The Lieb–Thirring inequalities give a bound on the negative eigenvalues of a Schrödinger operator in terms of an $L^p$-norm of the potential. These are dual to bounds on the $H^1$-norms of a system of orthonormal functions. Here we extend these bounds to analogous inequalities for perturbations of the Fermi sea of noninteracting particles (i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials).

We use $X^{s,b}$-inspired spaces to prove a uniqueness result for Calderón’s problem in a Lipschitz domain $\Omega$ under the assumption that the conductivity lies in the space $W^{1,\infty }\left(\bar{\Omega }\right)$. For Lipschitz conductivities, we obtain uniqueness for conductivities close to the identity in a suitable sense. We also prove uniqueness for arbitrary $C^1$ conductivities.

We study the collapsing behavior of Ricci-flat Kähler metrics on a projective Calabi–Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov–Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkähler manifolds in the large complex structure limit. In this setting, we prove an analogue of a result of Gross and Wilson for $K3$ surfaces, which is motivated by the Strominger–Yau–Zaslow picture of mirror symmetry.

Let $X$ be a smooth, connected, dimension $n$, quasi-projective variety embedded in ${\mathbb{P}}^N$. Consider integers $\{k_1,\dots ,k_r\}$, with $k_i>0$, and the Hilbert scheme $H_{\{k_1,\dots ,k_r\}}\left(X\right)$ of aligned, finite, degree $\sum k_i$ subschemes of $X$, with multiplicities $k_i$ at points $x_i$ (possibly coinciding). The expected dimension of $H_{\{k_1,\dots ,k_r\}}\left(X\right)$ is $2N-2+r-(\sum k_i)(N-n)$. We study the locus of points where $H_{\{k_1,\dots ,k_r\}}\left(X\right)$ is not smooth of expected dimension, and we prove that the lines carrying this locus do not fill up ${\mathbb{P}}^N$.

Starting from the quantitative stability result of Bianchi and Egnell for the $2$-Sobolev inequality, we deduce several different stability results for a Gagliardo–Nirenberg–Sobolev (GNS) inequality in the plane. Then, exploiting the connection between this inequality and a fast diffusion equation, we get stability for the logarithmic Hardy–Littlewood–Sobolev (Log-HLS) inequality. Finally, using all these estimates, we prove a quantitative convergence result for the critical mass Keller–Segel system.