talks

slides for many of the talks: slides.com/kmodin

2025

1. K. Modin

   The reversibility paradox in matrix hydrodynamics

   Mathematics Colloquium, University of Toronto, Canada, October 8, 2025

   Abs HTML Slides

   The “reversibility paradox” refers to the seeming contradiction between two major discoveries made during the end of the 19th century: the law of entropy, due to Boltzmann, and the recurrence theorem, due to Poincaré. The discourse is captured in the following question: How can reversible dynamics give rise to irreversible behavior? In this talk, I shall discuss how the same enigma shows up in a finite-dimensional model of the 2-D incompressible Euler equations. The model, suggested by V. Zeitlin in 1991, uses quantization theory to replace the vorticity formulation of the 2-D Euler equations by an isospectral flow of matrices. This way, a spatial discretization preserving all the geometric structure is obtained. Recent advances on Zeitlin’s model propose a connection between the long-time behavior of (generic) 2-D Euler flows and integrability conditions for “blob dynamics.” A major point at issue, however, is whether simulations based on Zeitlin’s model truly reflect the dynamics of 2-D Euler. It is within this context that the reversibility paradox enters. The work is joint with Milo Viviani.

2. K. Modin

   Matrix Hydrodynamics and Mixing

   ICMS Workshop: Blurring the lines between pure and applied through mixing, Isle of Skye, Scotland, June 1-6, 2025

   Abs HTML Slides

   In 1991, Vladimir Zeitlin realized that quantization naturally leads to spatial discretizations of the two-dimensional incompressible Euler equations. Contrary to other discretizations, this approach - matrix hydrodynamics - preserves the rich geometry in phase space (Lie-Poisson structure, co-adjoint orbits, Casimirs, spatial symmetries, etcetera). Matrix hydrodynamics is therefore suitable for long-time computer simulations aiming to study statistical properties of generic solutions. In this talk, I shall give an overview of recent developments in matrix hydrodynamics while emphasizing its relevance to mixing.

3. K. Modin

   Emmy Noether’s theorem

   CAM Seminar, Mathematical Sciences Chalmers, Gothenburg, Sweden, March 24, 2025

   Abs HTML Slides

   I wish to talk about Noether’s theorem, including its proof. The subject came up on the occasion of the newly published popular science book “Emmys teorem” by Julia Ravanis.

4. K. Modin

   Mini-course: Information geometry of diffeomorphism groups

   Infinite-dimensional Geometry: Theory and Applications, Erwin Schrödinger Institute (ESI), Vienna, Jan 13 - Feb 14, 2024

   Abs HTML Video

   The study of diffeomorphism groups and their applications to problems in analysis and geometry has a long history. Geometric hydrodynamics, pioneered by V. Arnold in the 1960s, considers an ideal fluid flow as the geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms of a flow domain in the energy metric. A similar consideration on the space of densities led to a geometric description of the optimal mass transport and the Kantorovich-Wasserstein metric. Likewise, the information geometry associated with the Fisher-Rao metric and Hellinger distance has an equally beautiful infinite-dimensional geometry and can be regarded as a higher Sobolev analogue of optimal transportation. In this mini-course, I shall describe various Riemannian metrics on diffeomorphism groups and describe the connection to information geometry. The material is based on a forthcoming essay with B. Khesin and G. Misiolek, which shall make up the course literature.

2024

1. K. Modin

   Statistical mechanics, 2-D fluids, and structure-preserving numerics

   Mathematics Colloquium, University of Pisa, Italy, September 12, 2024

   Abs Slides

   In a 1949 paper, the Norwegian American genius Lars Onsager demonstrated that Boltzmann’s theory of statistical mechanics provides insights to the long-time behavior of 2-D fluids described by Euler’s equations. The key notion is to “approximate” the fluid by a finite number of interacting point vortices. Since these are governed by Hamiltonian dynamics, the framework of statistical mechanics provides a characterization of the long-time asymptotic behavior. The approach successfully predicts clustering of equally signed vortices, which is part of Onsager’s legacy. Nevertheless, and as Onsager proclaimed in his paper, the point vortex approximation is highly non-regular and might therefore be poor. Furthermore, statistical mechanics assumes ergodicity, which fails for 2-D Euler. In this talk, I will present another finite-dimensional approximation of 2-D Euler due to Vladimir Zeitlin. Unlike point vortices, Zeitlin’s model gives smooth approximations. It thereby renders a refinement of Onsager’s statistical approach, and, together with a symplectic numerical time-integration scheme, provides a structure preserving numerical method. Simulations using this method suggest compelling connections between 2-D Euler and integrability of low-dimensional Hamiltonian systems with symmetry. The work is joint with Milo Viviani.

