\mathbb{R}^{n}$ and there exists an $m$-so that for $\mu$ almost every $x$ there exists an ellipse $E_{x}$ so that $0<\lim_{r \downarrow 0} \frac{ \mu(x + rE_{x})}{r^{m}} < \infty$, then $\mu$ is countably $m$-rectifiable. Surprisingly, no uniform ellipticity or continuity in $x$ is required of the ellipses $E_{x}$.