Beilinson defined the singular support of a constructible sheaf on a smooth scheme over a field as a closed conical subset on the cotangent bundle. He further proved its existence and fundamental properties, using Radon transform as a crucial tool. In first lectures, we formulate the definition in a slightly different but equivalent way, using an interpretation by Braverman–Gaitsgory of the local acycliciity. We also recall Beilinson’s proof of existence.
In mixed characteristics, the theory is still far from complete. As a replacement of the cotangent bundle, we introduce the Frobenius–Witt cotangent bundle, that has the correct rank but defined only on the characteristic p fiber. Using it, we define  the singular support and its relative variant. Finally, we show that Beilinson’s argument using the Radon transform gives a proof of the existence of the saturation of the relative variant.

Beilinson defined the singular support of a constructible sheaf on a smooth scheme over a field as a closed conical subset on the cotangent bundle. He further proved its existence and fundamental properties, using Radon transform as a crucial tool. In first lectures, we formulate the definition in a slightly different but equivalent way, using an interpretation by Braverman–Gaitsgory of the local acycliciity. We also recall Beilinson’s proof of existence.
In mixed characteristics, the theory is still far from complete. As a replacement of the cotangent bundle, we introduce the Frobenius–Witt cotangent bundle, that has the correct rank but defined only on the characteristic p fiber. Using it, we define  the singular support and its relative variant. Finally, we show that Beilinson’s argument using the Radon transform gives a proof of the existence of the saturation of the relative variant.

Beilinson defined the singular support of a constructible sheaf on a smooth scheme over a field as a closed conical subset on the cotangent bundle. He further proved its existence and fundamental properties, using Radon transform as a crucial tool. In first lectures, we formulate the definition in a slightly different but equivalent way, using an interpretation by Braverman–Gaitsgory of the local acycliciity. We also recall Beilinson’s proof of existence.
In mixed characteristics, the theory is still far from complete. As a replacement of the cotangent bundle, we introduce the Frobenius–Witt cotangent bundle, that has the correct rank but defined only on the characteristic p fiber. Using it, we define  the singular support and its relative variant. Finally, we show that Beilinson’s argument using the Radon transform gives a proof of the existence of the saturation of the relative variant.

The search for alternative compactifications of the moduli space of smooth curves has been central in the panorama of moduli spaces. A possible way to construct such compactifications is allowing curves with worse-than-nodal singularities in the moduli problem and imposing some stability conditions using the combinatorics of the curves to get the desired moduli space. We classify the open substacks inside the moduli stack $mathcal{G}_{1,n}$ of $n$-pointed Gorenstein curves of genus one which admits a proper good moduli space. They agree with those defined by Bozlee, Kuo and Neff. Moreover, we will prove that these spaces are actually projective and we will explain why the classification is a consequence of a wall-crossing phenomenon. This is a on-going project with Luca Battistella and Andrea Di Lorenzo.
 
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Tensor networks including Matrix Product States are powerful tools to investigate quantum many-body systems as well as classical statistical systems, both in conceptual and computational aspects. However, in many cases there are limitations to approach critical systems due to finite bond dimensions. This problem can be circumvented by combining the tensor-network calculation with finite-size scaling of Conformal Field Theory. On the other hand, the effect of the finite bond dimensions can be understood in terms of emergent relevant perturbation to the Conformal Field Theory. I will demonstrate this through a « universal » spectrum of the Matrix Product State transfer matrix.
Refs. A. Ueda and M. O., Phys. Rev. B 104, 165132 (2021); Phys. Rev. B 108, 024413 (2023). J. T. Schneider, A. Ueda, Y. Liu, A. M. Läuchli, M. O., and L. Tagliacozzo, SciPost Phys. 18, 142 (2025).
 
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We define an ‘t Hooft anomaly index for a group acting on a 2d quantum lattice system by finite-depth circuits. It takes values in the degree-4 cohomology of the group and is an obstruction to on-siteability of the group action. We introduce a 3-group (modeled as a crossed square) describing higher symmetries of a 2d lattice system and show that the 2d anomaly index is an obstruction for promoting a symmetry action to a morphism of 3-groups. This demonstrates that ‘t Hooft anomalies are a consequence of a mixing between ordinary symmetries and higher symmetries. Similarly, to any 1d lattice system we attach a 2-group (modeled as a crossed module) and interpret the Nayak-Else anomaly index as an obstruction for promoting a group action to a morphism of 2-groups. The meaning of indices of Symmetry Protected Topological states is also illuminated by higher group symmetry.
 
 
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Séminaire Laurent Schwartz — EDP et applications
 

Séminaire Laurent Schwartz — EDP et applications
 

Séminaire Laurent Schwartz — EDP et applications
 

A category of motives is an axiomatic framework in which several cohomology theories, which typically appear in algebraic geometry, are represented. While Voevodsky’s classical framework of motivic homotopy theory focused on $mathbb{A}^1$-invariant cohomology theories, such as $ell$-adic étale cohomology, the more recent developments in integral $p$-adic Hodge theory have motivated lots of progress towards a more general theory of non-$mathbb{A}^1$-motives in which $p$-adic cohomology theories, such as crystalline or prismatic cohomology, are also represented. In this talk, I want to explain how the Beilinson–Lichtenbaum phenomenon in non-$mathbb{A}^1$-invariant motivic cohomology can be used to shed some light on the proof of Fontaine’s crystalline conjecture in $p$-adic Hodge theory. This is based on a joint work with Arnab Kundu, where we develop a version in families of Gabber’s presentation lemma to prove such a Beilinson–Lichtenbaum phenomenon over general valuation rings. 
 
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Running Seminar
 
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In this talk, I will discuss various aspects of the Distance Conjecture in AdS/CFT. First, I will briefly introduce the Distance Conjecture and how to naturally translate it to the CFT side, as encoded in the CFT Distance Conjecture. In the second part, I will sketch a proof of the first statement in this conjecture, namely that higher-spin symmetry always lies at infinite distance in the conformal manifold of any local CFT in more than two dimensions. For the third part, we will change gears from model-independent proofs to asking more refined questions in well-known models. Specifically, we will focus on a mini-landscape of supersymmetric CFTs in four dimensions which feature three distinct infinite distance limits distinguished by the CFT Distance Conjecture parameter. Borrowing insights from the Swampland program, I will argue that these three limits correspond to three different strings becoming tensionless in AdS. To support this claim, I will discuss how some properties of these CFTs, such as their large N Hagedorn temperature, are determined solely by the CFT Distance Conjecture parameter. I will also discuss how one of these limits nicely fits with the Type IIB string, while another corresponds to a non-critical string theory in AdS.
 
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