2. K. Modin

   A matrix discretization for axisymmetric 3-D Euler

   Math in Maine, Merril House, Maine, August 18-24, 2024

   HTML Slides
3. K. Modin

   Matrix hydrodynamics and 2-D Euler

   EQUADIFF’24, Karlstad, Sweden, June 10-14, 2024

   Abs HTML Slides

   In 1991, Zeitlin showed how quantization theory naturally leads to a spatial discretization of the 2-D incompressible Euler equations. This finite dimensional model takes the form of an isospectral flow of matrices. It is the only known discretization that preserves the Lie–Poisson structure, i.e., the Hamiltonian structure associated with geodesic flows on Lie groups equipped with a right invariant Riemannian metric—the 1966 discovery of Arnold. Despite over three decades in presence, Zeitlin’s model is surprisingly underexplored. In the last couple of years, however, this beautiful approach has seen something of a renaissance, with several new developments, and applications to geophysical flows (quasi-geostrophic equations) as well as incompressible 2-D magnetohydrodynamics. In this talk, I wish to present some insights gained during these developments, particularly those related to the long-time behavior of 2-D Euler—the long-standing problem described as the “jewel of 2-D turbulence”. This is a joint work with Milo Viviani (Scuola Normale Superiore, Pisa, Italy).

4. K. Modin

   The reversibility paradox in matrix hydrodynamics

   Foundations of Numerical Differential Geometry, Trondheim, Norway, May 7-8, 2024

   Abs HTML Slides

   The “reversibility paradox” refers to the seeming contradiction between two major discoveries made during the end of the 19th century: the law of entropy, given by Boltzmann, and the recurrence theorem, given by Poincaré. The discourse is captured in the following question: how can reversible dynamics give rise to irreversible behavior? In this talk, I shall discuss how the same enigma shows up in a finite-dimensional model of the 2-D incompressible Euler equations. The model, suggested by V. Zeitlin in 1991, uses quantization theory to replace the vorticity formulation of the 2-D Euler equations by an isospectral flow of matrices. This way, a spatial discretization preserving all the geometric structure is obtained. Recent advances on Zeitlin’s model propose a connection between the long-time behavior of (generic) 2-D Euler flows and integrability conditions for “blob dynamics”. A major point at issue, however, is whether simulations based on Zeitlin’s model truly reflect the dynamics of 2-D Euler. It is within this context the reversibility paradox enters.

5. K. Modin

   On reversibility and matrix hydrodynamics

   Seminar at SNS Pisa, Italy, November 7, 2024

   Abs Slides

   Some time ago, Milo Viviani and I unveiled numerical simulations that indicate a connection between the long-time behavior of 2-D Euler equations and integrability conditions for “blob dynamics”. After presenting these results, I was asked an insightful question: The phase space underlying the numerical model in the simulations is compact. Because the dynamics is also Hamiltonian, we have Poincaré recurrence. But the dynamics in the simulations, leading to blob formations, seem contractive. Isn’t the mechanism for blob formations instead induced by fictitious dissipation, introduced via the numerical time discretization? I didn’t have a good answer at the time, but the question stayed with me. Today I have an answer, which is the subject of this talk.

