arXiv research

Analyzes tt*-structures from $ADE$-type Stokes data.

We compute Stokes matrices and monodromy for the quantum cohomology of projective spaces. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves.

Paper derives explicit expression of Alekseev-Meinrenken diffeomorphism.

Study extends symmetries of sphere points to surface mapping classes.

In the first part of the paper, we solve the boundary and monodromy problems for the isomonodromy equation of the $n\times n$ meromorphic linear system of ordinary differential equations with Poncaré rank $1$. In particular, we derive an explicit expression of the Stokes matrices of the linear system, via the boundary …

Study of WKB asymptotics of Stokes matrices and spectral curves, proving rhombus inequalities.

Constructs positive energy representations from Toda equations Stokes data.

Algorithm describes Fourier transform of Stokes data at infinity.

The abstract discusses resurgent functions in quantum knot invariants.

Study infinite-dimensional Toda manifold at irregular singularity, revealing non-uniqueness of formal solutions.

Complex Chern-Simons theory reveals peacock patterns in perturbative series.

We propose a Lie-theoretic definition of the tt*-Toda equations for any complex simple Lie algebra $\mathfrak{g}$, based on the concept of topological-antitopological fusion which was introduced by Cecotti and Vafa. Our main result concerns the Stokes data of a certain meromorphic connection, whose isomonodromic deform…

Stokes' theorem's boundary maximizes entropy.

Several intrinsic topological ways to encode connections on vector bundles on smooth complex algebraic curves will be described. In particular the notion of {\em Stokes decompositions} will be formalised, as a convenient intermediate category between the Stokes filtrations and the Stokes local systems/wild monodromy re…

We study moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint (generalising the At…

We point out, and draw some consequences of, the fact that the Poisson Lie group G* dual to G=GL_n(C) (with its standard complex Poisson structure) may be identified with a certain moduli space of meromorphic connections on the unit disc having an irregular singularity at the origin. The Riemann-Hilbert map for such co…

Lean 4 formalizes Stokes' theorem for smooth singular cubes.

Paper studies periodic solutions to Navier-Stokes equations on hyperbolic manifolds.

Study on Navier-Stokes equations on non-compact manifolds, proving existence and decay of solutions.

Paper compares five surface Navier-Stokes derivations and finds some are equivalent.

Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.

Stokes equations help uniquely identify manifold metrics from boundary data.

Paper connects Stokes phenomena to quantum groups and Poisson-Lie groups.

Categorifies Stokes coefficients in Chern-Simons theory models.

We consider the concept of Stokes-Dirac structures in boundary control theory proposed by van der Schaft and Maschke. We introduce Poisson reduction in this context and show how Stokes-Dirac structures can be derived through symmetry reduction from a canonical Dirac structure on the unreduced phase space. In this way, …

Stokes theorem holds for Lipschitz forms on a smooth manifold.

The paper validates Stokes' theorem for differential subcomplexes in positively graded Lie groups.

The author presents the generalized Stokes theorem for R-linear forms on Lie algebroids (which can be non-local). We apply the Stokes formula on forms to prove that two homotopic homomorphisms of Lie algebroids implies the existence of a chain operator joining their pullback operators.

Paper connects Painlevé VI equation to irregular systems, solving monodromy data.

Study well-posedness of generalized Stokes operator on cylindrical domains.

Proves invertibility of layer potentials for generalized Stokes operators on smooth domains.

In this paper, we show the existence of real-analytic stationary Navier-Stokes flows with isotropic streamlines in all latitudes in some simply-connected flow region on a rotating round sphere. We also exclude the possibility of having a Poiseuille's flow profile to be one of these stationary Navier-Stokes flows with i…

For a given bounded domain $Ω\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the heat trace associated with the Stokes operator as $t\to 0^+$. These coefficients (i.e., heat invariants) provide precise information for the volume of the domain $…

We present type $θ$ Stokes' theorem for type $θ$ $k$-chains which extends the fundamental theorem of calculus in higher dimensions.

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.

By proving graph theoretical versions of Green-Stokes, Gauss-Bonnet and Poincare-Hopf, core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting. In this pedagogical exposition we present the main proofs on a single page and add illustrations. While discrete Stokes is is old, the …

The Navier-Stokes equations on certain manifolds can perform universal computation.

Stokes-Dirac structures are infinite-dimensional Dirac structures defined in terms of differential forms on a smooth manifold with boundary. These Dirac structures lay down a geometric framework for the formulation of Hamiltonian systems with a nonzero boundary energy flow. Simplicial triangulation of the underlaying m…

This paper clarifies a global structure of Stokes-Dirac structures used for describing interconnected port-Hamiltonian systems defined on manifolds with non-trivial topology under consistent boundary condition.

The Wodzicki residue and the cut-off integral extend to classical symbol-valued forms. We show that they obey a Stokes' type property and that the extended Wodzicki residue can be interpreted as a complex residue like the ordinary one. In the case of cut-off integrals, Stokes' property (i.e. vanishing on exact forms) o…

What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic repr…

The purpose of this paper is to study the validity of Stokes' Theorem for singular submanifolds and differential forms with singularities in Euclidean space. The results are presented in the context of Lebesgue Integration, but their proofs involve techniques from gauge integration in the spirit of R.~Henstock, J.~Kurz…

For a projective algebraic variety $V$ with isolated singularities, endowed with a metric induced from an embedding, we consider the analysis of the natural partial differential operators on the regular part of $V$. We show that, in the complex case, the Laplacians of the de Rham and Dolbeault complexes are discrete op…

Proves generic existence of spectral networks for many cases.

In "Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data" (arxiv:1209.2045) we described all smooth solutions of the two-function tt*-Toda equations in terms of asymptotic data, holomorphic data, and monodromy data. In this supplementary article we focus on the holomorphic data and its interpret…

Optimizes shapes in uncertain Navier-Stokes flow problems.

Study reveals how to determine area and curvature from fluid flow resonances.