About

Ovidiu Costin is a professor in the Department of Mathematics at The Ohio State University. He earned his PhD from Rutgers University in 1995, specializing in applied mathematics, complex analysis, differential and partial differential equations, mathematical physics, real analysis, and related fields. His areas of expertise include analysis, ordinary and partial differential equations, mathematical physics, asymptotics, Borel summability, analyzable functions, integrable systems, complex dynamics, spectral theory, special functions, and random matrices. His research focuses on these mathematical disciplines, contributing to the understanding of complex systems and mathematical physics through his work. As a faculty member, he is involved in teaching, advising, and advancing mathematical research within his areas of specialization at Ohio State University.

Research topics

• Mathematics
• Mathematical analysis
• Pure mathematics
• Physics
• Applied mathematics

Selected publications

• Pade Approximants for Geodesy

  arXiv (Cornell University) · 2026-05-05

  preprintOpen access1st authorCorresponding

  In this note we analyze the use of Padé approximants for downward continuation beyond the radius of convergence of spherical harmonic expansions (SHEs), and for identifying the complex singularities of the gravitational potential. SHEs are, in essence, expansions in 1/r, i.e., expansions about the point at infinity. Their domain of convergence is generically the exterior of the Brillouin sphere. However, for synthetic models with analytic topography and density the region of convergence may be larger, with the deviation decreasing as the structural complexity of the planet increases.

  PublisherDOI

• Pade Approximants for Geodesy

  ArXiv.org · 2026-05-05

  articleOpen access1st authorCorresponding

  In this note we analyze the use of Padé approximants for downward continuation beyond the radius of convergence of spherical harmonic expansions (SHEs), and for identifying the complex singularities of the gravitational potential. SHEs are, in essence, expansions in 1/r, i.e., expansions about the point at infinity. Their domain of convergence is generically the exterior of the Brillouin sphere. However, for synthetic models with analytic topography and density the region of convergence may be larger, with the deviation decreasing as the structural complexity of the planet increases.

  PublisherOA PDF

• On Uniqueness of Mock Theta Functions

  arXiv (Cornell University) · 2026-04-21

  preprintOpen access1st authorCorresponding

  We develop a resurgent approach to the problem of unique continuation of mock theta functions across their natural boundary. The starting point is the representation of the associated Mordell-Appell integrals as Laplace transforms of resurgent functions, which serve as the primary analytic objects. By rotating the Laplace contour by $π$, i.e. onto the Stokes line, one obtains, in all known cases, the mock-modular relations between the Mordell-Appell integrals and the corresponding unary series in $\hat q=e^{-πi τ}$ and $\hat q_1=e^{-πi (-1/τ)}$. We then prove that these relations admit a unique solution on the $q$-side, expressed in terms of $q=e^{πi τ}$ and $q_1=e^{πi (-1/τ)}$, with coefficients determined by the corresponding Mordell-Appell integrals. This yields a canonical continuation across the natural boundary, given by a resurgent extension of the classical principle of permanence of relations, and singles out a distinguished family of mock theta functions in each group. We present a complete analysis for the order 3 and 5 cases (mf3 and mf5). The method extends naturally to higher orders; a general theory will appear in a separate paper.

  PublisherDOI

• ceff from resurgence at the Stokes line

  Journal of High Energy Physics · 2026-02-05

  articleOpen access

  A bstract In recent papers [1, 2], a new method to cross the natural boundary has been proposed, and applied to Mordell-Borel integrals arising in the study of Chern-Simons theory, based on decompositions into resurgent cyclic orbits . Resurgent analysis on the Stokes line leads to a unique transseries decomposition in terms of unary false theta functions, which can be continued across the natural boundary to produce dual q -series whose integer-valued coefficients enumerate BPS states. This constitutes a deeper new manifestation of resurgence in quantum field theoretic path integrals. In this paper we show that the algebraic structure of the resurgent cyclic orbits , combined with just the leading term of the q -series, completely determines the large order rate of growth of the dual q -series coefficients. The essential exponent of this asymptotic growth has a Cardy-like interpretation [12] of an effective central charge in a 3 dimensional quantum field theory with $$ \mathcal{N}=2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> supersymmetry related to the Chern-Simons theory through the 3 d -3 d correspondence.

  PublisherOA PDFDOI

• On Uniqueness of Mock Theta Functions

  arXiv (Cornell University) · 2026-04-21

  articleOpen access1st authorCorresponding

  We develop a resurgent approach to the problem of unique continuation of mock theta functions across their natural boundary. The starting point is the representation of the associated Mordell-Appell integrals as Laplace transforms of resurgent functions, which serve as the primary analytic objects. By rotating the Laplace contour by $π$, i.e. onto the Stokes line, one obtains, in all known cases, the mock-modular relations between the Mordell-Appell integrals and the corresponding unary series in $\hat q=e^{-πi τ}$ and $\hat q_1=e^{-πi (-1/τ)}$. We then prove that these relations admit a unique solution on the $q$-side, expressed in terms of $q=e^{πi τ}$ and $q_1=e^{πi (-1/τ)}$, with coefficients determined by the corresponding Mordell-Appell integrals. This yields a canonical continuation across the natural boundary, given by a resurgent extension of the classical principle of permanence of relations, and singles out a distinguished family of mock theta functions in each group. We present a complete analysis for the order 3 and 5 cases (mf3 and mf5). The method extends naturally to higher orders; a general theory will appear in a separate paper.