2023

1. K. Modin

   Tutorial talk: Wasserstein-Otto geometry

   Infinite Dimensional Geometry and Fluids, Banff International Research Station, Canada, November 5-10, 2023

   HTML Slides
2. K. Modin

   Mini-course: Geometric Hydrodynamics on Flatland

   XXXI Workshop on Geometry and Physics, Santiago de Compostella, Spain, September 4-8, 2023

   Abs HTML Slides

   Euler’s equations for the motion of an incompressible perfect fluid possess a rich geometric structure, as revealed over the years by Poincaré, Arnold, and many others. The structure is particularly striking in two dimensions, where it gives rise to infinitely many conservation laws (Casimir functions). In this mini-course I will give an overview of geometric hydrodynamics, with an emphasis on the two-dimensional case and why its long-time dynamics (turbulence) is so different from the three- and higher-dimensional setting. I will also discuss how results from quantization theory can be used to construct numerical discretizations that preserve the underlying geometry - an approach due to V. Zeitlin.

3. K. Modin

   Eulerian and Lagrangian stability in Zeitlin’s model of hydrodynamics

   Foundation of Computational Mathematics (FoCM), Paris, France, June 12-21, 2023

   Abs HTML Slides

   The 2-D Euler equations of a perfect fluid possess a beautiful geometric description: they are reduced geodesic equations on the infinite-dimensional Lie group of symplectomorphims with respect to a right-invariant Riemannian metric. This structure enables insights to Eulerian and Lagrangian stability via sectional curvature and Jacobi equations. The Zeitlin model is a finite-dimensional analog of the 2-D Euler equations; the only known discretization that preserves the rich geometric structure. Theoretical and numerical studies indicate that Zeitlin’s model provides consistent long-time behaviour on large scales, but to which extent it truly reflects the Euler equations is mainly open. Towards progress, I present here two new results. First, convergence of the sectional curvature in the Euler–Zeitlin equations on the Lie algebra su(N) to that of the Euler equations on the sphere. Second, L2-convergence of the corresponding Jacobi equations for Lagrangian and Eulerian stability. The new results allow geometric conclusions about Zeitlin’s model to be transferred to Euler’s equations and vice versa. Joint work with Manolis Perrot (Univ. Grenoble Alpes, France).

4. K. Modin

   Zeitlin’s model - old and new

   Optimal Transport: applications to physics, Les Houches, France, March 12-17, 2023

   HTML Slides
5. K. Modin

   On the gradient flow structure of the continuous Toda and incompressible porous medium equation

   Optimal Transport: applications to physics, Les Houches, France, March 12-17, 2023

   HTML Slides
6. K. Modin

   A Brief History of Geometric Mechanics

   Mathematics Colloquium, Lund University, Sweden, February 1, 2023

   Abs HTML Slides

   In this talk I wish to give an overview of the tight connection between differential geometry and classical mechanics. I shall take a historical route, going from Newton and Euler, via Lagrange and Hamilton, Poincaré and the Prize of King Oscar II, to Arnold and the Riemannian description of hydrodynamics.

2022

1. K. Modin

   Lie-Poisson integrators

   HB60 workshop, Trysil, Norway, December 12-17, 2022

   HTML Slides
2. K. Modin

   Link between integrability of point-vortex dynamics and long-time behaviour of 2D Euler equations

   Mathematics Colloquium of Tulane University, December 2, 2022

   Slides
3. K. Modin

   Numerical integration of classical spin systems

   UppASD Autumn School, Stockholm, October 11-13, 2022

   HTML Slides
4. K. Modin

   To Craft Knowledge

   Bildung: The Humanities, Arts, and Sciences (LINA Days), Trollhättan, October 3-7, 2022

   HTML
5. K. Modin

   Geometry and structure-preserving discretization of 2D Euler equations

   Workshop in Geometry and Geometric integration (Geo2Int), Bergen, September 7-9, 2022

   HTML Slides
6. K. Modin

   Optimal transport and quantization

   Optimal transport, geometric and stochastic hydrodynamics, Lisbon, June 21-24, 2022

   HTML
7. K. Modin

   Zeitlin’s model – old and new

   Simons Center for Geometry and Physics (SCGP), June 13, 2022

   Video
8. K. Modin

   Mini-course: Lectures on geometric hydrodynamics

   Winter School and Workshop Wisla (online), January 24-30, 2022

   HTML

2021

1. K. Modin

   Zeitlin’s model for ideal hydrodynamics on the sphere

   CUNY Einstein Chair Mathematics Seminar, December 1, 2021

   Video Slides
2. K. Modin

   Quantized hydrodynamics on the sphere

   Online-presentation given 2021-10 in the Hamiltonian Seminar Series (online), University of Toronto and University of Arizona, October 19, 2021

