Related papers: Unrectifiable normal currents in Euclidean spaces
We consider the problem of determining the class of continuous-time dynamical systems that can be globally linearized in the sense of admitting an embedding into a linear system on a higher-dimensional Euclidean space. We solve this problem…
We study some fundamental properties of real rectifiable currents and give a generalization of King's theorem in characterizing currents defined by positive real holomorphic chains. Our proof uses Siu's semicontinuity theorem and largely…
We present some results concerning currents of integration on finite-dimensional analytic spaces in Hilbert spaces, using the setting of metric currents. In particular, we obtain the characterization of such currents as positive closed…
Many generative models originally developed in finite-dimensional Euclidean space have functional generalizations in infinite-dimensional settings. However, the extension of rectified flow to infinite-dimensional spaces remains unexplored.…
In this paper, we study the two-dimensional steady compactly supported incompressible Euler equations with free boundaries. We consider flows with constant vorticity that are perturbations of annular equilibria, in contrast to the laminar…
We introduce a new family of gauge invariant regularizations of Chern-Simons theories which generate one-loop renormalizations of the coupling constant of the form $k\to k+2 s c_v$ where $s$ can take any arbitrary integer value. In the…
The zero curvature representation for two dimensional integrable models is generalized to spacetimes of dimension d+1 by the introduction of a d-form connection. The new generalized zero curvature conditions can be used to represent the…
The main difficulty of quantum field theory is the problem of divergences and renormalization. However, realistic models of quantum field theory are renormalized within the perturbative framework only. It is important to investigate…
A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures $\mu$ in $n$-dimensional Euclidean space for all $n\geq 2$ in terms of…
We construct a branched center manifold in a neighborhood of a singular point of a $2$-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the…
We show that in two dimensions the incompressible Euler equations can be re-expressed in terms of an abelian gauge theory with a Chern-Simons term. The magnetic field corresponds to fluid vorticity and the electric field is the product of…
A comprehensive study of one-dimensional metric currents and their relationship to the geometry of metric spaces is presented. We resolve the one-dimensional flat chain conjecture in this general setting, by proving that its validity is…
In Part I of the paper, we prove non-uniqueness of the solution to the Cauchy problem of the Euler equations of an ideal incompressible fluid in dimension two with vorticity in some Lebesgue space. The radially symmetric external force is…
We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the…
We introduce an invariant linked to some foundational questions in geometric measure theory and provide bounds on this invariant by decomposing an arbitrary cycle into uniformly rectifiable pieces. Our invariant measures the difficulty of…
The most general version of a renormalizable $d=4$ theory corresponding to a dimensionless higher-derivative scalar field model in curved spacetime is explored. The classical action of the theory contains $12$ independent functions, which…
For nice functions, invariant means over integral currents (certain generalized surfaces), can be uniquely defined.
Let $X$ be a (reduced) pure-dimensional analytic space. We prove that direct images of principal value and residue currents on $X$ are smooth outside sets that are small in a certain sense. We also prove that the sheaf of such currents,…
In two lectures, we overview the renormalon and renormalon-related techniques and their phenomenological applications. We begin with a single renormalon chain which is a well defined and systematic way to specify the character of…
Abandoning dimensional regularization allows important simplifications in loop calculations and gives a handle to interpret non-renormalizable Quantum Field Theories. I review the current status of FDR, a fully four-dimensional approach to…