PIICQ workshop "Excursions in Integrability"

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Monday, May 23, 20229:30 AM Matrix valued orthogonality and random tilings - Arno Kuijlaars (Katholieke Universiteit Leuven)Matrix valued orthogonality and random tilings
- Arno Kuijlaars (Katholieke Universiteit Leuven)

9:30 AM - 10:15 AMRoom: 128 Matrix valued orthogonal polynomials play a role in random tiling models with periodic weightings. The talk will be focused on lozenge tilings of a hexagon, and it will be shown that the matrix valued orthogonality can be related to orthogonality for meromorphic functions on a Riemann surface. The higher genus cases are of particular interest since these are believed to correspond to random tiling models with three different phases in the large size limit.10:15 AM Biased doubly periodic Aztec diamond and an elliptic curve - Maurice Duits (KTH)Biased doubly periodic Aztec diamond and an elliptic curve- Maurice Duits (KTH)

10:15 AM - 11:00 AMRoom: ZOOM ROOM I will report on a recent joint work with Alexei Borodin on the biased doubly periodic Aztec diamond. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of a linear flow on an elliptic curve. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. I will discuss how, in these periodic cases, one can compute the local correlations in the smooth disordered (or gaseous) region.11:00 AM Coffee BreakCoffee Break11:00 AM - 11:30 AMRoom: 12811:30 AM Asymptotic Analysis of the Interaction Between a Soliton and a Regular Gas of Solitons (a.k.a. Gulliver and the Lilliputians) - Manuela Girotti (Saint Mary's University, Mila Institute)Asymptotic Analysis of the Interaction Between a Soliton and a Regular Gas of Solitons (a.k.a. Gulliver and the Lilliputians)- Manuela Girotti (Saint Mary's University, Mila Institute)

11:30 AM - 12:15 PMRoom: 128 We analyze the case of a (dense) mKdV soliton gas and its large time behaviour in the presence of a single tracer soliton. The solution, which can be expressed in terms of Fredholm determinants, can be decomposed as the sum of the background gas solution (an elliptic wave), plus a soliton solution: the individual expressions are however quite convoluted due to the interaction dynamics. Additionally, we are able to derive the kinetic velocity equations and the local phase shift of the gas after the passage of the soliton, and we can trace the location of the soliton peak as the dynamics evolves. This is a joint work with T. Grava (SISSA, Bristol), R. Jenkins (UCF), K. McLaughlin (CSU) and A. Minakov (U. Karlova).12:15 PM The complex elliptic Ginibre ensemble at weak non-Hermiticity - Thomas Bothner (University of Bristol)The complex elliptic Ginibre ensemble at weak non-Hermiticity- Thomas Bothner (University of Bristol)

12:15 PM - 1:00 PMRoom: ZOOM ROOM In this talk we will focus on the complex elliptic Ginibre ensemble (eGinUE) and analyze the statistical behavior of its eigenvalues in a suitable scaling limit, known as the weak nonHermiticity limit. In this limit the asymmetry parameter in the model scales with the matrix dimension and the so obtained 2D limiting point processes generalize the well-known sine and Airy processes from the Gaussian unitary ensemble. Using integro-differential Painlevé transcendents we will show how the gap functions of the 2D limiting point processes can be evaluated in closed form and how Riemann-Hilbert techniques can subsequently yield precise asymptotic information for the same functions. Based on current joint work with Alex Little.1:00 PM Lunch BreakLunch Break1:00 PM - 2:30 PMRoom: 1282:30 PM On the longtime behavior of solutions of the KdV equation on the line - Percy Deift (NYU Courant)On the longtime behavior of solutions of the KdV equation on the line- Percy Deift (NYU Courant)

2:30 PM - 3:15 PMRoom: ZOOM ROOM The speaker will describe recent work on this long-standing problem and the issues involved in their resolution. This is joint work with Ken McLaughlin and Thomas Kriecherbauer and incorporates works with people over many years, going back to 1994 with Xin Zhou and Stephanos Venakides, 2000 with Irina Nenciu and also some useful discussions with Deniz Bilman3:15 PM On the longtime behavior of solutions of the KdV equation on the line: use of Riemann-Hilbert and d-bar methods for the analysis - Ken McLaughlin (Colorado State University)On the longtime behavior of solutions of the KdV equation on the line: use of Riemann-Hilbert and d-bar methods for the analysis- Ken McLaughlin (Colorado State University)

