Related papers: A Guided Tour to Normalized Volume
Inspired by regularization techniques in statistics and machine learning, we study complementary composite minimization in the stochastic setting. This problem corresponds to the minimization of the sum of a (weakly) smooth function endowed…
We present here a brief review and discussion on recent developments in the theory of spacetime singularities. After mentioning some key motivations on the main ideas and concepts involved, we take the approach that the singularities will…
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 review realistic models that reproduce quantum theory in some limit and yield potentially new physics outside that limit. In particular, we consider deterministic hidden-variables theories (such as the pilot-wave model) and their…
A new renormalization scheme for theories with nontrivial internal symmetry is proposed. The scheme is regularization independent and respects the symmetry requirements.
The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.
In this paper we consider the problem of minimizing area subject to a volume constraint in a given convex set.
We extend the framework of K-stability (Tian, Donaldson) to more general algebro-geometric setting, such as partial desingularisations of (fixed) singularities, (not necessarily flat) families over higher dimensional base and the classical…
We examine four dimensional magnetically charged extremal black holes in certain non-linear U(1) gauge theories coupled to two derivative gravity. For a given coupling, one can tune the magnetic charge (or vice versa) so that the curvature…
We present a brief description of the ``consistent discretization'' approach to classical and quantum general relativity. We exhibit a classical simple example to illustrate the approach and summarize current classical and quantum…
This is neither an elementary introduction to singularity theory nor a specialized treatise containing many new theorems. The purpose of this little book is to invite the reader on a mathematical promenade. We pay a visit to Hipparchus,…
2+1 gravity coupled to a massless scalar field has an initial singularity when the spatial slices are compact. The quantized model is used here to investigate several issues of quantum gravity. The spectrum of the volume operator is studied…
The aim of the paper is to characterize Kawamata log terminal singularities and log canonical singularities by dimensions of jet schemes. It is a generalization of Mustata's result.
We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and…
The intrinsic volumes of Gaussian polytopes are considered. A lower variance bound for these quantities is proved, showing that, under suitable normalization, the variances converge to strictly positive limits. The implications of this…
In this paper we will explore fundamental constraints on the evolution of certain symplectic subvolumes possessed by any Hamiltonian phase space. This research has direct application to optimal control and control of conservative mechanical…
This is an expository survey article on compound Du Val (cDV) singularities, with emphasis on recent homological approaches, including: noncommutative resolutions, tilting theory, contraction algebras, classification, derived categories,…
We consider the problem of minimizing the sum of submodular set functions assuming minimization oracles of each summand function. Most existing approaches reformulate the problem as the convex minimization of the sum of the corresponding…
This is a survey article on $F$-singularities and their applications.
The end of LEP and SLC is a good moment to review the way to summarize search results in order to exploit at best, in future analyses and speculations, the pieces of information coming from all experiments. Some known problems with the…