arXiv subjects

Nikita Nikolaev

Publications and source records attributed to Nikita Nikolaev.

10 recordsLinked to original sources

arXiv2026

Chewing gums, snakes and candle cakes

The aim of these lecture notes, based on lectures given by the second author at the CIME school in Cetraro, is to illustrate a range of ideas surrounding higher Teichmuller spaces of Riemann surfaces with marked boundaries through explicit and computationally tractable examples. After reviewing the classical Teichmuller space of hyperbolic Riemann surfaces with boundary and its combinatorial description in terms of Thurston shear coordinates on a fat-graph, we explain how the bordered cusped Teichmuller space arises as a confluent limit when two boundary components in the Riemann surface collide via the so-called chewing-gum move giving rise to a candle cake. We then revisit these constructions from the Fock-Goncharov perspective, explaining snake calculus for transport matrices in PSL_n(R) and explain how the chewing gum move is the inverse of amalgamation. Rather than focusing on formal proofs, our goal is to illustrate the underlying theorems and constructions in a concrete and intuitive way.

math.GT↗

arXiv2026

Decorated Local Systems and Character Varieties

The focus of this paper is the study of the moduli space of representations of fundamental groupoids of surfaces $Σ$ with boundaries with values in $G:=GL_n(\mathbb C)$. In absence of marked points on the boundary, this moduli space is realized in many equivalent ways: as the moduli space of linear local systems on $Σ$, as the moduli space of representations of the fundamental groupoid $Π_1 (Σ)$, as the space of monodromy data and as character variety. By adding marked points to the boundary of $Σ$ in order to capture irregular singularities, the Betti moduli space has been generalized in several ways by many authors. Although it is clear that these different approaches describe essentially the same spaces of mathematical objects, exactly how they fit together has not yet been established. Motivated by the broader programme of establishing an explicit and conceptually coherent relationship between the existing approaches to the study of the decorated Betti moduli space, in this paper, we develop a categorical framework that allows for a systematic definition of the \dfn{decorated Betti moduli spaces} space, in the presence of higher order poles, designed to specialize to the different points of view encountered in the literature.

math.AG↗

arXiv2025

Resurgence of Tritronquées Solutions of the Deformed Painlevé I Equation

We prove that the formal $\hbar$-power series solution of the deformed Painlevé I equation is resurgent, which means it is generically Borel summable and its Borel transform admits endless analytic continuation. In particular, we find that the Borel transform defines a global multivalued holomorphic function on a singular algebraic surface isomorphic to the Fermat quintic surface $x^5 + y^5 + z^5 = 0$ modulo an involution. This surface is an algebraic fibration over the complex plane of the differential equation with generic fibre a smooth quintic curve. Each fibre is equipped with a fivefold covering map over another complex plane (the Borel plane) with ten ramification points (the Borel singularities) spread equally over two branch points giving two opposite Stokes rays.

math.DG↗

arXiv2024

An Exact Perturbative Existence and Uniqueness Theorem

We investigate singularly perturbed nonlinear complex differential systems of the form $\hbar \partial_x f = F (x, \hbar, f)$ where $\hbar$ is a small complex perturbation parameter. Under a geometric assumption on the eigenvalues of the Jacobian matrix of $F$, we prove an Existence and Uniqueness Theorem for exact perturbative solutions; i.e., holomorphic solutions with prescribed perturbative expansions in $\hbar$. In fact, these solutions are the Borel resummation of the formal perturbative solutions.

math.CA↗

arXiv2024

Geometry and Resurgence of WKB Solutions of Schrödinger Equations

We prove that formal WKB solutions of Schrödinger equations on Riemann surfaces are resurgent. Specifically, they are Borel summable in almost all directions and their Borel transforms admit endless analytic continuation away from a discrete subset of singularities. Our approach is purely geometric, relying on understanding the global geometry of complex flows of meromorphic vector fields using techniques from holomorphic Lie groupoids and the geometry of spectral curves. This framework provides a fully geometric description of the Borel plane, Borel singularities, and the Stokes rays. In doing so, we introduce a geometric perspective on resurgence theory.

math.DG↗

arXiv2021

Gevrey Asymptotic Implicit Function Theorem

We prove an Asymptotic Implicit Function Theorem in the setting of Gevrey asymptotics with respect to a parameter. The unique implicitly defined solution admits a Gevrey asymptotic expansion and furthermore it is the Borel resummation of the corresponding implicitly defined formal power series solution. The main theorem can therefore be rephrased as an Implicit Function Theorem for Borel summable power series. As an application, we give a diagonal or Jordan decomposition for holomorphic matrices in Gevrey asymptotic families.

math.CV↗

arXiv2021

Triangularisation of Singularly Perturbed Logarithmic Differential Systems of Rank 2

We study singularly perturbed linear systems of rank two of ordinary differential equations of the form $\hbar x\partial_x ψ(x, \hbar) + A (x, \hbar) ψ(x, \hbar) = 0$, with a regular singularity at $x = 0$, and with a fixed asymptotic regularity in the perturbation parameter $\hbar$ of Gevrey type in a fixed sector. We show that such systems can be put into an upper-triangular form by means of holomorphic gauge transformations which are also Gevrey in the perturbation parameter $\hbar$ in the same sector. We use this result to construct a family in $\hbar$ of Levelt filtrations which specialise to the usual Levelt filtration for every fixed nonzero value of $\hbar$; this family of filtrations recovers in the $\hbar \to 0$ limit the eigen-decomposition for the $\hbar$-leading-order of the matrix $A (x, \hbar)$, and also recovers in the $x \to 0$ limit the eigen-decomposition of the residue matrix $A (0, \hbar)$.

math.CA↗

arXiv2021

Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs

We prove an existence and uniqueness theorem for exact WKB solutions of general singularly perturbed linear second-order ODEs in the complex domain. These include the one-dimensional time-independent complex Schrödinger equation. Notably, our results are valid both in the case of generic WKB trajectories as well as closed WKB trajectories. We also explain in what sense exact and formal WKB solutions form a basis. As a corollary of the proof, we establish the Borel summability of formal WKB solutions for a large class of problems, and derive an explicit formula for the Borel transform.

math.AP↗

arXiv2021

Exact Solutions for the Singularly Perturbed Riccati Equation and Exact WKB Analysis

The singularly perturbed Riccati equation is the first-order nonlinear ODE $\hbar \partial_x f = af^2 + bf + c$ in the complex domain where $\hbar$ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as $\hbar \to 0$ in a halfplane. These exact solutions are constructed using the Borel-Laplace method; i.e., they are Borel summations of the formal divergent $\hbar$-power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrödinger equation with a rational potential.

math.CA↗

arXiv2021

Abelianisation of Logarithmic $\mathfrak{sl}_2$-Connections

We prove a functorial correspondence between a category of logarithmic $\mathfrak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $π: Σ\to X$. The proof is by constructing a pair of inverse functors $π^{\text{ab}}, π_{\text{ab}}$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $π_\ast$.

math.AG↗