Lectures on Geometry of Hydrodynamics and String Amplitudes by Nikita Nekrasov

During the academic year 2026–2027, Professor Nekrasov will spend part of his time at the Centre for Geometry and Physics as a Wallenberg Guest Professor. His lecture series will provide a unique opportunity to learn about recent developments at the interface of geometry, mathematics, and fundamental physics, directly from one of the leading figures in the field.

The lectures are aimed at a general mathematical physics audience. Everyone interested in the topics, including students, is very welcome to attend.

Three lectures on Geometry of Hydrodynamics

12 October, 10:15–12:00
Å80127
Lecture I: From Spinning Tops to Fluid Dynamics

13 October, 10:15–12:00
Å80127
Lecture II: From Poisson Geometry to Paths and Intersections

16 October, 10:15–12:00
Å101121 (Sonja Lyttkens)
Lecture III: Relativistic Hydrodynamics, Spacetime Geometry and Gravity

Two lectures on String Amplitudes

23 October, 10:15–12:00
Å90101
Lecture I: String Amplitudes – From Lorentzian Propagation to Moduli Spaces

23 October, 13:15–15:00
Å90101
Lecture II: String Amplitudes – Integration Cycles and Complex Metrics

Abstracts

Geometry of Hydrodynamics

Lecture I: From Spinning Tops to Fluid Dynamics

Hamiltonian dynamics does not require a symplectic phase space: it can be naturally formulated on a Poisson manifold, whose symplectic leaves carry the actual motion. We will introduce this point of view through the classical spinning top and its Lie-Poisson geometry, and then pass to its infinite-dimensional analogues. The Euler equations of ideal incompressible and compressible hydrodynamics emerge as spinning-top equations on groups of diffeomorphisms and their extensions. We will also discuss two beautiful finite-dimensional relatives of these systems: Kirchhoff's equations for a rigid body interacting with an ideal fluid and the dynamics of point vortices.

Lecture II: From Poisson Geometry to Paths and Intersections

What is the relation between the Poisson geometry underlying the equations of motion and the symplectic geometry familiar from classical mechanics? We will explore this question by replacing instantaneous configurations by paths and histories. Spaces of paths naturally connect Poisson and symplectic structures and provide a geometric interpretation of equations of motion and boundary conditions. This leads to a complementary viewpoint in which classical dynamics can be formulated as an intersection problem. We will explain how this perspective arises in mechanics and field theory and why it is particularly natural for hydrodynamics.

Lecture III: Relativistic Hydrodynamics, Spacetime Geometry and Gravity

Relativistic hydrodynamics invites a reformulation in which spacetime, rather than a space of initial conditions, plays the central geometric role. We will review the covariant description of ideal relativistic fluids and explain how their equations reveal unexpected Poisson and symplectic structures associated directly with spacetime. We will discuss extensions of this framework and conclude by exploring several ways in which hydrodynamics meets gravity: from fluids propagating in curved spacetime and analogue geometries to the emergence of hydrodynamic equations from gravitational dynamics.

String Amplitudes

Lecture I: String Amplitudes – From Lorentzian Propagation to Moduli Spaces

I will introduce string scattering amplitudes beginning with a basic question already present in ordinary quantum field theory: what is the geometry behind the Feynman propagator? The Schwinger proper-time representation makes its Lorentzian origin and the role of the iϵ prescription explicit, while the usual Wick rotation tends to hide them. For strings, proper time is replaced by the geometry of long worldsheet tubes, and perturbative amplitudes become integrals over moduli spaces of punctured Riemann surfaces. We will see, already for the four-point amplitude, that the naive Euclidean moduli-space integral diverges for physical Lorentzian kinematics. Its divergences occur precisely at degenerations corresponding to propagating intermediate strings. This will lead us to scattering equations, complex saddles, and the question of what integration cycle should define the physical amplitude.

Lecture II: String Amplitudes – Integration Cycles and Complex Metrics

The standard string amplitudes are obtained by analytic continuation from a region where the Euclidean moduli-space integral converges. But can this continuation be understood directly as a geometric integration cycle for the string path integral at physical kinematics? I will discuss the string iϵ prescription, complexification of the space of worldsheet metrics, and the Picard–Lefschetz viewpoint in which amplitudes are decomposed into contributions from complex saddles. I will then describe ongoing work with David Gross exploring a more geometric possibility: an integration cycle consisting of real worldsheet metrics with both Euclidean and Lorentzian regions, separated by loci where the metric changes signature. This leads naturally to two chiral complex structures, scattering equations, and a potentially much richer collection of string saddles. The goal is to understand whether causality, convergence, and the integration contour of string theory can all be encoded directly in worldsheet geometry.