The magic triangle due to Cvitanovic and Deligne-Gross is an extension of the Freudenthal-Tits magic square of semisimple Lie algebras. In a recent work with Kimyeong Lee, we identify all 2d rational conformal field theories associated to the magic triangle. These include various Wess-Zumino-Witten models, Virasoro minimal models, compact bosons and their non-diagonal modular invariants. At level one, we find a two-parameter family of modular linear differential equation of fourth order whose solutions produce the affine characters of all elements in the magic triangle. We find a universal coset relation for the whole triangle which generalizes the dual pairs with respect to (E8)_1 in the Cvitanovic-Deligne exceptional series. At level two, we find a special row of the triangle – the subexceptional series has novel N=1 supersymmetry, and the super characters satisfy a one-parameter family of fermionic modular linear differential equations. Moreover, we find many new coset constructions involving WZW models at higher levels.
 
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Séminaire de géométrie arithmétique
The Cartier transform of Ogus and Vologodsky can be seen as a generalization of Cartier descent. It is an equivalence between modules with integrable connections on a smooth scheme over a perfect field of positive characteristic and Higgs modules on the Frobenius base change of this scheme. We discuss a generalization of this transform to log smooth schemes. More precisely, we discuss two generalizations of Shiho’s local version and Oyama’s crystalline-type version of this transform. For a log smooth scheme $X$ over a perfect field $k$ of positive characteristic, we obtain, under the assumption that the exact relative Frobenius lifts to the Witt vectors, a fully faithful functor from the category of quasi-coherent modules on the base change $X’=Xtimes_{k,F_k}k$ of $X$ equipped with a quasi-nilpotent Higgs field, to the category of quasi-coherent modules on $X$ equipped with a quasi-nilpotent integrable connection. In another direction and without any lifting assumptions, we construct a crystalline-type interpretation of this functor. To address the issue of essential surjectivity, we refine the topoi and crystals mentioned above by endowing them with an indexed structure, inspired by Lorenzon’s extension of Cartier descent to smooth logarithmic schemes.
 
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The absolute Galois group of the rational number field is, of course, a central object in number theory.  However, it is known to be deficient in some respects.  In 1951, André Weil defined what came to be known as the Weil group.  This is a topological group refining the Galois group: it surjects onto the absolute Galois group with nontrivial connected kernel.  The Weil group provides an extension of the theory of Galois representations, allowing for a closer connection with automorphic forms.
In this course, I will explain that there remain further deficiencies of the Weil group, which must be corrected by a further refinement.  Our motivation comes from cohomological considerations, and the refinement we discuss is homotopy-theoretic in nature and goes in an orthogonal direction from the conjectural refinement proposed by Langlands (known as the Langlands group).  Yet, as we will explain, it does have relevance for the Langlands program.

The absolute Galois group of the rational number field is, of course, a central object in number theory.  However, it is known to be deficient in some respects.  In 1951, André Weil defined what came to be known as the Weil group.  This is a topological group refining the Galois group: it surjects onto the absolute Galois group with nontrivial connected kernel.  The Weil group provides an extension of the theory of Galois representations, allowing for a closer connection with automorphic forms.
In this course, I will explain that there remain further deficiencies of the Weil group, which must be corrected by a further refinement.  Our motivation comes from cohomological considerations, and the refinement we discuss is homotopy-theoretic in nature and goes in an orthogonal direction from the conjectural refinement proposed by Langlands (known as the Langlands group).  Yet, as we will explain, it does have relevance for the Langlands program.

The absolute Galois group of the rational number field is, of course, a central object in number theory.  However, it is known to be deficient in some respects.  In 1951, André Weil defined what came to be known as the Weil group.  This is a topological group refining the Galois group: it surjects onto the absolute Galois group with nontrivial connected kernel.  The Weil group provides an extension of the theory of Galois representations, allowing for a closer connection with automorphic forms.
In this course, I will explain that there remain further deficiencies of the Weil group, which must be corrected by a further refinement.  Our motivation comes from cohomological considerations, and the refinement we discuss is homotopy-theoretic in nature and goes in an orthogonal direction from the conjectural refinement proposed by Langlands (known as the Langlands group).  Yet, as we will explain, it does have relevance for the Langlands program.

