The purpose of this lecture series is to introduce the audience to basic ideas of specific areas of contemporary pure mathematics. Each lecture shall present an area: where it comes from, where it currently is, where it goes. Lectures will be given by prominent mathematicians twice a year: in the Spring and in the Autumn. Before and after each lecture we will organize reading seminars to prepare the audience for the lecture and to dig further into the topic of the lecture. With the lecturer’s consent, lectures will be recorded, slides and/or lecture notes will be provided if available.

Upcoming lecture

 

Time: 14h00, Thursday, November 6, 2025
Venue: Lecture hall 301, A5, Institute of Mathematics, VAST
Lecturer: Takuro Mochizuki (Kyoto)

Title: An aspect of the non-abelian Hodge theory: around wild harmonic bundles and twistor D-modules

Abstract: A harmonic bundle is an interesting geometric object on a complex manifold, defined as a flat bundle with a pluri-harmonic metric or, equivalently, as a Higgs bundle with a pluri-harmonic metric. The concept of the harmonic bundle was discovered in the mid-1980s in the studies of instantons, harmonic maps, and polarized variation of Hodge structure. Corlette, Donaldson, Hitchin, and Simpson established that harmonic bundles on any projective manifold are equivalent to Higgs bundles and flat bundles, respectively, satisfying some algebraic conditions. Since then, many interesting related works have been done, called the non-abelian Hodge theory.

One of the branches of the non-abelian Hodge theory is the research of wild harmonic bundles and twistor $D$-modules on higher dimensional projective manifolds. A harmonic bundle with a wild singularity is called a wild harmonic bundle. The Corlette-Donaldson-Hitchin-Simpson equivalences have been generalized to equivalences for wild harmonic bundles on any complex projective manifold. Also, as flat bundles are generalized to $D$-modules, harmonic bundles are generalized to pure twistor $D$-modules and extended to mixed twistor $D$-modules. One of the main theorems in the theory of wild harmonic bundles and twistor $D$-modules is the functoriality of pure and mixed twistor $D$-modules with respect to standard operations for algebraic holonomic $D$-modules. Another main theorem is an equivalence between semisimple algebraic holonomic $D$-modules and pure twistor $D$-modules. These results are useful in proving the decomposition theorem for semisimple algebraic holonomic $D$-modules. The study also led to a deeper understanding of irregular singularities of meromorphic flat bundles. More recently, ``irregular Hodge filtration'' has been studied based on twistor $D$-modules. There are also applications for the rigidity and boundedness of meromorphic flat bundles.

In this lecture, we plan to overview this aspect of the non-abelian Hodge theory.

Poster

Registration for participation: form

Online participation: Join Zoom Meeting

https://us06web.zoom.us/j/82593112883?pwd=mHKEOY6K68tzbQEr0RpVVzl4BnrcaA.1

Meeting ID: 825 9311 2883

Passcode: 123456

Organizers: Phung Ho Hai, Doan Trung Cuong,

Scientific secretary: Dao Van Thinh

Scientific advisor: Hélène Esnault

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