3:15 PM - 4:00 PMRoom: 128 This is intended to be part 2 of the previous talk (of Percy Deift), and the collaborations are as described by Percy in his abstract . The presentation was crafted together with Thomas Kriecherbauer. By way of introduction, we will consider the stationary phase analysis of an integral, when the steepest descent method doesn't immediately work. The basic d-bar operator will play a useful rule. We will then carry out some analysis of the Riemann-Hilbert problem for the KdV equation in the longtime limit, explaining the use of the d-bar method.4:00 PM Integrability Of Integro-Differential Painlevé Equations - Mattia CafassoIntegrability Of Integro-Differential Painlevé Equations- Mattia Cafasso

4:00 PM - 4:45 PMRoom: 1284:45 PM4:45 PM - 5:45 PMRoom: 128 -
Tuesday, May 24, 20229:30 AM Generalized Gibbs states for the Calogero fluid - Herbert Spohn (TU München)Generalized Gibbs states for the Calogero fluid
- Herbert Spohn (TU München)

9:30 AM - 10:15 AMRoom: 128 The Calogero fluid is an integrable many-particle system with interaction potential 1/sinh^2. I will explain how to compute the generalized free energy and the associated density of states of the Lax matrix. As a novel method, scattering coordinates will be used. Also the classical version of the Bethe equations will be pointed out.10:15 AM From the Ablowitz-Ladik lattice to the Circular $\beta$-ensemble - Guido Mazzuca (sissa)From the Ablowitz-Ladik lattice to the Circular $\beta$-ensemble- Guido Mazzuca (sissa)

10:15 AM - 11:00 AMRoom: 128 The main focus of this talk is the interplay between the theory of integrable systems, and random matrix theory. Specifically, I consider the Ablowitz-Ladik lattice, which is an integrable system, I introduce the Generalized Gibbs ensemble for this lattice, and I relate it with a random matrix model, the Circular $\beta$ ensemble. This connection allows me to compute explicitly the density of states for the Ablowitz-Ladik lattice in terms of the one of the random matrix ensemble. This talk is mainly based on my recent paper with Tamara Grava, 2107.02303, and the one with Ronan Memin, 2201.0342911:00 AM Coffee BreakCoffee Break11:00 AM - 11:30 AMRoom: 12811:30 AM Marking and conditioning of determinantal point processes - Tom Claeys (UCLouvain)Marking and conditioning of determinantal point processes- Tom Claeys (UCLouvain)

11:30 AM - 12:15 PMRoom: 128 The Its-Izergin-Korepin-Slavnov procedure is a method that allows to express Fredholm determinants of integrable kernel operators in terms of Riemann-Hilbert method, and hence paves the road for asymptotic analysis and for the study of underlying integrable systems. I will explain that this procedure allows more generally to characterize the correlation kernels of certain conditional determinantal point processes. I will show how these conditional ensembles are connected with a remarkable rigidity result, with tail distributions of the KPZ equation, and with integrable PDEs. The talk will be based on joint work with Gabriel Glesner.12:15 PM Determinantal point processes: quasi-symmetries and interpolation - Alexander BufetovDeterminantal point processes: quasi-symmetries and interpolation- Alexander Bufetov

