Denseness conditions, morphisms and equivalences of toposes

We systematically investigate morphisms and equivalences of toposes from multiple points of view. We establish a dual adjunction between morphisms and comorphisms of sites, introduce the notion of weak morphism of toposes and characterize the functors which induce such morphisms. In particular, we examine continuous comorphism of sites and show that this class of comorphisms notably includes all fibrations as well as morphisms of fibrations. We also establish a characterization theorem for essential geometric morphisms and locally connected morphisms in terms of continuous functors, and a relative version of the comprehensive factorization of a functor.
Then we prove a general theorem providing necessary and sufficient explicit conditions for a morphism of sites to induce an equivalence of toposes. This stems from a detailed analysis of arrows in Grothendieck toposes and denseness conditions, which yields results of independent interest. We also derive site characterizations of the property of a geometric morphism to be an inclusion (resp. a surjection, hyperconnected, localic), as well as site-level descriptions of the surjection-inclusion and hyperconnected-localic factorizations of a geometric morphism.

A characterization of complex quasi-projective manifolds uniformized by unit balls

In 1988 Simpson extended the Donaldson-Uhlenbeck-Yau theorem to the context of Higgs bundles, and as an application he proved a uniformization theorem which characterizes complex projective manifolds and quasi-projective curves whose universal coverings are complex unit balls. In this paper we give a necessary and sufficient condition for quasi-projective manifolds to be uniformized by complex unit balls. This generalizes the uniformization theorem by Simpson. Several byproducts are also obtained in this paper.

A non-commutative Bayes' theorem

Using a diagrammatic reformulation of Bayes' theorem, we provide a necessary and sufficient condition for the existence of Bayesian inference in the setting of finite-dimensional $C^*$-algebras. In other words, we prove an analogue of Bayes' theorem in the joint classical and quantum context. Our analogue is justified by recent advances in categorical probability theory, which have provided an abstract formulation of the classical Bayes' theorem. In the process, we further develop non-commutative almost everywhere equivalence and illustrate its important role in non-commutative Bayesian inversion. The construction of such Bayesian inverses, when they exist, involves solving a positive semidefinite matrix completion problem for the Choi matrix. This gives a solution to the open problem of constructing Bayesian inversion for completely positive unital maps acting on density matrices that do not have full support. We illustrate how the procedure works for several examples relevant to quantum information theory.

Backbone free energy estimator applied to viral glycoproteins

Proteins, Backbone Hydrogen Bonds, Backbone Free Energy,
Viral Glycoproteins, Antiviral Vaccine/Drug TargetsEarlier analysis of the Protein Data Bank derived the distribution of rotations from the plane
of a protein hydrogen bond donor peptide group to the plane of its acceptor peptide group. The quasi Boltzmann formalism of Pohl-Finkelstein is employed to estimate free energies of protein elements with these hydrogen bonds pinpointing residues with high propensity for conformational change. This is applied to viral glycoproteins as well as capsids, where the 90th-plus percentiles of free energies determine residues that correlate well with viral fusion peptides
and other functional domains in known cases and thus provide a novel method for predicting these sites of importance as antiviral drug or vaccine targets in general. The method is implemented at
https://bion-server.au.dk/hbonds/
from an uploaded Protein Data Bank file.

Superconnection in the spin factor approach to particle physics

The notion of superconnection devised by Quillen in 1985 and used in gauge-Higgs field theory in the 1990's is applied to the spin factors (finite-dimensional euclidean Jordan algebras) recently considered as representing the finite quantum geometry of one generation of fermions in the Standard Model of particle physics.

Les suites spectrales de Hodge-Tate

This book presents two important results in p-adic Hodge theory following the approach initiated by Faltings, namely (i) his main p-adic comparison theorem, and (ii) the Hodge-Tate spectral sequence. We establish for each of these results two versions, an absolute one and a relative one. While the absolute statements can reasonably be considered as well understood, particularly after their extension to rigid varieties by Scholze, Faltings' initial approach for the relative variants has remained much less studied. Although we follow the same strategy as that used by Faltings to establish his main p-adic comparison theorem, part of our proofs is based on new results. The relative Hodge-Tate spectral sequence is new in this approach.

The relative Hodge-Tate spectral sequence - an overview

We give in this note an overview of a recent work leading to a generalization of the Hodge-Tate spectral sequence to morphisms. The latter takes place in Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work.

Graphon Models in Quantum Physics

In this work we explain some new applications of Infinite Combinatorics to Quantum Physics. We investigate the use of the theory of graphons in non-perturbative Quantum Field Theory and Deformation Quantization which lead us to discover some new interrelationships between these fundamental topics. In one direction, we study Dyson--Schwinger equations in the context of the graph function theory of sparse graphs which enables us to analyze non-perturbative parameters of strongly coupled Quantum Field Theories via cut-distance compact topological regions of Feynman diagrams, Kontsevich's $\star$-product and other new mathematical settings. In another direction, we initiate a theory of graph function representations for Kontsevich admissible graphs to formulate a new topological Hopf algebraic formalism for the study of these graphs which
brings some new useful mathematical tools to relate Deformation Quantization program with non-perturbative renormalization program in Quantum Field Theory models.

Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

For a complex smooth log pair \((Y,D)\), if the quasi-projective manifold \(U=Y-D\) admits a complex polarized variation of Hodge structures with local unipotent monodromies around \(D\) or admits an integral polarized variation of Hodge structures, whose period map is quasi-finite, then we prove that \((Y,D)\) is algebraically hyperbolic in the sense of Demailly, and that the generalized big Picard theorem holds for \(U\): any holomorphic map \(f:\Delta-\{0\}\to U\) from the punctured unit disk to \(U\) extends to a holomorphic map of the unit disk \(\Delta\) into \(Y\). This result generalizes a recent work by Bakker-Brunebarbe-Tsimerman, in which they proved that if the monodromy group of the above variation of Hodge structures is arithmetic, then \(U\) is Borel hyperbolic: any holomorphic map from a quasi-projective variety to \(U\) is algebraic.

Inverses, disintegrations, and Bayesian inversion in quantum Markov categories

We analyze three successively more general notions of reversibility and statistical inference: ordinary inverses, disintegrations, and Bayesian inferences. We provide purely categorical definitions of these notions and show how each one is a strictly special instance of the latter in the cases of classical and quantum probability. This provides a categorical foundation for Bayesian inference as a generalization of reversing a process. To properly formulate these ideas, we develop quantum Markov categories by extending recent work of Cho–Jacobs and Fritz on classical Markov categories. We unify Cho–Jacobs’ categorical notion of almost everywhere (a.e.) equivalence in a way that is compatible with Parzygnat–Russo’s C∗-algebraic a.e. equivalence in quantum probability.