The absolute Galois group of the rational number field is, of course, a central object in number theory.  However, it is known to be deficient in some respects.  In 1951, André Weil defined what came to be known as the Weil group.  This is a topological group refining the Galois group: it surjects onto the absolute Galois group with nontrivial connected kernel.  The Weil group provides an extension of the theory of Galois representations, allowing for a closer connection with automorphic forms.
In this course, I will explain that there remain further deficiencies of the Weil group, which must be corrected by a further refinement.  Our motivation comes from cohomological considerations, and the refinement we discuss is homotopy-theoretic in nature and goes in an orthogonal direction from the conjectural refinement proposed by Langlands (known as the Langlands group).  Yet, as we will explain, it does have relevance for the Langlands program.

Seed Seminar of Mathematics and Physics
Winter ’26: Flavors of Amplitudes 
The direct detection of gravitational waves has put the relativistic two-body problem in the spotlight and stimulated progress in perturbative approaches that provide analytic insight into its dynamics. Two strategies that have been witnessing interesting developments to this end are the ones based on scattering amplitudes, which apply to binary scatterings at large impact parameter, and on black-hole perturbation theory, which applies to extreme-mass-ratio binaries. In this talk, I will discuss how two very different kinds of differential equations have been playing a key role in such recent developments, in particular with the goal of characterizing the gravitational waves emitted and (re)absorbed by these systems.
Plus d’informations : https://seedseminar.apps.math.cnrs.fr/program/#february-18-2026
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Seed Seminar of Mathematics and Physics
Winter ’26: Flavors of Amplitudes 
To compute properties of phase transitions in condensed matter or the interactions of elementary particles, quantum field theory is typically solved perturbatively. This expansion produces divergent series, so the extraction of meaningful results (resummation) is not straightforward. In fact, very little is known about the actual asymptotic behaviour of these series. In this talk, I will introduce a new limit of quantum field theory (the „tropical“ limit), which is easily computable to very high orders in perturbation theory, yet at the same time captures the full complexity of subdivergences, renormalization, and scheme dependence. I will illustrate that the values of Feynman integrals and their tropical limit are highly correlated. Based on data up to 400 loops, we can precisely determine the asymptotic growth of the (tropical) beta function in different renormalization schemes. In particular, we find unexpectedly complicated instantons, and we confirm the absence of renormalons in the minimal subtraction scheme.
Plus d’informations : https://seedseminar.apps.math.cnrs.fr/program/#february-18-2026
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Seed Seminar of Mathematics and Physics
Winter ’26: Flavors of Amplitudes 
Perturbative calculations of string amplitudes are twofold: an expansion in the string coupling (the genus expansion of the worldsheet) and a low-energy expansion in the momenta. In this talk, I will focus on the low-energy expansion of closed string amplitudes at genus one, specifically for four- and five-point massless states of type IIB superstrings in flat spacetime. Evaluating these amplitudes involves integrating over the moduli space of punctured tori. I will demonstrate how the formalism of equivariant iterated Eisenstein integrals can be used to systematically calculate these integrals. Additionally, I will discuss the implications of these results for the S-duality of type IIB and conclude by exploring the underlying number-theoretic aspects of string amplitudes.
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Seed Seminar of Mathematics and Physics
Winter ’26: Flavors of Amplitudes 
In this presentation, I will first review what purposes axiomatic quantum field theory serves and what the Wightman framework is and achieves. With that being done, I will address the question of the existence of the S-matrix in this framework, following a modern and dimension-independent approach to the Haag–Ruelle scattering; in doing so, I will put a special emphasis on the single-particle problem. With all this discussion having taken place in flat space, I will conclude by presenting some ongoing work on an AdS space version of these arguments.
 
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Seed Seminar of Mathematics and Physics
Winter ’26: Flavors of Amplitudes 
I will talk about recent progress on the computation of gravitational waveforms directly from scattering amplitudes and field-theoretic techniques. I will present the general formalism and give an overview of new results. I will emphasise some technical aspects, such as the computation of the relevant integrals up to next-to-leading order in perturbation theory, and their structure in various limits.
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Seed Seminar of Mathematics and Physics
Winter ’26: Flavors of Amplitudes 
Feynman integrals whose associated geometries extend beyond the Riemann sphere, such as elliptic and Calabi–Yau geometries, are becoming increasingly relevant in modern precision calculations. They arise not only in collider cross-section computations, but also in gravitational-waves scattering.                                                A powerful approach to compute such integrals is based on systems of differential equations, in particular when these can be brought into a canonical form, in which their singularity structure is manifest. In this talk, I will show that canonical Feynman integrals do enjoy similar properties, albeit different associated geometries, and I will illustrate how intersection theory can be used to further study and constrain the functions appearing in the amplitudes.
Plus d’informations : https://seedseminar.apps.math.cnrs.fr/program/#february-4-2026
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