This talk is about a mixture of old and new work. First I will talk about how you can iterate Pappus’s theorem and construct a 2-parameter family of relatively Anosov representations of the modular group into Isom(X), where X = SL(3,R)/SO(3). Then I will explain how to interpret these representations as symmetry groups of patterns of geodesics in X that have the same asymptotic properties as the Farey graph in the hyperbolic plane. Finally I will say a few words about how this picture allows for a complete classification of the Barbot component of discrete faithful representations of the modular group into Isom(X).

A quasifuchsian manifold is a hyperbolic structure on the product of a surface and the line that is naturally compactified by two conformal structures at infinity. By a classical result of Bers, curves that have bounded length in the hyperbolic structures on these surfaces also have bounded length in the hyperbolic 3-manifold. However, the converse fails — one can construct examples of quasifuchsian manifolds that contain curves of bounded length in the 3-manifold while the curves are arbitrarily long in the hyperbolic structures at infinity. To rectify this, Minsky gave a description of the bounded length curves in the 3-manifold in terms of the data at infinity using the curve complex. These a priori bounds played a central role in the Brock-Canary-Minsky proof of the ending lamination conjecture. Bowditch later gave a new proof of these bounds. We will describe another proof of this result. While it uses many of the ideas of the approaches of Minsky and Bowditch, unlike their proofs the result is effective.

Human postnatal development is typically viewed from the perspective of our ‘human’ organs.  As we come to appreciate how our microbial communities are assembled following birth, there is an opportunity to determine how this microbial facet of our developmental biology is related to healthy growth as well as to the risk for and manifestations of disorders that produce abnormal growth. We are testing the hypothesis that perturbations in the normal development of the gut microbiome are causally related to childhood undernutrition, a devastating global health problem whose long-term sequelae, including stunting, neurodevelopmental abnormalities, plus metabolic and immune dysfunction, remain largely refractory to current therapeutic interventions. The journey to preclinical proof-of-concept, and the path forward to clinical proof-of-concept emphasize the opportunities as well as the experimental, analytic and other challenges encountered when developing microbiota-directed therapeutics.
 
 

The widely anticipated revolution in biology, driven by the superpowers of GAI, has been slow to materialize. In this talk, I will discuss the concept, the obstacles, and examine two specific projects involving language and image analysis. The first case study involves extracting information from a specific type of biological literature—model organisms’ lifespan extension under pharmacological perturbation—where we aim to agglomerate experimental parameters over an extensive body of published literature and automatically review and assess the quality of emerging studies in that context. The second project focuses on novel high-resolution image analysis, where we aim to characterize protein and lipid remodeling in various organs and tissues, to detect changes in a nested hierarchy of repetitive elements of tissue architectures and elucidate changes reflecting sex specificity, aging, and disease. We will anticipate what will be needed for a productive man-machine symbiosis to emerge in systems biology.
 

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.

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