Scientists / Open questions

阿兰·孔涅

Alain Connes

非交换几何、算子代数

1982 年菲尔兹奖

17 个问题

Modular curvature of the noncommutative two-torus: intrinsic meaning of the skew term, analytic theory of the modular Ricci flow, and extensions beyond the conformally flat toric class
Renormalization-group flow of the spectral action in the framework of spectral triples
The BC-system and L-functions: geometric counterpart of the adele class space, canonical constructions at finite primes, p-adic KMS theory, and characteristic one
Hochschild and cyclic (co)homology of the quadratic self-duality quotients of the Yang-Mills algebra
Computing the K-theory and analytic index of foliation C*-algebras and elaborating a type III transverse differential calculus (open problems of Connes' Bourbaki exposé 551)
Spectral Pati-Salam model from noncommutative geometry: vacuum structure, quantum stability of the inner-fluctuation connection, and quantitative phenomenology
Geometric realization of the relative discrete series and decomposition of L²-kernels of invariant elliptic operators on homogeneous spaces of unimodular Lie groups
An adequate Riemann–Roch theorem in characteristic one for the arithmetic site and its square: the missing intersection-theoretic input of the Weil-style strategy for the Riemann zeta function
Classification of spherical noncommutative manifolds beyond the $\theta$ -deformation class and construction of their smooth and metric structures
Open problems in the spectral action for noncommutative spaces with boundary: gauge-field boundary conditions, Lorentzian extension, and applications
Two-sided higher Heisenberg quantization of four-dimensional geometry: particle-picture dynamics and noncommutative representation theory beyond manifolds
Perturbation semigroup structure and generalized inner fluctuations for genuinely noncommutative spectral triples without the first-order condition
General construction of endomotives in arbitrary characteristic: the unreduced, correspondence-based framework
Characteristic classes for Hopf algebra actions and the Hopf-cyclic cohomology of quantum groups: the surviving core of the Connes-Moscovici program
The abelian Singer-group problem for finite projective planes and the unfinished case (3) of the classification of finite hyperfield extensions of the Krasner hyperfield
A noncommutative manifold whose diffeomorphism group is the Connes–Kreimer character group, and a reconstruction formula for its elements
Open issues in Connes' axiomatic foundation of noncommutative geometry (1996): finite-dimensional realization of the algebra-Dirac constraints, the quantum-group origin of the Standard Model algebra, hypoelliptic (quartic) extension, and the Planckian fading of commutativity

返回开放问题

Modular curvature of the noncommutative two-torus: intrinsic meaning of the skew term, analytic theory of the modular Ricci flow, and extensions beyond the conformally flat toric class

Renormalization-group flow of the spectral action in the framework of spectral triples

The BC-system and L-functions: geometric counterpart of the adele class space, canonical constructions at finite primes, p-adic KMS theory, and characteristic one

Hochschild and cyclic (co)homology of the quadratic self-duality quotients of the Yang-Mills algebra

Computing the K-theory and analytic index of foliation C*-algebras and elaborating a type III transverse differential calculus (open problems of Connes' Bourbaki exposé 551)

Spectral Pati-Salam model from noncommutative geometry: vacuum structure, quantum stability of the inner-fluctuation connection, and quantitative phenomenology

Geometric realization of the relative discrete series and decomposition of L²-kernels of invariant elliptic operators on homogeneous spaces of unimodular Lie groups

An adequate Riemann–Roch theorem in characteristic one for the arithmetic site and its square: the missing intersection-theoretic input of the Weil-style strategy for the Riemann zeta function

Classification of spherical noncommutative manifolds beyond the θ\thetaθ-deformation class and construction of their smooth and metric structures

Open problems in the spectral action for noncommutative spaces with boundary: gauge-field boundary conditions, Lorentzian extension, and applications

Two-sided higher Heisenberg quantization of four-dimensional geometry: particle-picture dynamics and noncommutative representation theory beyond manifolds

Perturbation semigroup structure and generalized inner fluctuations for genuinely noncommutative spectral triples without the first-order condition

General construction of endomotives in arbitrary characteristic: the unreduced, correspondence-based framework

Characteristic classes for Hopf algebra actions and the Hopf-cyclic cohomology of quantum groups: the surviving core of the Connes-Moscovici program

The abelian Singer-group problem for finite projective planes and the unfinished case (3) of the classification of finite hyperfield extensions of the Krasner hyperfield

A noncommutative manifold whose diffeomorphism group is the Connes–Kreimer character group, and a reconstruction formula for its elements

Open issues in Connes' axiomatic foundation of noncommutative geometry (1996): finite-dimensional realization of the algebra-Dirac constraints, the quantum-group origin of the Standard Model algebra, hypoelliptic (quartic) extension, and the Planckian fading of commutativity

Classification of spherical noncommutative manifolds beyond the $\theta$ -deformation class and construction of their smooth and metric structures