   Slides
3. K. Modin

   Zeitlin’s model for ideal hydrodynamics on the sphere

   Conference on the Numerical Solution of Differential and Differential-Algebraic Equations (NUMDIFF-16), Martin Luther University, Halle, September 6-10, 2021

   HTML Slides
4. B. Khesin, G. Misiolek, and K. Modin

   Geometric Hydrodynamics and Infinite-dimensional Newton’s Equations

   5th Conference on Geometric Science of Information (GSI’21), Sorbonne University, Paris, July 21-23, 2021

   Video Slides
5. K. Modin

   Mini-course: Introduction to geometric hydrodynamics, Part II

   Young Researchers between Geometry and Stochastic Analysis (online), June 16-18, 2021

   HTML Video
6. K. Modin

   Mini-course: Introduction to geometric hydrodynamics, Part I

   Young Researchers between Geometry and Stochastic Analysis (online), June 16-18, 2021

   HTML Video Slides
7. K. Modin

   Hydrodynamics on the Sphere: Numerical Methods based on Quantization

   SIAM Conference on Applications of Dynamical Systems (DS21), May 23-27, 2021

   HTML Slides
8. K. Modin

   What makes nonholonomic integrators work?

   Geometry, Dynamics, and Mechanics Seminar (online), May 25, 2021

   HTML Video Slides
9. K. Modin

   Two-dimensional incompressible inviscid hydrodynamics: old questions and new insights

   Mathematics Colloquium of Florida State University, February 12, 2021

   HTML Slides

2020

1. K. Modin

   Toeplitz-Berezin quantization and Lie-Poisson integrators for quasi-geostrophic equations

   FoCM Workshop on Geometric Integration and Computational Mechanics (online-only), June 15-18, 2020

   Slides
2. K. Modin

   Isospectral flows and a Casimir preserving scheme for long-time simulation of 2D hydrodynamics

   Structure-Preserving Geometric Discretization of Physical Systems, Princeton Center for Theoretical Science (PCTS), USA, February 17-18, 2020

   Slides

2019

1. K. Modin

   Long time behaviour of 2D spherical ideal hydrodynamics

   HIM, Bonn, Germany, November 13, 2019

   Video Slides
2. K. Modin

   Information geometry and diffeomorphisms

   Information Geometry Workshop, IMT Toulouse, October 18, 2019

   HTML Slides
3. K. Modin and M. Viviani

   Hamiltonian isospectral flows and geophysical hydrodynamics

   Seminar at Lund University, Sweden, April 25, 2019

   Slides

2018

1. M. Bauer and K. Modin

   Semi-invariant Riemannian metrics on diffeomorphisms

   BIRS, Banff, Canada, December 10, 2018

   Video Slides
2. K. Modin

   Link between hydrodynamics and quantum mechanics

   CAM seminar, October 17, 2018

   HTML Slides

2017

1. K. Modin

   Symplectic integrators for classical spin systems

   Physics seminar in Uppsala, December 8, 2017

   Slides
2. K. Modin

   Riemannian Gradient Flows in Shape Analysis

   Isaac Newton Institute (INI), Cambridge, UK, Nov 13, 2017

   Video Slides
3. B. Khesin, G. Misiolek, and K. Modin

   Newton’s Equations on Diffeomorphisms and Densities

   3rd Conference on Geometric Science of Information (GSI2017), Paris, France, Nov 7-9, 2017

   Video Slides
4. M. Bauer, S. Joshi, and K. Modin

   Diffeomorphic Random Sampling Using Optimal Information Transport

   3rd Conference on Geometric Science of Information (GSI2017), Paris, France, Nov 7-9, 2017