12:15 PM - 1:00 PMRoom: 128 For the sine-process, it is proved that almost every realization with one particle removed is a complete and minimal set for the Paley-Wiener space, whereas if two particles are removed, then one obtains a zero set for the Paley-Wiener space. In joint work with Qiu and Shamov, it is proved that almost every realization of a determinantal point process is a uniqueness set for the underlying Hilbert space. To every realization of the sine-process we assign a random entire function, an infinite product with zeros at the particles of our configuration, the scaling limit of the ratio of two values of the characteristic polynomial of a random matrix. In order to study the behaviour of the logarithm of this product, we introduct a Gaussian field under which the variance of the random variable assigned to a given function is proportional to the square of its Sobolev 1/2-norm. More specifically now, we consider the Gaussian process, indexed by point in the upper half-plane, of Cauchy transforms of our configuration, as well as the process of anti-derivatives of the Cauchy transforms: precisely, the differences of the logarithms occurring in the logarithm of our random entire function. The key rôle is played by the remark that the Gaussian process formed by the antiderivatives of the Cauchy transforms is invariant under the Lobachevskian isometries of the upper half-plane. The argument then proceeds by taking the scaling limit in the Borodin-Okounkov-Geronimo-Case formula and obtaining an analogue of the Johansson change of variable formula for the sine-process, using the quasi-invariance of the sine-process under the group of diffeomorphisms with compact support. In joint work with Qiu, the Patterson-Sullivan construction is used to interpolate Bergman functions from a realization of the determinantal point process with the Bergman kernel, in other words, by the Peres-Virág theorem, the zero set of a random series with independent complex Gaussian entries. The invariance of the zero set under the isometries of the Lobachevsky plane again plays a key rôle.1:00 PM Lunch BreakLunch Break1:00 PM - 2:30 PMRoom: 1282:30 PM Integrable differential equations for KPZ fixed point with narrow-wedge initial condition - Jinho Baik (University of Michigan)Integrable differential equations for KPZ fixed point with narrow-wedge initial condition- Jinho Baik (University of Michigan)

2:30 PM - 3:15 PMRoom: ZOOM ROOM The KPZ fixed point is a two-dimensional random field that is the conjectured limit of the height functions of the KPZ universality class for random growth models. The one-point distribution of the KPZ fixed point is the Tracy-Widom distribution which is related to the Painlevel II equation. The equal-time, multi-position distributions are also known to be related to integrable differential equations. We will discuss integrable differential equations for multi-time distributions. We also discuss similar results for the periodic KPZ fixed point.3:15 PM The Maxwell-Bloch system in the sharp-line limit - Peter Miller (University of Michigan)The Maxwell-Bloch system in the sharp-line limit- Peter Miller (University of Michigan)

3:15 PM - 4:00 PMRoom: ZOOM ROOM We study the characteristic Cauchy problem for the Maxwell-Bloch system that describes the interaction of an optical pulse with an active quantum medium. It is well known that in the sharp-line limit that the atoms in the medium are not Doppler-shifted in frequency, this system can be embedded in the integrable hierarchy of the nonselfadjoint Zakharov-Shabat spectral problem. However, it is also known that there are certain difficulties with formulating and using the inverse-scattering transform based on this spectral problem in the usual way. We construct a Riemann-Hilbert problem that returns the unique causal solution of the Cauchy problem and use it to explain features of solutions such as the stimulated decay by a suitable optical pulse of an unstable medium to its stable state and the spontaneous generation of a dispersive tail of the optical pulse with positive time that ruins absolute integrability that would be needed for the standard inverse-scattering transform to make sense. This tail is related to a specific self-similar solution of the Maxwell-Bloch system that in turn is connected with a concrete special solution of the Painlevé-III equation that has become important in several recent application problems for the focusing nonlinear Schrödinger equation. This is joint work with Sitai Li (Xiamen).4:00 PM Open TASEP and the Ribosome Flow Model - Thomas KriecherbauerOpen TASEP and the Ribosome Flow Model- Thomas Kriecherbauer

4:00 PM - 4:45 PMRoom: 128 -
Wednesday, May 25, 20229:30 AM Director's speechDirector's speech9:30 AM - 9:40 AMRoom: 1289:40 AM Laudatio by Danilo LewanskiLaudatio by Danilo Lewanski9:40 AM - 9:45 AMRoom: 1289:45 AM Curves on Hyperkähler varieties and modular forms - Greog Oberdieck (Universität Bonn)Curves on Hyperkähler varieties and modular forms
- Greog Oberdieck (Universität Bonn)

