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.