  PublisherOA PDF

• Correction to: The Likely Maximum Size of Twin Subtrees in a Large Random Tree

  Annals of Combinatorics · 2025-05-17

  articleOpen access
  PublisherOA PDFDOI

• Orientation reversal and the Chern-Simons natural boundary

  Journal of High Energy Physics · 2025-08-20 · 2 citations

  articleOpen access

  A bstract We show that the fundamental property of preservation of relations, underlying resurgent analysis, provides a new perspective on crossing a natural boundary, an important general problem in theoretical and mathematical physics. This reveals a deeper rigidity aspect of resurgence in a quantum field theory path integral. The physical context here is the non-perturbative completion of complex Chern-Simons theory that associates to a 3-manifold a collection of q -series invariants labeled by Spin c structures, for which crossing the natural boundary corresponds to orientation reversal of the 3-manifold. Our new resurgent perspective leads to a practical numerical algorithm that generates q -series which are dual to unary q -series composed of false theta functions. Until recently, these duals were only known in a limited number of cases, essentially based on Ramanujan’s mock theta functions, and the common belief was that the duals might not even exist in the general case. Resurgence analysis identifies as primary objects Mordell integrals: up to changes of variables, they are Laplace transforms of resurgent functions. Their unique Borel summed transseries decomposition on either side of the Stokes line is simply the unique decomposition into real and imaginary parts. In turn, the latter are combinations of unary q -series in terms of q and its modular counterpart $$ \overset{\sim }{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> , and are resurgent by construction. The Mordell integral is analytic across the natural boundary of the q and $$ \overset{\sim }{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> series, and uniqueness of a similar decomposition which preserves algebraic relations on the other side of the boundary defines the unique boundary crossing of the q series. We demonstrate that this continuation can be efficiently implemented numerically. In the cases where unique mock modular identities are known, they are found by this numerical procedure, but the procedure can go well beyond the known list of identities. A particularly interesting feature of the resurgent approach is that it reveals new aspects, and is very different from other known approaches based on indefinite theta series, Appell-Lerch sums, and representation theory of logarithmic vertex operator algebras.

  PublisherOA PDFDOI

• Gaussian Fluctuation in Random Matrices

  WORLD SCIENTIFIC eBooks · 2025-08-01

  book-chapter1st authorCorresponding
  PublisherDOI

• Long time evolution of the Hénon-Heiles system for small energy

  arXiv (Cornell University) · 2024-11-25

  preprintOpen access1st authorCorresponding

  The Hénon-Heiles system, initially introduced as a simplified model of galactic dynamics, has become a paradigmatic example in the study of nonlinear systems. Despite its simplicity, it exhibits remarkably rich dynamical behavior, including the interplay between regular and chaotic orbital dynamics, resonances, and stochastic regions in phase space, which have inspired extensive research in nonlinear dynamics. In this work, we investigate the system's solutions at small energy levels, deriving asymptotic constants of motion that remain valid over remarkably long timescales -- far exceeding the range of validity of conventional perturbation techniques. Our approach leverages the system's inherent two-scale dynamics, employing a novel analytical framework to uncover these long-lived invariants. The derived formulas exhibit excellent agreement with numerical simulations, providing a deeper understanding of the system's long-term behavior.

  PublisherOA PDFDOI

• The Likely Maximum Size of Twin Subtrees in a Large Random Tree

  Annals of Combinatorics · 2024-07-27

  article
  PublisherDOI

Recent grants

• Non-Perturbative Analysis of Physical and Mathematical Models

  NSF · $284k · 2022–2026

• Structure of Solutions of the Time Dependent Schroedinger Equation and of Certain Classes of Evolution Nonlinear PDEs

  NSF · $139k · 2006–2011

• Borel Summation and Applications to PDEs

  NSF · $280k · 2008–2012

• Development of Non-Perturbative Approaches to Partial Differential Equations Arising in Physical Applications

  NSF · $470k · 2015–2020

• Collaborative Research: Nonlinear PDE's and Integro-Differential Equations in the Complex Plane

  NSF · $47k · 2005–2008

Frequent coauthors

• Rodica D. Costin

  The Ohio State University

  37 shared

• Joel L. Lebowitz

  33 shared

• S. Tanveer

  The Ohio State University

  33 shared

• Min Huang

  China University of Geosciences

  26 shared

• Martin D. Kruskal

  15 shared

• Stavros Garoufalidis

  Southern University of Science and Technology

  14 shared

• Louis Dupaigne

  Université Claude Bernard Lyon 1

  13 shared

• A. Rokhlenko

  Rutgers, The State University of New Jersey

  10 shared

Awards & honors

• Graduate Teaching Awards