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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List of topics I want to present in a mathematically palatable language.
Life processes and information systems: from the cell to large language models (LLMs).Viruses: information storage, transmission, and evolution.Information warfare, organic poisons, infections, and immune systems.The central dogma (Crick): principal information flow in the cell; information reduction under protein folding.Metastability, catalysis, and signaling: signaling in cells, plants, animals, and humans.Information reduction and energy dissipation in epigenesis / development: the emergence of a chick from an egg.Information flows in animal nervous systems and the human brain: (broken) symmetries in olfaction, proprioception, somatosensation, vision, and audition.Modularity of circuitry in cells and in the brain.Poincaré–Sturtevant stochastic representations of geometry; object-dominated life-related (visual and non-visual) signals vs. stochastic continuity of abiotic signals.Stochastic generative combinatorics in languages.Information spaces and Kanerva’s sparse distributed memory (SDM) model.Artificial neural networks, information geometry, and orthogonal symmetries in LLMs.A mathematical perspective on learning, knowledge, and understanding.
 

List of topics I want to present in a mathematically palatable language.
Life processes and information systems: from the cell to large language models (LLMs).Viruses: information storage, transmission, and evolution.Information warfare, organic poisons, infections, and immune systems.The central dogma (Crick): principal information flow in the cell; information reduction under protein folding.Metastability, catalysis, and signaling: signaling in cells, plants, animals, and humans.Information reduction and energy dissipation in epigenesis / development: the emergence of a chick from an egg.Information flows in animal nervous systems and the human brain: (broken) symmetries in olfaction, proprioception, somatosensation, vision, and audition.Modularity of circuitry in cells and in the brain.Poincaré–Sturtevant stochastic representations of geometry; object-dominated life-related (visual and non-visual) signals vs. stochastic continuity of abiotic signals.Stochastic generative combinatorics in languages.Information spaces and Kanerva’s sparse distributed memory (SDM) model.Artificial neural networks, information geometry, and orthogonal symmetries in LLMs.A mathematical perspective on learning, knowledge, and understanding.
 

List of topics I want to present in a mathematically palatable language.
Life processes and information systems: from the cell to large language models (LLMs).Viruses: information storage, transmission, and evolution.Information warfare, organic poisons, infections, and immune systems.The central dogma (Crick): principal information flow in the cell; information reduction under protein folding.Metastability, catalysis, and signaling: signaling in cells, plants, animals, and humans.Information reduction and energy dissipation in epigenesis / development: the emergence of a chick from an egg.Information flows in animal nervous systems and the human brain: (broken) symmetries in olfaction, proprioception, somatosensation, vision, and audition.Modularity of circuitry in cells and in the brain.Poincaré–Sturtevant stochastic representations of geometry; object-dominated life-related (visual and non-visual) signals vs. stochastic continuity of abiotic signals.Stochastic generative combinatorics in languages.Information spaces and Kanerva’s sparse distributed memory (SDM) model.Artificial neural networks, information geometry, and orthogonal symmetries in LLMs.A mathematical perspective on learning, knowledge, and understanding.
 

List of topics I want to present in a mathematically palatable language.
Life processes and information systems: from the cell to large language models (LLMs).Viruses: information storage, transmission, and evolution.Information warfare, organic poisons, infections, and immune systems.The central dogma (Crick): principal information flow in the cell; information reduction under protein folding.Metastability, catalysis, and signaling: signaling in cells, plants, animals, and humans.Information reduction and energy dissipation in epigenesis / development: the emergence of a chick from an egg.Information flows in animal nervous systems and the human brain: (broken) symmetries in olfaction, proprioception, somatosensation, vision, and audition.Modularity of circuitry in cells and in the brain.Poincaré–Sturtevant stochastic representations of geometry; object-dominated life-related (visual and non-visual) signals vs. stochastic continuity of abiotic signals.Stochastic generative combinatorics in languages.Information spaces and Kanerva’s sparse distributed memory (SDM) model.Artificial neural networks, information geometry, and orthogonal symmetries in LLMs.A mathematical perspective on learning, knowledge, and understanding.
 

List of topics I want to present in a mathematically palatable language.
Life processes and information systems: from the cell to large language models (LLMs).Viruses: information storage, transmission, and evolution.Information warfare, organic poisons, infections, and immune systems.The central dogma (Crick): principal information flow in the cell; information reduction under protein folding.Metastability, catalysis, and signaling: signaling in cells, plants, animals, and humans.Information reduction and energy dissipation in epigenesis / development: the emergence of a chick from an egg.Information flows in animal nervous systems and the human brain: (broken) symmetries in olfaction, proprioception, somatosensation, vision, and audition.Modularity of circuitry in cells and in the brain.Poincaré–Sturtevant stochastic representations of geometry; object-dominated life-related (visual and non-visual) signals vs. stochastic continuity of abiotic signals.Stochastic generative combinatorics in languages.Information spaces and Kanerva’s sparse distributed memory (SDM) model.Artificial neural networks, information geometry, and orthogonal symmetries in LLMs.A mathematical perspective on learning, knowledge, and understanding.
 

List of topics I want to present in a mathematically palatable language.
Life processes and information systems: from the cell to large language models (LLMs).Viruses: information storage, transmission, and evolution.Information warfare, organic poisons, infections, and immune systems.The central dogma (Crick): principal information flow in the cell; information reduction under protein folding.Metastability, catalysis, and signaling: signaling in cells, plants, animals, and humans.Information reduction and energy dissipation in epigenesis / development: the emergence of a chick from an egg.Information flows in animal nervous systems and the human brain: (broken) symmetries in olfaction, proprioception, somatosensation, vision, and audition.Modularity of circuitry in cells and in the brain.Poincaré–Sturtevant stochastic representations of geometry; object-dominated life-related (visual and non-visual) signals vs. stochastic continuity of abiotic signals.Stochastic generative combinatorics in languages.Information spaces and Kanerva’s sparse distributed memory (SDM) model.Artificial neural networks, information geometry, and orthogonal symmetries in LLMs.A mathematical perspective on learning, knowledge, and understanding.
 

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.