arXiv research

Geometrically proves WKB solutions of Schrödinger equations are resurgent.

Paper describes a new method for character varieties of surface groups.

In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the ${\cal N}=(2,2)$ 2d Landau-Ginzburg theory in models describing link embeddings in ${\mathbb{R}}^3$ to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert sp…

Study non-perturbative quantum geometry of string theories using finite difference equations and resurgence analysis.

Lectures detail field theory dynamics and exact WKB analysis.

QTAML models quantum tunneling errors for AI robustness.

Introduces sheaf quantization, a topological approach to geometric quantization.

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

Study of free particle's geometry and its perturbations using complex projective structures.

New estimates for Hitchin's equations at high energy.

We show that the Borel sums of the Voros symbols considered in the theory of exact WKB analysis arise naturally as Fock-Goncharov coordinates of framed $PGL_2(\mathbb{C})$-local systems on a marked bordered surface. Using this result, we show that these Borel sums can be meromorphically continued to any point of $\math…

Study on rank 2 Higgs bundles on 5-punctured sphere, proving $P=W$ conjecture in lowest degree.

Analyzes tunneling effects for Schrödinger operators on vector bundles.

The Fokker-Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker-Planck equation we describe a construction of exact …

Proves generic existence of spectral networks for many cases.

Analyzes how quadratic differential trajectories change with variation, proving a wall-crossing formula.

In this paper, we are interested in the location of conjugate points along a geodesic in the volumorphism group of a compact three-dimensional manifold without boundary (the configuration space of an ideal fluid). As shown in the author's previous work, these are typically pathological, i.e., they can occur in clusters…

A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$. We prove that the degree of a $q$-holonomic sequence is eventually a quadratic quasi-polynomial. Our proof uses differential Galois theory (adapting proofs regarding hol…

We consider BPS states in a large class of d=4, N=2 field theories, obtained by reducing six-dimensional (2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S^1 yields sigma models, whose target spaces are moduli spac…

Holomorphic curves found in compact quotients of SL(2,C).

In this paper we introduce efficient Monte Carlo estimators for the valuation of high-dimensional derivatives and their sensitivities (''Greeks''). These estimators are based on an analytical, usually approximative representation of the underlying density. We study approximative densities obtained by the WKB method. Th…

A quasiclassical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is not given by an exact 2-form. For this, the multidimensional WKB method in the form of Maslov canonical operator is applied. In this case, the canon…

We consider the eigenvalue equation for the Laplace-Beltrami operator acting on scalar functions on the non-compact Eguchi-Hanson space. The corresponding differential equation is reducible to a confluent Heun equation with Ince symbol [0,2,1_2]. We construct approximations for the eigenfunctions and their asymptotic s…

Based on our previous study [IS2] we develop fully the stationary scattering theory for the Schrodinger operator on a manifold possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends, possibly with unbounded and non-smooth obstacles. We develop the theory largel…

In the limit $\hbar\to 0$, we analyze a class of Schrödinger operators $H_\hbar = \hbar^2 L + \hbar W + V\cdot \mathrm{id}$ acting on sections of a vector bundle $\mathcal{Eh}$ over a Riemannian manifold $M$ where $L$ is a Laplace type operator, $W$ is an endomorphism field and the potential energy $V$ has a non-degene…

The Hitchin-Simpson equations are first-order non-linear equations for a pair consisting of a connection and a Higgs field. In this paper, we study the behavior of sequences of solutions to the Hitchin-Simpson equations on closed Kähler manifolds with unbounded $L^2$ norms of the Higgs fields. We prove a compactness re…

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 …

Paper derives closed-form solutions for CEV model using semiclassical approximation.

This is the first in a series of papers in which we study an efficient approximation scheme for solving the Hamilton-Jacobi-Bellman equation for multi-dimensional problems in stochastic control theory. The method is a combination of a WKB style asymptotic expansion of the value function, which reduces the second order …

In this paper we study a proposal of Nekrasov, Rosly and Shatashvili that describes the effective twisted superpotential obtained from a class S theory geometrically as a generating function in terms of certain complexified length-twist coordinates, and extend it to higher rank. First, we introduce a higher rank analog…

Let $(E,\overline{\partial}_E,θ)$ be a stable Higgs bundle of degree $0$ on a compact connected Riemann surface. Once we fix the flat metric $h_{\det(E)}$ on the determinant of $E$, we have the harmonic metrics $h_t$ $(t>0)$ for the stable Higgs bundles $(E,\overline{\partial}_E,tθ)$ such that $\det(h_t)=h_{\det(E)}$. …

The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by TTQ. Le and the author that such sequences are $q$-holonomic, that is, they satisfy linear $q$-difference equations with coefficients Laurent polynomials in $q$ and $q^n$. We show from first principles that $q$-holonomic sequence…

Optimizes financial decisions with illiquid assets using Kelly criterion.

We consider the $q$-nonabelianization map, which maps links $L$ in a 3-manifold $M$ to links $\widetilde{L}$ in a branched $N$-fold cover $\widetilde{M}$. In quantum field theory terms, $q$-nonabelianization is the UV-IR map relating two different sorts of defect: in the UV we have the six-dimensional $(2,0)$ superconf…

Two Alexander polynomials emerge in a Kashaev limit, resolving a paradox.

Efficient private matrix analysis algorithms for recent variants.

This paper introduces compositional data analysis for financial ratios, improving industry-level analysis.

In this dissertation, the main goal is visualisation of financial time series. We expect that visualisation of financial time series will be a useful auxiliary for technical analysis. Firstly, we review the technical analysis methods and test our trading rules, which are built by the essential concepts of technical ana…

Paper combines geometry and time-series analysis for spatiotemporal data.

In this paper the exact linear relation between the leading eigenvectors of the modularity matrix and the singular vectors of an uncentered data matrix is developed. Based on this analysis the concept of a modularity component is defined, and its properties are developed. It is shown that modularity component analysis …

This paper investigates to identify the requirement and the development of machine learning-based mobile big data analysis through discussing the insights of challenges in the mobile big data (MBD). Furthermore, it reviews the state-of-the-art applications of data analysis in the area of MBD. Firstly, we introduce the …

Interactive DR framework for comparing datasets.

Combines topological and geometric approaches to data analysis.

Proposes a multivariate regression model for better analysis of multiple datasets.

New method uses topological data analysis to study stock market crashes.

Analyzes stock trends and e-commerce user behavior using Twitter data.

Study analyzes Disney stock market performance using machine learning.

This study analyzes data science vocabulary changes over 13 years.