Related papers: On the motivic measure on the space of functions

We define a motivic measure on the Berkovich analytification of an algebraic variety defined over a trivially valued field, and introduce motivic integration in this setting. The construction is geometric with a similar spirit as…

A survey of work on motivic integration.

We define a motivic analogue of the Haar measure for groups of the form G(k((t))), where k is an algebraically closed field of characteristic zero, and G is a reductive algebraic group defined over k. A classical Haar measure on such groups…

In this paper, we study harmonic functions on metric measure spaces with Riemannian Ricci curvature bounded from below, which were introduced by Ambrosio-Gigli-Savar\'e. We prove a Cheng-Yau type local gradient estimate for harmonic…

We elaborate notions of integration over the space of arcs factorized by the natural $C^*$-action and over the space of non-parametrized arcs (branches). There are offered two motivic versions of the zeta function of the classical monodromy…

There are two definitions of the measurable functional on the topological vector space: as a linear and measurable real-valued function and as a pointwise limit of the sequence of the continious linear functionals. In general case they are…

We show how a metric space induces a linear functional (a "mean") on real-valued functions with domains in that metric space. This immediately induces a "relative" measure on a collection of subsets of the underlying set.

We give an explicit formula for the motivic integrals related to the Milnor number over spaces of parametrised arcs on the plane with fixed tangency orders with the axis. These integrals are rational functions of the parameters and the…

We show that every operator in $L^{2}$ has an associated measure on a space of functions and prove that it can be used to find solutions to abstract Cauchy problems, including partial differential equations. We find explicit formulas to…

Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence theory of metric measure spaces. We investigate several…

We introduce a generalization of symmetric functions and apply the resulting theory to compute the class in the Grothendieck ring of varieties of the space of geometrically irreducible hypersurfaces of a fixed degree in projective space.

In this survey article we discuss a framework of noncommutative geometry with differential graded categories as models for spaces. We outline a construction of the category of noncommutative spaces and also include a discussion on…

A functional equation for the motivic integral corresponding to the Milnor number of an arc is derived using the Denef-Loeser formula for the change of variables. Its solution is a function of five auxiliary parameters, it is unique up to…

We introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative cases. This allows us to define a motivic Fourier transformation for which we get…

We consider some measure-theoretic properties of functions belonging to a Sobolev-type class on metric measure spaces that admit a Poincar\'e inequality and are equipped with a doubling measure. The properties we have selected to study are…

We prove a recognition principle for motivic infinite P1-loop spaces over a perfect field. This is achieved by developing a theory of framed motivic spaces, which is a motivic analogue of the theory of E-infinity-spaces. A framed motivic…

Some fixed point results are given for a class of functional contractions over partial metric spaces. These extend some contributions in the area due to Ilic et al [Math. Comput. Modelling, 55 (2012), 801-809].

We develop a "motivic integration" version of the Poisson summation formula for function fields, with values in the Grothendieck ring of definable exponential sums. We also study division algebras over the function field, and obtain…

We propose a definition of magnitude for a length space with a Borel measure, which involves integrals over the set of geodesics. This quantity agrees with the magnitude of finite metric spaces, up to re-scaling the metric to ensure the…

We introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative cases. This allows us to define a motivic Fourier transformation for which we get…