85 problems

Tensor-product conjecture. The full energy trans-series can be written as a tensor product of the minimal trans-series and the medium, namely

The relativistic Boltzmann equation in the relaxation-time model describes non-equilibrium dynamics, and its hydrodynamic gradient expansion is generally asymptotic rather than con…

In quantum Chern–Simons theory, the Kashaev invariant of a knot and the WRT invariant of a $3$-manifold are examples of quantum invariants with asymptotic expansions at roots of un…

Let $I$ be a homogeneous ideal, and let $I^{[k]}$ denote the generalized power defined in the cited source. Let $\rho(J)$ denote the resurgence of an ideal $J$. Unbounded resurgenc…

Let the trans-series be organized in a resurgence triangle whose rows are indexed by instanton number and whose columns are indexed by independent topological-$\Theta$ sectors. Eac…

Consider the Fokker–Planck Hamiltonian … with … Its energy eigenvalues are denoted by $E_n(g)$, with $n=0$ the ground state and $n\geq 1$ excited states. Generalized Bohr–Sommerfel…

Let $M$ be a matroid and let $I_{\Delta(M)}$ be its Stanley–Reisner ideal. Write $\widehat{\rho}(I_{\Delta(M)})$ for its asymptotic resurgence, $\omega(I_{\Delta(M)})$ for the larg…

Stokes–BPS correspondence. The Stokes constants of the topological string free energy can be identified with the BPS invariants of the associated four-dimensional theory.

Integral Stokes coefficient conjecture. Each $\gamma_i$ determines such a normalized formal heat solution, and there exists a coefficient $S_{ij}^{+}\in\mathbb Z$ such that

Picard–Lefschetz/Alien correspondence. The pointed Alien derivative of $\widetilde K_0$ at $\omega$ is

Kontsevich's conjecture. Singularities of the Borel transform of $K(\tau;a,b)$ correspond to relative critical values of the complexified energy functional…

Stokes-constant conjecture. The Stokes constant associated with the singularity … gammainGamma$.

In kinetic theory, consider the factorially divergent gradient series for the hydrodynamic expansion and its Borel resummation. Borel-resummation conjecture. Hydrodynamics can be s…

The localized expressions in the discussion define the perturbative and non-perturbative contributions associated with error functions, with the relevant cancellation leaving only…

Resurgence conjecture. For every analytic section of $E$ and each $\gamma\in\Gamma$, one can naturally associate a resurgent series in the standard coordinate $z$ on $\mathbb{C}$.

Let $Z(t,\nu,\hbar)$ be the perturbative partition function for the Painlevé I system, and let $\mathfrak S$ denote the Stokes automorphism associated with the $B$-periods. Write…

Let $M$ be a closed oriented 3-manifold, and consider analytically continued Chern–Simons theory with complex level $k$. The relevant flat connections include flat $\mathrm{SU}(2)$…

Conjecture 1. For large $k$, there are real constants $(C,R)$ such that

Let $M$ be an oriented three-dimensional manifold and let $G=\operatorname{SL}(2,\mathbb{C})$. The moduli space of flat $G$-connections on $M$ may have infinite asymptotic ends, ar…

Let $p$ be a positive integer, let $a$ be an index for the Mordell–Borel integrals $JS_{(p,a)}(t)$ and $JC_{(p,a)}(t)$, and let $q$ and $\widetilde{q}$ be the corresponding variabl…

The instanton amplitudes are associated with non-perturbative corrections to the perturbative partition function of topological string theory, and they are functionals of that part…

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…

Let $G(\Lambda,a)$ be the formal series for the two-point function of an asymptotically free quantum field theory, where $\Lambda$ is the kinematical parameter. Suppose its $\beta$…

Let $N$ be the modulus and $k$ an index for the $q$-Pochhammer symbols $g_{k,N}$ and $f_{k,N}$, with asymptotic expansion $\tilde{g}_{k,N}(y)$ as in the source. The Borel--Laplace…

Let $p$ be an odd integer of the form $p=6k\pm1$, with $k\in\mathbb Z$, and let $K_p^{(j)}(t)$ and $L_p^{(j)}(t)$ be the linear combinations of Mordell–Borel integrals defined in t…