Gravitational instantons are non-compact Calabi--Yau metrics with L^2 bounded curvature and are categorized into six types. I will describe several projects on gravitational instantons including: Fredholm theory and deformation, non-collapsing degeneration limits, and existence of stable non-holomorphic minimal spheres in certain types. These projects utilize geometric microlocal analysis in different settings, where the analysis of singular operators is obtained through geometric resolutions. Based on works jointly with Rafe Mazzeo, Yu-Shen Lin, Sidharth Soundararajan, and Alia Yusaini.

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Fractal uncertainty principle (FUP) is an estimate on the $L^2$ operator norm of $1_X B_h 1_Y$ where $X,Y$ are fractal sets down to some small scale $h$ and $B_h$ is a unitary operator whose integral kernel oscillates at wave length $h$. This has applications to quantum chaos, including spectral gaps for convex co-compact hyperbolic surfaces. Many FUP results are available, typically reducing the case of general $B_h$ to the Fourier transform. In this talk I will give a new FUP with a reasonably large exponent for $1/2$-dimensional sets $X,Y$, relying on the difference between the operator $B_h$ used for hyperbolic surfaces and the Fourier transform. The key ingredients come from projection theory. Joint work with Alex Cohen, and some help from AI.

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We study the classical problem of sectional curvature pinching in the class of 2-step nilmanifolds, which are necessarily of mixed curvature. We show that the pinching constant of any 2-step nilmanifold lies in the compact interval $[-3, -\frac{3}{2}]$. The upper bound $-\frac{3}{2}$ is achieved by the complex Heisenberg group $\mathrm{Heis}_3(\mathbb{C})$ with a Ricci soliton metric. The upper bound exhibits rigidity: if a simply connected 2-step nilmanifold $N$ has the pinching constant $-\frac{3}{2}$, then $N$ admits a Ricci soliton complex Heisenberg group as a totally geodesic subgroup. On the other hand, the lower bound satisfies non-rigidity: any 2-step nilpotent Lie group admits a metric with pinching constant $-3$. This is derived by showing that there is an open neighborhood $U$ of $\mathrm{Heis}_3(\mathbb{R})\times \mathbb{R}^{n-3}$ in the space of $n$-dimensional 2-step nilmanifolds such that the pinching constant is $-3$ on $U$, and any 2-step nilpotent Lie group $N$ has a metric $g$ such that $(N,g)$ lies in $U$. In fact, if $N$ is not isomorphic to $\mathrm{Heis}_3(\mathbb{R})\times \mathbb{R}^{n-3}$, then there is a curve $g_t$ of metrics on $N$ with $(N,g_t)\in U$, showing that there are uncountably many left-invariant metrics on $N$ such that the pinching constant is $-3$.

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