Discussion of topics related to the Stokes phenomenon.
Specific topics will include: moduli spaces of meromorphic connections, spaces of Stokes data, abelianisation of connections, the WKB method, quantisation of isomonodromic deformations
There will be two main strands to our seminar series: the Research Seminar (consisting of research talks mainly aimed at specialists) and the Working Seminar (consisting of introductory talks mainly aimed at students).
|14 October 2019|
|The Stokes Phenomenon: Introduction||working|
|15 October 2019|
(Columbia & Skoltech)
|Isomonodromy equations on algebraic curves and canonical transformations (part 1)||research|
|16 October 2019|
(Columbia & Skoltech)
|Isomonodromy equations on algebraic curves and canonical transformations (part 2)||research|
|29 October 2019|
|Quantisation of isomonodromy systems (part 1)||working|
|31 October 2019|
|Quantisation of isomonodromy systems (part 2)||working+research|
|12 November 2019|
|Singularly Perturbed Levelt Filtrations||research|
12 November 2019
Nikita Nikolaev (University of Geneva)
Singularly Perturbed Levelt Filtrations
It is well-known that when a differential equation has a singular point, the vector space of initial conditions at any nearby point is naturally filtered by growth rates of solutions as they are analytically continued into the singularity. Such filtrations are often called Levelt filtrations, and they turn out to be essential invariants for parameterising moduli spaces of singular differential equations or more generally meromorphic connections.
What about differential equations that contain a singular perturbation parameter ℏ? Such equations are ubiquitous in mathematics and physics: the Schrödinger equation is the most famous example. Most notably, in the limit as ℏ → 0, a singularly perturbed differential system degenerates simply to a linear system of equations (i.e., no differential equations). This is the nature of singular perturbation theory. This linear system in the limit as ℏ → 0 is often called a Higgs field.
Clearly, for any fixed nonzero value of the perturbation parameter ℏ, we have a single usual singular differential equation, and therefore can construct a Levelt filtration. But what is the behaviour in ℏ of these filtrations, especially with respect to the limit ℏ → 0?
In my recent paper [arXiv:1909.04011], I study singularly perturbed systems of rank 2 with a regular singular point. The main result is that there is a well-defined filtration which on the vector space of initial conditions over the generic point that varies holomorphically over a sector in the ℏ-plane, which specialises to the Levelt filtration at the generic point (i.e., ℏ ≠ 0), and which converges to the eigen-decomposition of the Higgs field as ℏ → 0 as well as to the eigen-decomposition of the residue as the coordinate z → 0.
29, 31 October 2019
Gabriele Rembado (University of Geneva)
Quantisation of isomonodromy systems
We will at first contemplate the quantisation of the following geometric picture. Consider a vector bundle on the sphere equipped with a logarithmic connection. The connection defines systems of differential equations with meromorphic coefficients (having simple poles) for the local sections of the bundle. Solutions to such systems are multi-valued, and transform by monodromy upon analytic continuation around a pole: we are interested in deformations of the logarithmic connection such that this monodromy is fixed (up to gauge), and call them isomonodromic deformations. It turns out they are controlled by an Hamiltonian system.
Importantly, a natural quantisation of this system yields the Knizhnik--Zamolodchikov connection in conformal field theory. Thus the final output is a (very important) integrable quantum Hamiltonian system constructed out of the quantisation of a classical system that controls isomonodromic deformations (or an isomonodromy system, for short).
In the first part we will try to review this construction and to introduce the relevant formalism of deformation quantisation (and give some motivation in TQFT if time allows).
In the second more technical part we will consider extensions of this story where higher order poles in the meromorphic connections are allowed. This will involve quantising the symplectic (Nakajima) varieties of representations of certain quivers, as well as introducing the formalism of quantum Hamiltonian reduction.
15, 16 October 2019
Igor Krichever (Columbia & Skoltech)
Isomonodromy equations on algebraic curves and canonical transformations
In the talk we will discuss the Hamiltonian theory of isomonodromy equations for meromorphic connections with irregular singularities on algebraic curves and an explicit formula for the symplectic structure on the space of monodromy and Stokes matrices. We will show that the isomondromy equations induce a flat connection on the space of the spectral curves of the Hitchin systems.
14 October 2019
Nikita Nikolaev (University of Geneva)
The Stokes Phenomenon: Introduction
There is an interesting and surprising phenomenon in complex analysis: the asymptotic behaviour of some functions in the complex plane at (say) infinity depends on the direction of approach towards infinity; moreover, matching asymptotic expansions with functions comes with abrupt discontinuities at specific discrete directions. This phenomenon was first discovered by Sir George Stokes in 1847 in the study of geometric optics, and it is now known as the Stokes Phenomenon.
We now know that the Stokes phenomenon occurs in the analysis of solutions to meromorphic differential equations with an irregular singular point. In the last few decades, it was recognised that similar phenomena appear in a wide range of subjects in mathematics and physics, from the study of stability conditions in the geometry of moduli spaces of sheaves to counting black holes in string theory. These and many other phenomena are now more generally known as Wall-Crossing Phenomena.
In this introductory lecture, I want to demonstrate the Stokes phenomenon through a very explicit example. The prerequisites will be kept at a minimum, so this talk is especially suitable for students and non-experts.