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Irie Lab
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The lab studies periodic orbits of dynamical systems using symplectic geometry.
AboutAbout the lab
Approaching periodic orbits of dynamical systems with the tools of geometry.
The Irie Lab studies symplectic geometry, in the fields of topology and differential geometry.
It focuses in particular on symplectic capacities and periodic orbits of Hamiltonian dynamical systems.
Building on Floer homology, a theory that arose from pseudoholomorphic curves, the lab investigates the computation of capacities, periodic orbits of billiard dynamical systems, closing lemmas, and more.
In the future, the lab also has in view establishing a method to compute Floer homology by finite-dimensional Morse theory in situations with a torus action.
Key terms
- Symplectic geometry
- The geometry that studies the structure of the space combining the positions and momenta of objects (phase space)
- Periodic orbit
- The path of a motion that returns to its original state after a fixed time and repeats the same movement
- Floer homology
- A theory that uses pseudoholomorphic curves to summarize information such as periodic orbits into algebraic quantities
NewsLab news
Associate Professor Irie (at the time) received the Japan Academy Medal
Associate Professor Irie (at the time) received the JSPS Prize (Japan Society for the Promotion of Science)
ResearchResearch themes
The lab studies symplectic geometry, especially symplectic capacities, periodic orbits of Hamiltonian dynamical systems, and related topics. Based on Floer homology, it works on computing capacities, string topology, closing lemmas, and more.
*Illustrative imageComputing Capacities and Billiard Dynamical Systems
The lab derived a method that reduces the computation of capacities of “fiberwise convex” domains in cotangent bundles to computing the (relative) homology of loop spaces, and gave several applications.
*Illustrative imageString Topology and Floer Theory
The lab rigorously carried out part of Fukaya's program connecting Floer theory of Lagrangian submanifolds with string topology (at the chain complex level).
*Illustrative imageClosing Lemmas for Dynamical Systems
Using embedded contact homology (ECH), the lab proved a C∞ closing lemma for Reeb dynamical systems on closed three-dimensional contact manifolds. It also showed that a similar argument works with Morse theory for minimal surfaces.
*Illustrative imageExtending the Closing Lemma to Higher Dimensions
Motivated by the question of whether the closing lemma extends to Reeb dynamical systems on contact manifolds of dimension five and higher, the lab defined a property called the strong closing property.
MembersFaculty
- ProfessorKei Irie
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Overtwisted contact structure” Pmassot / CC BY-SA 3.0 (cropped)
- Computing Capacities and Billiard Dynamical Systems: “Billiard trajectories in an ellipse (three behaviors)” Dixy52 / CC0 (cropped)
- String Topology and Floer Theory: “Homotopy of pointed circle maps” KSmrq / CC BY-SA 3.0 (cropped)
- Closing Lemmas for Dynamical Systems: “Standard contact structure r 3” Pmassot / CC BY-SA 3.0 (cropped)
- Extending the Closing Lemma to Higher Dimensions: “Hopfkeyrings” 作者不詳 / Public domain (cropped)
- Related lab (Ono Lab): “Torus vectors radial” RokerHRO / CC BY-SA 3.0 (cropped)
- Related lab (Fundamental Mathematical Sciences Lab (Mochizuki)): “Hopf Fibration” Niles Johnson / CC BY 4.0 (cropped)
- Related lab (Yamashita Lab): “Riemann zeta function absolute value vi” Conscious / CC BY 2.5 (cropped)
- Related lab (Nakanishi Lab): “Korteweg-de Vries equation for solitons” Diederik Korteweg. Gustav de Vries. Sander Bais. Gijs Mathijs Ontwerpers, Amsterdam / Public domain (cropped)
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