The exact WKB method for solving linear ODEs with a small parameter is one of the most classical examples of Exact Perturbation Theory. ODEs — such as the stationary Schrödinger equation or the Berk-Nevins-Roberts equation — can be easily solved in exponential power series: this is the famous WKB ansatz. The resulting formal WKB solutions, however, are always divergent and therefore have no direct analytic meaning. Attempting to apply the Borel resummation to get true analytic solutions turns out to be the correct approach, but a difficult mathematical problem, especially for equations of higher-order. I will describe a solution to this problem that I have developed through a series of recent works. The key new insights come from developing a careful understanding of the geometry that controls the underlying Stokes phenomenon. Another advantage of this solution is that the constructions involved in the proof are purely geometric, allowing for an uplift to an algebro-geometric formulation of the exact WKB method for meromorphic connections in terms of finding invariant splittings of extensions of vector bundles.

One of the most classical settings for Exact Perturbation Theory is the exact WKB method for solving singularly perturbed linear ODEs such as the Schrödinger equation. Such ODEs can be easily solved in exponential power series: this is the famous WKB ansatz. The resulting formal WKB solutions, however, are always divergent and therefore have no direct analytic meaning. Attempting to apply the Borel resummation to get true analytic solutions turns out to be the correct approach, but a difficult mathematical problem, especially for equations of higher-order. I will describe a solution to this problem that I have developed through a series of recent works. Another outcome of this solution is that the constructions involved in the proof can be made completely geometrically invariant. So I will also describe an algebro-geometric formulation of the WKB method for meromorphic connections in terms of invariant splittings of bundle extensions.

One of the most classical settings for Exact Perturbation Theory is the exact WKB method for solving singularly perturbed linear ODEs such as the Schrödinger equation. Such ODEs can be easily solved in exponential power series: this is the famous WKB ansatz. The resulting formal WKB solutions, however, are always divergent and therefore have no direct analytic meaning. Attempting to apply the Borel resummation to get true analytic solutions turns out to be the correct approach, but a difficult mathematical problem, especially for equations of higher-order. I will describe a solution to this problem that I have developed through a series of recent works. Another outcome of this solution is that the constructions involved in the proof can be made completely geometrically invariant. So I will also describe an algebro-geometric formulation of the WKB method for meromorphic connections in terms of invariant splittings of bundle extensions.

Perhaps surprisingly, flat meromorphic connections on holomorphic bundles over complex manifolds — objects that you’d probably consign to a purely differential-geometric context — can be equivalently studied using the representation theory of naturally associated groups or, better yet, Lie groupoids. The moduli spaces of these representations are really remarkable geometric objects that carry many interesting structures, including a symplectic structure and a cluster algebra structure. I will talk about various aspects of my ongoing series of projects to study these spaces using a method called abelianisation which one can think of as a kind of spectral theory for connections.

One of the most classical settings for Exact Perturbation Theory is the exact WKB method for solving singularly perturbed linear ODEs such as the Schrödinger equation. Such ODEs can be easily solved in exponential power series: this is the famous WKB ansatz. The resulting formal WKB solutions, however, are always divergent and therefore have no direct analytic meaning. Attempting to apply the Borel resummation to get true analytic solutions turns out to be the correct approach, but a difficult mathematical problem, especially for equations of higher-order. I will describe a solution to this problem that I have developed through a series of recent works. Another outcome of this solution is that the constructions involved in the proof can be made completely geometrically invariant. So I will also describe an algebro-geometric formulation of the WKB method for meromorphic connections in terms of invariant splittings of bundle extensions.

 Divergent series expansions naturally arise in many parts of mathematics and physics from dynamical systems, to gauge theory, to quantisation of Poisson structures, to mirror symmetry and stability in algebraic geometry, to QFT and string theory, to name a few. This course is an introduction to an amazing emerging subject of Exact Perturbation Theory, also known as the theory of Resurgence. This theory provides powerful methods to upgrade divergent expansions to some meaningful analytic information. One such method is called Borel resummation: it involves a detailed analysis (via the Borel transform) of a given divergent expansion in order to recover exponentially small terms (a.k.a., nonperturbative corrections) that otherwise cannot be captured by the ordinary perturbation theory. The astounding big-picture upshot of this theory is that — contrary to Freeman Dyson’s conclusion that perturbation theory is incomplete — the divergent sector of perturbation theory actually already encodes all the necessary nonperturbative information, and it is therefore only a matter of applying suitable methods to extract it.

