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A majority of established quantum generalizations of discrete structures are shown to be instances of a single quantum generalization. In particular, the quantum graphs of Duan, Severini and Winter, the quantum metric spaces of Kuperberg…
Following Robert's [26], we study the structure of unitary groups and groups of approximately inner automorphisms of unital $C^*$-algebras, taking advantage of the former being Banach-Lie groups. For a given unital $C^*$-algebra $A$, we…
This is the second in a sequence of three articles exploring the relationship between commutative algebras and $E_\infty$-algebras in characteristic $p$ and mixed characteristic. Given a topological space $X,$ we construct, in a manner…
To every minimal model of a complete local isolated cDV singularity Donovan--Wemyss associate a finite dimensional symmetric algebra known as the contraction algebra. We construct the first known standard derived equivalences between these…
We construct Hopf algebra isomorphisms of discrete multiplier Hopf C*-algebras, and Hopf AF C*-algebras (generalized quantum UHF algebras), from K-theoretical data. Some of the intermediate results are of independent interest, such as a…
The real Calogero--Moser space $\mathcal{C}_n^\mathbb{R}$ is a noncompact, totally real submanifold of the complex Calogero--Moser space $\mathcal{C}_n$. We prove that every symplectic diffeomorphism of $\mathcal{C}_n^\mathbb{R}$ smoothly…
We define quantum automorphism groups of a wide range of discrete structures. The central tool for their construction is a generalisation of the Tannaka-Krein reconstruction theorem. For any direct sum of matrix algebras $M$, and any…
We give examples of minimal diffeomorphisms of compact connected manifolds which are not topologically orbit equivalent, but whose transformation group C*-algebras are isomorphic. The examples show that the following properties of a minimal…
We define Hamiltonian simplex differential graded algebras (DGA) with differentials that deform the high energy symplectic homology differential and wrapped Floer homology differential in the cases of closed and open strings in a Weinstein…
Given any symplectomorphism on $D^{2n} (n\geq 1)$ which is $C^{\infty}$ close to the identity, and any completely integrable Hamiltonian system $\Phi^t_H$ in the proper dimension, we construct a $C^{\infty}$ perturbation of $H$ such that…
We propose an algebraic viewpoint of the problem of deformation quantization of the so called almost Poisson algebras, which are algebras with a commutative associative product and an antisymmetric bracket which is a bi-derivation but does…
In the present paper, we introduce the notion of nearly holomorphic Drinfeld modular forms and study an analogue of Maass-Shimura operators in this context. Furthermore, for a given nearly holomorphic Drinfeld modular form, we show that its…
This paper investigates if a differential graded algebra can have more than one $A_\infty$-structure extending the given differential graded algebra structure. We give a sufficient condition for uniqueness of such an $A_\infty$-structure up…
Connectivity is a homotopy invariant property of a separable C*-algebra A which has three important consequences: absence of nontrivial projections, quasidiagonality and realization of the Kasparov group KK(A,B) as homotopy classes of…
We present some results concerning the generalized homologies associated with nilpotent endomorphisms $d$ such that $d^N=0$ for some integer $N\geq 2$. We then introduce the notion of graded $q$-differential algebra and describe some…
In this work, we present two alternative yet equivalent representation formulae for Whitney forms that are valid for any choice of coordinates, and generalizes the original characterization of Whitney forms in Whitney (1957) that requires…
We define $A_{\infty}$-structures -- algebras, coalgebras, modules, and comodules -- in an arbitrary monoidal DG category or bicategory by rewriting their definitions in terms of unbounded twisted complexes. We develop new notions of strong…
We introduce and study certain deformations of Drinfeld quasi-modular forms by using rigid analytic trivialisations of corresponding Anderson's t-motives. We show that a sub-algebra of these deformations has a natural graduation by the…
In this work we construct Calabi quasi-morphisms on the universal cover of the group Ham(M) of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the…
Branched rough paths, defined as paths with values in the character group of the Connes-Kreimer Hopf algebra $\mathcal{H}_\mathrm{CK}$, constitute integration theories that may fail to satisfy the usual integration by parts identity. Using…