Lectures on selected areas in Pure Mathematics

• March 2026: Kęstutis Česnavičius (Orsay)
• 2025, November, Takuro Mochizuki , An aspect of the non-abelian Hodge theory: around wild harmonic bundles and twistor D-modules
• 2025, March: Eva Viehmann, Affine Deligne-Lusztig varieties
• 2024, November: Michel Brion, Around the inverse Galois problem
• 2024, March: Christopher Deninger, Zeta functions and cohomology
• 2023, December: Hélène Esnault, Local systems in algebraic-arithmetic geometry

 Detailed information on the lectures

The Grothendieck--Serre conjecture and its relationship to purity results

Time: 14h00, Thursday, March 26, 2026

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

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

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

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Meeting ID: 825 9311 2883

Passcode: 123456

Affine Deligne-Lusztig varieties
Time: March 20, 2025 at 15:00 (GMT +07)
Lecturer: Eva Viehmann (Muenster)

Abstract: Affine Deligne-Lusztig varieties (ADLVs) have first been introduced by Rapoport and are an analog in affine Grassmannians or affine flag varieties of classical Deligne-Lusztig varieties, a central tool to study representations of finite groups of Lie type. ADLVs play an important role to describe the reduction of Shimura varieties or of moduli spaces of shtukas. In my lecture I will report on what is known about the geometry of ADLVs and on applications to Shimura varieties.

Poster

Around the inverse Galois problem
Time: November 21, 2024, 14:00
Venue: Room 301, Building A5, Institute of Mathematics, VAST

by Michel Brion (Grenoble)

Video Clip

Abstract: The inverse Galois problem asks whether every finite group can be realized as the Galois group of some finite Galois extension of the field Q of rational numbers. This problem is still open, even if many partial results have been obtained. The lectures will present some of these results and a general approach due to Hilbert, via the construction of appropriate Galois extensions of the field of rational functions Q(t). Then we will turn to an analog of the inverse Galois problem for function fields in one variable over an algebraically closed field, where a positive answer is known. Finally, we will discuss a further analog, which asks whether every algebraic group can be realized as the automorphism group of some projective variety. Here the answer is generally negative, as shown by Lombardo and Maffei. But it is positive for linear algebraic groups, due to the work of Florence. 

Registration for participation: form

Poster

Preparation seminar: follow this link

Zeta functions and cohomology

by Christopher Deninger (Muenster)
Title: Zeta functions and cohomology
Time: March 7, 2024, 14:00
Venue: Room 301, Building A5, Institute of Mathematics, VAST