9:45 AM - 10:30 AMRoom: 128 The enumerative geometry of curves on algebraic surfaces is a fascinating subject with roots in the 19-th century. For example, Cayley and Salmon in 1849 proved the remarkable fact that every cubic surface in P3 contains precisely 27 lines. A more modern highlight is the Yau-Zaslow formula, proven by Beauville and Bryan-Leung in the 1990s, which says that the counts of rational curves on K3 surfaces are the Fourier coefficients of the reciprocal of the discriminant modular form. The result links together the geometry of K3 surfaces, modular forms and insights from string theory in a beautiful way. In this talk, I will review these results and then discuss an analogue of K3 surfaces in higher dimension, the mysterious class of hyperkähler varieties. For the most prominent deformation family of these varieties, the K3[n]-type hyperkähler varieties, I will discuss how counting of curves is linked to Jacobi forms. In particular, we obtain new constructions of vector-valued Jacobi forms and gain new tools to attack difficult enumerative problems about hyperkähler varieties and CHL Calabi-Yau threefolds.10:30 AM Laudatio by Alessandro TanziniLaudatio by Alessandro Tanzini10:30 AM - 10:35 AMRoom: 12810:35 AM Kyiv formula, its generalizations and applications - Pavlo GavrylenkoKyiv formula, its generalizations and applications- Pavlo Gavrylenko

10:35 AM - 11:20 AMRoom: 128 I'm going to give an overview in the growing field that is developing after the formula by Gamayun, Iorgov, Lisovyy that expresses the general solution of Painlevé VI in terms of conformal blocks (Kyiv formula). This field now includes higher-rank generalizations, multi-point and toric generalizations, q-deformations, quantum deformations, expansions in the irregular region. I will tell some short stories about these generalizations done in different combinations, and also about related problems where the Kyiv formula can be used.11:20 AM Coffee BreakCoffee Break11:20 AM - 11:50 AMRoom: 12811:50 AM Laudatio by Herbert SpohnLaudatio by Herbert Spohn11:50 AM - 11:55 AMRoom: 12811:55 AM Large Genus Asymptotics for Intersection Numbers - Amol Aggarwal (Columbia University and Institute for Advanced Study)Large Genus Asymptotics for Intersection Numbers- Amol Aggarwal (Columbia University and Institute for Advanced Study)

11:55 AM - 12:40 PMRoom: 128 In this talk we explain results on the large genus asymptotics for intersection numbers between $\psi$-classes on the moduli space of curves. By combining this result with a combinatorial analysis of recently proven formulas of Delecroix-Goujard-Zograf-Zorich, we further describe some features about how random flat surfaces of large genus look. The proof uses a comparison between the recursive relations (Virasoro constraints) that uniquely determine them with the jump probabilities of a certain asymmetric simple random walk.12:40 PM Closing ceremony with Gianluigi Rozza (Director of Mathematics Area)Closing ceremony with Gianluigi Rozza (Director of Mathematics Area)12:40 PM - 12:50 PMRoom: 12812:50 PM Lunch BreakLunch Break12:50 PM - 2:30 PMRoom: 1282:30 PM Neural network, spin glasses and random matrices - Francesco Mezzadri (University of Bristol)Neural network, spin glasses and random matrices- Francesco Mezzadri (University of Bristol)

2:30 PM - 3:15 PMRoom: ZOOM ROOM Neural networks have been extremely successful when applied to machine learning problems, like computer vision, speech processing and media generation. Mathematically, training a neural network is achieved by optimising a loss function. For large neural networks this is a substantially complex problem. In a landmark paper, Choromanska et al. (2015) argued that training loss surfaces of large networks can be modelled by spherical multi-spin glasses. In this talk we present recent results about the loss surfaces of neural networks and generative adversarial networks using supersymmetric techniques from random matrix theory. Our results shed light on the strengths of spin glass models for neural networks. This is work in collaboration with Nicholas Baskerville, Jonathan Keating and Joseph Najnudel.3:15 PM Probabilistic conformal block and its semi-classical limit - Promit Ghosal (MIT)Probabilistic conformal block and its semi-classical limit- Promit Ghosal (MIT)

3:15 PM - 4:00 PMRoom: ZOOM ROOM Conformal blocks are fundamental objects in the conformal bootstrap program of 2D conformal field theory and are closely related to four dimensional super-symmetric (SUSY) gauge theory via Alday-Gaitto-Tachikawa (AGT) correspondence. In this talk, I will demonstrate a probabilistic construction of 1-point torus conformal block and discuss its semi-classical limit. This talk will be based on two separate works. The first one with Guillaume Remy, Xin Sun and Yi Sun and the second one with Harini Desiraju and Andrei Prokhorov.7:30 PM Social DinnerSocial Dinner7:30 PM - 8:30 PMRoom: 128 -
Thursday, May 26, 20229:30 AM TT* equations of Cecotti and Vafa, Riemann-Hilbert method and Iwasawa factorization. - Alexander ItsTT* equations of Cecotti and Vafa, Riemann-Hilbert method and Iwasawa factorization.
- Alexander Its

