21 problems

Let a formal series be Borel–Écalle resummed using a choice of well-behaved average. A well-behaved average is the averaging choice used to define the resummed function across the…

Ecalle's proof concerns the injection of accelero-summable transseries into germs of functions near infinity, as used to establish the quasianalyticity needed in the analysis of po…

The quantization condition gives $\widetilde R(x)=\nu$, with $\nu=\pi\left(n+\frac{1}{2}\right)$, and perturbative inversion produces…

Asymptotic-sectors conjecture. All non-perturbative sectors of the full Adler function are asymptotic.

Let $\Lbb$ denote the class of log-atomic surreal numbers, and let $\kappa_x$ be the kappa-number associated with a surreal number $x$ by the recursive definition … The iterated ex…

Model completeness conjecture. The structures $\mathbb{T}_{\operatorname{an,exp}}$ and $\mathbb{T}_{\operatorname{an}}$ are both model complete.

Let $\mathbb T$ be the differential field of logarithmic-exponential transseries, and let $\mathbb R$ be the real field. Write $\mathbb T[\imag]$ and $\mathbb C=\mathbb R[\imag]$ f…

A formal series is resummed by the Borel-Ecalle method using a choice of a well-behaved average, and different well-behaved averages may be used in this construction. Average-indep…

Let $Sk$ be the smallest set of functions $f:\mathbb R^{>0}\to\mathbb R^{>0}$ containing the constant function $1$ and the identity function $\mathbf{x}$, and such that if…

Let $\textbf{Hy}$ be the field of hyperseries and $\textbf{No}$ the field of surreal numbers. For each hyperseries $f(x)\in\textbf{Hy}$, consider its value…

Let $\textbf{Hy}$ be the intended field of hyperseries, equipped with field operations, differentiation, and composition. Let $\Phi$ be a partial function from $\textbf{Hy}$ into i…

Let $\mathbb{T}^{\operatorname{da}}$ denote the field of accelero-summable transseries, and let $H$ be a Hardy field. In real accelero-summation, use the organic average whenever s…

Let $\mathbb{L}$ be the class of log-atomic surreal numbers, and let $\mathbb{R}\langle\langle\mathbb{L}\rangle\rangle$ denote the associated field of exponential-logarithmic trans…

Let $\mathbb{L}$ be the class of log-atomic surreal numbers, namely those $\lambda\in\mathbf{No}$ such that every iterated logarithm $\log_n(\lambda)$ is of the form $\exp(\gamma)$…

Let $\mathbf{No}$ be Conway's field of surreal numbers, equipped with the exponential function of Kruskal and Gonshor. A compatible differential structure of Hardy-type means a der…

Let $\mathds{K}=\mathbb{R}((\Gamma))$ be a field of generalized series with a partial logarithm, and let $\mathbb{L}\subset\mathds{K}$ be a complete subfield containing $\Gamma$ an…

Let $\mathds{K}$ be a subfield of generalized series such that the surreal numbers $\mathbf{No}$ can be described as an exponential-logarithmic series field…

Let $\mathcal L^*$ be the language obtained by extending the natural language of ordered valued differential rings for $\mathbb{T}$ with a unary function symbol naming the specifie…

Let $\mathbb{T}$ be the differential field of transseries, and let an $H$-field with small derivation be an ordered differential field satisfying the relevant $H$-field axioms and…

Consider $\mathbb{T}$ as an ordered valued differential field. The $\mathbb{T}$-conjecture. The field $\mathbb{T}$ is model complete. Equivalently, in the language of differential…

Let $\mathbb{T}$ be the ordered valued differential field of transseries. Let $H$-fields be ordered differential fields equipped with the associated $H$-field structure, and let sm…