   Video Slides
5. K. Modin, R. McLachlan, and O. Verdier

   The Spherical Midpoint Method

   Foundations of Computational Mathematics, Barcelona, Spain, July 10-19, 2017

   Slides
6. M. Bauer, S. Joshi, and K. Modin

   Global results for geodesic equations on densities

   CRiSP workshop, Trondheim, July 2, 2017

   Slides
7. K. Modin

   Computational anatomy and PDE

   Guest seminar in the Nordic Graduate Course on Computational Mathematical Modeling, April 26, 2017

   Slides
8. K. Modin

   Diffeomorphisms and imaging

   Manifolds and Geometric Integration Colloquia (MaGIC), Vatnahalsen, Norway, March 1-5, 2017

   Slides
9. K. Modin

   Silly method for QR factorization

   CAM seminar, February 22, 2017

   HTML Slides

2016

1. K. Modin

   The polar decomposition as the limit of a lifted entropy gradient flow

   CAVALIERI Workshop on Optimal Transport and Optimization in Imaging, Inria Research Center, Paris, October 11-12, 2016

   Slides
2. K. Modin

   Deformation gradient flows

   Math in the Mine - Shape Analysis Meeting in Tende, France, June 26-July 2, 2016
3. K. Modin

   Diffeomorphic density transport - a numerical challenge

   MFO Workshop on Geometric Numerical Integration, Oberwolfach, Germany, March 20-26, 2016

2015

1. K. Modin

   Matrix decompositions and associated Riemannian gradient flows

   Math on the Rocks - Shape Analysis Meeting in Grundsund, Sweden, July 2015
2. K. Modin, M. Bauer, and S. Joshi

   Diffeomorphic density matching using Fisher-Rao geodesics

   ESI workshop: Infinite-dimensional Riemannian Geometry, Vienna, Austria, January 12-16, 2015

2014

1. K. Modin, R. McLachlan, and O. Verdier

   Spherical midpoint method

   Foundations of Computational Mathematics FoCM’14, Montevideo, Uruguay, December 11-20, 2014
2. K. Modin

   Optimal information transport

   AIMS conference 2014, workshop on Geometric Mechanics, Madrid, Spain, July 7-11, 2014
3. K. Modin, R. McLachlan, and O. Verdier

   Spherical midpoint method

   MaGIC 2014, Meråker, Norway, February 24-27, 2014

2013

1. K. Modin, R. McLachlan, and O. Verdier

   Collective integrators for point vortex dynamics on the sphere

   SciCADE 2013, Valladolid, Spain, September 16-20, 2013
2. K. Modin and O. Verdier

   Structure preserving methods for non-holonomic systems: examples and counter-examples

   ANODE 2013 in celebration of the 80th birthday of John Butcher, Auckland, New Zealand, January 7-11, 2013

2011

1. K. Modin, S. Marsland, R. McLachlan, and M. Perlmutter

   On a Geodesic Equation for Planar Conformal Template Matching

   MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA’11), September 22, 2011
2. K. Modin and G. Söderlind

   Geometric integration of Hamiltonian systems perturbed by Rayleigh damping

   Foundations of Computational Mathematics FoCM’11, Budapest, Hungary, July 4-14, 2011

2009

1. K. Modin and G. Söderlind

   Geometric Integration of Hamiltonian Systems Perturbed by Rayleigh Damping

   New Zealand Mathematics Colloquium 2009, Massey University - Albany, Auckland, New Zealand, December 8-10, 2009

2008

1. K. Modin and G. Söderlind

   Nambu Mechanics and its Connection to Adaptive Numerical Integration

   CoPS/Nordita Seminar, Stockholm, Sweden, December 2, 2008
2. K. Modin

   Nambu Mechanics and Time-transformation with Application to Adaptive Numerical Integration

   Nordic Öresund Workshop, Lund, Sweden, November 17-18, 2008

2006

1. K. Modin, C. Führer, and G. Söderlind

   Adaptivity in mechanical integrators

   11th Seminar “NUMDIFF” on Numerical Solution of Differential and Differential-Algebraic equations, Halle, Germany, September 4-8, 2006
2. K. Modin and C. Führer

   Time-step adaptivity in variational integrators

   China-Norway-Sweden Workshop on Computational Mathematics, Lund, Sweden, June 5-7, 2006