9:30 AM - 10:15 AMRoom: 128 In this talk, some interesting (we believe !) new features of the Riemann-Hilbert method of the asymptotic analysis of integrable systems will be discussed. These features have emerged during the study of the global solution of the tt* equations of Cecotti and Vafa which the speaker has been pursuing, for some time already, together with M. Guest and Chang-Shou Lin. The method which we use in this study is based on a combination of the isomonodrtomy technique and Iwasawa factorization from the theory of loop groups. This allows us to simplify significantly the asymptotic analysis of the tt* equations and, simultaneously, to bring a new light on some aspects of the well known relation between the Birkhoff-Grothendieck and Iwasawa factorizations. Another important link - to a 1980 paper by I. Krichever on the nonlinear analog of the d'Alembert's formula, will be also highlighted. The talk represents an ongoing joint project of M. Guest, I. Krichever, and the speaker.10:15 AM Quantum KdV and quasimodular forms - Giulio Ruzza (Université catholique de Louvain)Quantum KdV and quasimodular forms- Giulio Ruzza (Université catholique de Louvain)

10:15 AM - 11:00 AMRoom: 128 A result of Dubrovin provides a full description of the spectrum of the dispersionless quantum Korteweg-de Vries (qKdV) hierarchy (in the first Poisson structure). Eigenvalues are shifted-symmetric functions of partitions, related, by a result of Bloch and Okounkov, to quasimodular forms on the full modular group SL(2,Z). We extend this relation to the dispersive case by showing that q-weighted traces of the Hamiltonian operators of the qKdV (in the first Poisson structure) are quasimodular of homogeneous weight. This is achieved by first establishing a general criterion for quasimodularity, which remarkably simplifies the recursion formula (in the form obtained by Buryak and Rossi) for the qKdV Hamiltonian densities. These results naturally extend to the Intermediate Long Wave hierarchy. Joint work with Jan-Willem van Ittersum.11:00 AM Coffee BreakCoffee Break11:00 AM - 11:30 AMRoom: 12811:30 AM Borel-Laplace multi-transform, and integral representations of solutions of qDEs - Giordano Cotti (Scuola Internazionale Superiore di Studi Avanzati (SISSA))Borel-Laplace multi-transform, and integral representations of solutions of qDEs- Giordano Cotti (Scuola Internazionale Superiore di Studi Avanzati (SISSA))

11:30 AM - 12:15 PMRoom: 128 The quantum differential equation (qDE) is a rich object attached to a smooth projective variety X. It is an ordinary differential equation in the complex domain which encodes information of the enumerative geometry of X, more precisely its Gromov-Witten theory. Furthermore, the monodromy of its solutions conjecturally rules also the topology and complex geometry of X. These differential equations were introduced in the middle of the creative impetus for mathematically rigorous foundations of Topological Field Theories, Supersymmetric Quantum Field Theories and related Mirror Symmetry phenomena. Special mention has to be given to the relation between qDE's and Dubrovin-Frobenius manifolds, the latter being identifiable with the space of isomonodromic deformation parameters of the former. The study of qDE’s represents a challenging active area in both contemporary geometry and mathematical physics. In this talk I will introduce some analytic integral multitransforms of Borel-Laplace type, and I will use them to obtain Mellin-Barnes integral representations of solutions of qDEs.12:15 PM A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of B_n - Paolo Lorenzoni (Università di Milano-Bicocca)A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of B_n- Paolo Lorenzoni (Università di Milano-Bicocca)

