We consider the problem of counting polynomial curves on analytic or definable subsets over the field $\mathbb{C}((t))$ as a function of the degree r. A result of this type could be expected by analogy with the classical Pila–Wilkie counting theorem in the Archimedean situation.

Some non-Archimedean analogues of this type have been developed in the work of Cluckers, Comte, and Loeser for the field ${\mathbb{Q}}_p$, but the situation in $\mathbb{C}((t))$ appears to be significantly different. We prove that the set of polynomial curves of a fixed degree r on the transcendental part of a subanalytic set over $\mathbb{C}((t))$ is automatically finite, but we give examples that show their number may grow arbitrarily quickly even for analytic sets. Thus no analogue of the Pila–Wilkie theorem can be expected to hold for general analytic sets. On the other hand, we show that if one restricts to varieties defined by Pfaffian or Noetherian functions, then the number grows at most polynomially in r, thus showing that the analogue of the Wilkie conjecture does hold in this context.

Using the Donaldson–Auroux theory, we construct complete intersections in complex projective manifolds, which are negatively curved in various ways. In particular, we prove the existence of compact simply connected Kähler manifolds with negative holomorphic bisectional curvature. We also construct hyperbolic hypersurfaces, and we obtain bounds for their Kobayashi hyperbolic metric.

Consider the Hopf–Cole solution $h(t,x)$ of the Kandar–Parisi–Zhang (KPZ) equation with the narrow wedge initial condition. Regarding $t\to \mathrm{\infty }$ as a scaling parameter, we provide the first rigorous proof of the large deviation principle (LDP) for the lower tail of $h(2t,0)+\frac{t}{12}$, with speed $t^2$ and an explicit rate function ${\mathrm{\Phi }}_{-}(z)$. This result confirms existing physics predictions made by Corwin (2011); Sasorov, Meerson, and Prolhac (2017); and Krajenbrink, Le Doussal, and Prolhac (2018). Our analysis utilizes a formula from Borodin and Gorin (2016) to convert the LDP for the KPZ equation to calculating an exponential moment of the Airy point process (PP). To estimate this exponential moment, we invoke the stochastic Airy operator (SAO) and use the Riccati transform, comparison techniques, and certain variational characterizations of the relevant functional.

We give an explicit formula for the arithmetic intersection number of complex multiplication (CM) cycles on Lubin–Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions $K_1,K_2$ over a non-Archimedean local field F. Our formula works for all cases: $K_1$ and $K_2$ can be either the same or different, ramified or unramified over F. This formula translates the linear arithmetic fundamental lemma (linear AFL) into a comparison of integrals. As an example, we prove the linear AFL for ${\mathrm{GL}}_2(F)$.