My aim is to give a systematic treatment of the subject to build a solid foundation. Most of this course therefore will focus on the basic theory of divergent series and Borel resummation. At heart, the subject is just an extension of the ordinary complex analysis, and I will repeatedly stress this sentiment because I think it helps to demystify the subject. We will see, however, that this extension is exceptionally rich and full of remarkable new phenomena. Another feature that I find extraordinary (and that I will emphasise throughout the course) is that, although at first this subject may appear to be confined purely within the realm of analysis, in fact it has as much an algebro-geometric flavour as an analytic one.

I hope to cover the following topics: basic theory of asymptotic expansions, the Borel-Laplace transform, asymptotic expansions with factorial growth, Borel resummation and the Borel-Laplace Method, Stokes phenomenon, endless analytic continuation, algebra of resurgent transseries.

LECTURE NOTES | Dropbox folder

"industry talks and panel sessions, where experts in their fields will discuss how they use mathematics in their everyday work."

Perhaps surprisingly, flat meromorphic connections on holomorphic bundles over complex manifolds — objects that you’d probably consign to a purely differential-geometric context — can be equivalently studied using the representation theory of naturally associated groups or, better yet, Lie groupoids. The moduli spaces of these representations are really remarkable geometric objects that carry many interesting structures, including a symplectic structure and a cluster algebra structure. I will talk about various aspects of my ongoing series of projects to study these spaces using a method called abelianisation which one can think of as a kind of spectral theory for connections.

I will present recent progress in the amazing newly emerging subject of exact perturbation theory. This theory provides methods to upgrade divergent perturbative expansions to analytic information. One such method is called Borel resummation: it involves a detailed analysis (via a Borel transform) of a given divergent perturbative expansion in order to recover the nonperturbative corrections (i.e., exponentially small terms) that otherwise cannot be captured by the ordinary perturbation theory. The astounding big-picture upshot of this theory is that — contrary to Freeman Dyson’s conclusion that perturbation theory is incomplete — the divergent sector of perturbation theory actually already encodes all the necessary nonperturbative information, and it is therefore only a matter of applying suitable methods to extract it.

One of the most classical settings for exact perturbation theory is the so-called exact WKB method for solving singularly perturbed linear ODEs such as the Schrödinger equation. Such ODEs can be easily solved in (exponential) power series (the so-called WKB ansatz), but they are almost always divergent. Attempting to apply Borel resummation to get true analytic solutions turns out to be the correct approach, but a difficult mathematical problem. I will describe a solution I have developed through a series of recent works.

I will present recent progress in the amazing newly emerging subject of exact perturbation theory. This theory provides methods to upgrade divergent perturbative expansions to analytic information. One such method is called Borel resummation: it involves a detailed analysis (via a Borel transform) of a given divergent perturbative expansion in order to recover the nonperturbative corrections (i.e., exponentially small terms) that otherwise cannot be captured by the ordinary perturbation theory. The astounding big-picture upshot of this theory is that — contrary to Freeman Dyson’s conclusion that perturbation theory is incomplete — the divergent sector of perturbation theory actually already encodes all the necessary nonperturbative information, and it is therefore only a matter of applying suitable methods to extract it.

One of the most classical settings for exact perturbation theory is the so-called exact WKB method for solving singularly perturbed linear ODEs such as the Schrödinger equation. Such ODEs can be easily solved in (exponential) power series (the so-called WKB ansatz), but they are almost always divergent. Attempting to apply Borel resummation to get true analytic solutions turns out to be the correct approach, but a difficult mathematical problem. I will describe a solution I have developed through a series of recent works.

Perhaps surprisingly, flat meromorphic connections on holomorphic bundles over complex manifolds — objects that you’d probably consign to a purely differential-geometric context — can be equivalently studied using the representation theory of naturally associated groups or, better yet, Lie groupoids. The moduli spaces of these representations are really remarkable geometric objects that carry many interesting structures, including a symplectic structure and a cluster algebra structure. I will talk about various aspects of my ongoing series of projects to study these spaces using a method called abelianisation which one can think of as a kind of spectral theory for connections.

One of the most classical settings for Exact Perturbation Theory is the exact WKB method for solving singularly perturbed linear ODEs such as the Schrödinger equation. Such ODEs can be easily solved in exponential power series: this is the famous WKB ansatz. The resulting formal WKB solutions, however, are always divergent and therefore have no direct analytic meaning. Attempting to apply the Borel resummation to get true analytic solutions turns out to be the correct approach, but a difficult mathematical problem, especially for equations of higher-order. I will describe a solution to this problem that I have developed through a series of recent works. Another outcome of this solution is that the constructions involved in the proof can be made completely geometrically invariant. So I will also describe an algebro-geometric formulation of the WKB method for meromorphic connections in terms of invariant splittings of bundle extensions.