12:15 PM - 1:00 PMRoom: 128 We show that the orbit space of B_2 less the image of coordinate lines under the quotient map is equipped with two Dubrovin-Frobenius manifold structures which are related respectively to the defocusing and the focusing nonlinear Schrodinger equations. Motivated by this example, we study the case of B_n and we show that the defocusing case can be generalized to arbitrary n leading to a Dubrovin-Frobenius manifold structure on the orbit space of the group. The construction relies on the existence of a non-degenerate and non-constant invariant bilinear form that plays the role of the Euclidean metric in the Dubrovin-Saito standard setting. Up to n=4 the solutions of WDVV equations we get coincide with those associated with constrained KP equations. The talk is based on a joint work with Alessandro Arsie, Igor Mencattini and Guglielmo Moroni.1:00 PM Lunch BreakLunch Break1:00 PM - 2:30 PMRoom: 1282:30 PM Janossy densities of the thinned Airy point process: the Schrödinger and (c)KdV equations - Sofia Tarricone (UCLouvain)Janossy densities of the thinned Airy point process: the Schrödinger and (c)KdV equations- Sofia Tarricone (UCLouvain)

2:30 PM - 3:15 PMRoom: 128 The aim of the talk is to show that the Janossy densities of a suitably thinned Airy kernel point process are governed by the Schrödinger and (cylindrical) KdV equations; moreover, we prove that the associated wave functions satisfy a system of coupled integro-differential Painlevé II equations. These results are obtained by characterizing the Janossy densities in terms of a Riemann-Hilbert matrix factorization problem with poles which is analysed by the theory of Darboux-Schlesinger transformations. This approach also allows to investigate asymptotics for the Janossy densities in various regimes (in progress). The talk is based on ongoing work with T. Claeys, G. Glesner and G. Ruzza.3:15 PM Universality for free fermions - Gaultier LambertUniversality for free fermions- Gaultier Lambert

3:15 PM - 4:00 PMRoom: 128 The ground state of a free Fermi gas is a classical example of determinantal processes introduced by Macchi (1975) whose correlation kernel is associated with a Schrödinger operator on R^n. In this talk, I will explain how one proves universality of local correlations for these models using semiclassical analysis. If time permits, I will also mention a central limit theorem for a one-dimensional Fermi gas and explain the connection with random matrix theory. Joint work with Alix Deleporte. -
Friday, May 27, 20229:30 AM Solutions of the Bethe Ansatz Equations as Spectral Determinants - Davide Masoero (Universidade de Lisboa)Solutions of the Bethe Ansatz Equations as Spectral Determinants
- Davide Masoero (Universidade de Lisboa)

9:30 AM - 10:15 AMRoom: 128 The Quantum KdV model is a conformal field theory, which is integrable via the Bethe Ansatz Equations. It can be regarded as a deformation of the second KdV hamiltonian structure or as the scaling limit of the XXZ chain. In 1998,Dorey and Tateo discovered that the Bethe roots for the ground state of the Quantum KdV model coincide with the eigenvalues of certain anharmonic oscillators (ODE/IM correspondence).In 2004,Bazhanov, Lukyanov & Zamolodhchikov conjectured that the Bethe roots of every state of the model are the eigenvalues of a linear differential operator, namely an anharmonic oscillator with a monster potential. In this talk I provide an outline of the proof -- conditional on the existence of a certain Puiseux series -- of the BLZ conjecture, that I have recently obtained in collaboration with Riccardo Conti.10:15 AM Coffee BreakCoffee Break10:15 AM - 10:45 AMRoom: 12810:45 AM Modular transformation of tau-functions on the torus. - Harini Desiraju (SISSA)Modular transformation of tau-functions on the torus.- Harini Desiraju (SISSA)

10:45 AM - 11:30 AMRoom: 128 The connection constant of the tau-function on the torus corresponds to its modular transformation and is related to the modular kernel of conformal blocks. In this talk, I will show the connection constant in terms of the monodromy data for a torus with one simple pole as an example, but the techniques extend to any number of simple poles.11:30 AM Perturbative connection formulas for Heun equations - Oleg Lisovyy (LMPT, Tours University)Perturbative connection formulas for Heun equations- Oleg Lisovyy (LMPT, Tours University)

11:30 AM - 12:15 PMRoom: 128 The connection problem for Heun equation seeks to relate bases of Frobenius solutions associated to different Fuchsian singularities. In a recent paper 2201.04491 [hep-th], Bonelli, Iossa, Lichtig and Tanzini have proposed a conjecture relating the relevant connection coefficients to classical conformal blocks of the Virasoro algebra. In practical terms, their conjecture allows to compute the connection coefficients in the form of a perturbative expansion in a suitable parameter. I will explain how to derive the corresponding perturbative formulas directly from Heun’s equation.

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