Video: https://www.youtube.com/watch?v=RnPFsjfA5uo

Online participation: Join Zoom Meeting

https://us06web.zoom.us/j/82927090825?pwd=sCz1LoTwwU9lBgM74B7Q1G1jytdD3m.1

Meeting ID: 829 2709 0825
Passcode: 123456

Abstract: The Riemann zeta function ζ (s), first written down by Euler, is of basic importance for the study of the distribution of prime numbers. The Riemann hypothesis, arguably the most famous unsolved conjecture in mathematics, makes the simple prediction that the zeroes of ζ (s) with 0 ≤ Re s ≤ 1 lie on the line Re s = 1/2. Knowing this would have great consequences for the study of prime numbers. The special values of ζ (s) for integers s = n have also attracted a great deal of interest since Euler's proof that ζ (2) = ∑^∞_n=1 n^-2 = π² / 6. Similar formulas exist for even values s = 2 ν with ν ≥ 1 but an understanding of the nature of the values ζ (3) , ζ (5) , … emerged only much later in the work of Borel on higher regulators of algebraic K-groups. The theory of the Riemann zeta function was extended by Dedekind to rings of integers in number fields. It governs the distribution of the prime ideals of the number ring. Around 1920 Artin studied an analogous zeta function for certain smooth projective curves over finite fields and verified an analogue of the Riemann hypothesis for them. More generally, Hasse and Weil introduced a zeta function for all algebraic schemes over spec ℤ (in modern terminology). Hasse proved the Riemann hypothesis for elliptic curves over finite fields and Weil succeeded to do the same for all smooth projective curves over finite fields. Later Weil studied the Hasse-Weil zeta function for higher dimensional varieties over finite fields and in the smooth projective case formulated his famous Weil conjectures. He also suggested a path to approach his conjectures by interpreting the zeta function as an alternating product of characteristic polynomials of the Frobenius homomorphism acting on an as yet undefined cohomology theory for algebraic varieties with similar formal properties as singular cohomology of manifolds. A good deal of Grothendieck's revolutionary new approach to algebraic geometry via schemes was devoted to the development of such a "Weil cohomology'' for algebraic varieties in order to prove the Weil conjectures. This program was successful and culminated in Deligne's proof of the generalized Riemann hypothesis for smooth projective varieties over finite fields in 1973. He used l-adic cohomology for his proof. Later developments also allowed a proof via crystalline cohomology, which was the second Weil cohomology that Grothendieck invented. While l-adic cohomology is based on the study of étale coverings of algebraic varieties, crystalline cohomology is closer in spirit to de~Rham's idea of describing cohomology in terms of differential forms. Over finite fields the cohomological expression of the zeta-function has also been used to prove formulas for their special values. Moreover, cohomology has been a vital tool in Drinfeld's and Lafforgue's proofs of the Langlands conjecture for GL₂ resp. GL_n over function fields of curves. For algebraic schemes over spec ℤ with infinitely many residue characteristics the situation is much less satisfactory. There are precise conjectures on the Hasse-Weil zeta function concerning its analytic continuation, functional equation and the location of its zeros and poles. Moreover Birch-Swinnerton Dyer, Bloch, Beilinson, Kato, Lichtenbaum, Flach-Morin, Soulé and others have predicted the orders of vanishing and the leading terms of the zeta function using cohomology theories (motivic, Deligne, syntomic, Weil-étale and several others). All progress on these conjectures today relies on first expressing the Hasse-Weil zeta function by a product of automorphic L-functions and using ingenious arguments which are specific for the situation. What is missing, even for the Riemann zeta function is an infinite dimensional complex cohomology theory with an operator that could serve the same purposes for general algebraic schemes over spec ℤ as l-adic cohomology with Frobenius endomorphisms for varieties over finite fields. We will explain the formalism that such an infinite dimensional cohomology theory should satisfy and some of its expected properties. We will also mention recent ideas by ourselves and by Clausen-Scholze on a geometry for algebraic schemes over spec ℤ which carries an ℝ⁺-action instead of Frobenius that would give rise to the desired cohomology. 

Registration for participation: form

Poster

Preparation seminar:

l-adic cohomology and Deligne’s proof of Weil conjecture

Time: February 29- March 2, 2024

Venue:  Flamingo Đại Lải Resort, Hà Nội

Registration for participation: form

Plan of seminar: download here

Local systems in Algebraic-Arithmetic Geometry

by Hélène Esnault (Berlin, Harvard and Copenhagen)

Thursday, December 21, 2023 (14:00, GMT 07)
Venue: 301 Lecture hall, A5 Building

Video: https://www.youtube.com/watch?v=Basa5XLQjcM

Online participation:

Join Zoom Meeting

https://us06web.zoom.us/j/82927090825?pwd=sCz1LoTwwU9lBgM74B7Q1G1jytdD3m.1

Meeting ID: 829 2709 0825
Passcode: 123456

Abstract: Riemann proved that a topological cover of a Riemann surface is endowed with an analytic structure of a Riemann surface so that the covering is analytic. This is Riemann’s existence theorem. A Riemann surface underlies an algebraic structure which realizes it as the complex points of an algebraic curve. The topological cover is not only analytic, but algebraic as well. This goes back to Grothendieck in the proper case, to Deligne in general, who introduces the notion a regular singularities at infinity. Poincaré defined the topological fundamental group of topological manifold, in particular of Riemann surfaces. Grothendieck defined the étale fundamental group, which bridges Riemann’s existence theorem with Galois theory of fields. Those groups, defined with the help of a base point,  are difficult to understand, our knowledge is limited. To study them we consider their linear representations, modulo isomorphisms as we want an invariant of the variety only, not of the chosen base point.  A linear representation modulo isomorphism is a local system. 

Our two hours lecture shall make a small journey through some properties of local systems which intertwine topological, analytic, arithmetic properties, some of them being, in the current state of understanding, dreams.

 Registration for participation: form

Poster

Preparation seminar: Local systems.

Time: Thursday 7 and 14, November 2023, 14:00-18:00
Venue: 612 A6, Institute of Mathematics
Online participation:  follow the link above
Plan of seminar: download here