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Fundamental Mathematical Sciences Lab (Mochizuki)
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The lab studies differential geometry and algebraic geometry, centered on harmonic bundles and twistor D-modules.
AboutAbout the lab
Growing the theory of harmonic bundles and twistor D-modules into something that can be put to use.
Professor Takuro Mochizuki studies differential geometry and algebraic geometry in the Fundamental Mathematical Sciences Division of the Research Institute for Mathematical Sciences.
Beginning with research on the singularities of harmonic bundles, Professor Mochizuki established the theory of objects known as mixed twistor D-modules.
In recent years, Professor Mochizuki has been trying to apply the results and insights obtained so far to related objects and more concrete subjects.
Specifically, the lab studies solutions of the two-dimensional Toda equation, and instantons and monopoles with periodicity.
Key terms
- Harmonic bundle
- A Higgs bundle or flat bundle equipped with a special metric called a pluriharmonic metric.
- D-module
- A framework for handling differential equations in algebraic language. A module on which differential operators act.
- Kobayashi–Hitchin correspondence
- A correspondence linking the algebraic condition of "stability" for vector bundles with the existence of a special metric.
- Stokes phenomenon
- A phenomenon in which the form of the approximation to a differential equation's solution around a singularity switches depending on the direction of approach.
ResearchResearch themes
The lab aims to refine the theory of harmonic bundles and mixed twistor D-modules and grow it into something that can be used in more settings. In recent years, it has been trying to apply earlier results and insights to related objects and more concrete subjects, such as the Toda equation and monopoles.
*Illustrative imageHarmonic Bundles and Twistor D-modules
By studying the singularities of harmonic bundles, the lab classified their asymptotic behavior. It established the theory of mixed twistor D-modules, and in the process also worked on irregular singularities and the Stokes phenomenon.
*Illustrative imageToda Equations and Higgs Bundles
The lab uses the Kobayashi–Hitchin correspondence to classify harmonic bundles on the Higgs bundles that arise in mathematical physics. It has classified real-valued solutions of the two-dimensional Toda equation, and is extending this to the case of noncompact Riemann surfaces through joint research.
*Illustrative imageInstantons and Monopoles
The lab studies periodic instantons and monopoles by viewing them as infinite-dimensional wild harmonic bundles.
*Illustrative imageLimits of Harmonic Bundles on Riemann Surfaces
The lab investigates the phenomena that occur for harmonic bundles on compact Riemann surfaces when the number (scalar) multiplied to the Higgs field is made larger and larger.
MembersFaculty
- ProfessorTakuro Mochizuki
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: “Hopf Fibration” Niles Johnson / CC BY 4.0 (cropped)
- Harmonic Bundles and Twistor D-modules: “AiryAi Arg Surface” Inductiveload / Public domain (cropped)
- Toda Equations and Higgs Bundles: “Riemann sqrt” Wlongqi / CC BY-SA 3.0 (cropped)
- Instantons and Monopoles: “Curvature of BPST Instanton” Tazerenix / CC BY-SA 4.0 (cropped)
- Limits of Harmonic Bundles on Riemann Surfaces: “Orange Genus 2 Surface” Gelada / CC BY-SA 3.0 (cropped)
- Related lab (基礎数理研究部門
（中山 昇） ): “Sarti dodectic 360” Salix alba / CC BY-SA 4.0 (cropped) - Related lab (基礎数理研究部門
（小野 薫） ): “Torus vectors radial” RokerHRO / CC BY-SA 3.0 (cropped) - Related lab (基礎数理研究部門
（入江 慶） ): “Overtwisted contact structure” Pmassot / CC BY-SA 3.0 (cropped) - Related lab (基礎数理研究部門
（中西 賢次） ): “Korteweg-de Vries equation for solitons” Diederik Korteweg. Gustav de Vries. Sander Bais. Gijs Mathijs Ontwerpers, Amsterdam / Public domain (cropped)
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