A monotonicity formula for minimal sets with a sliding boundary condition

We prove a monotonicity formula for minimal or almost minimal sets for the Hausdorff measure $\cal{H}^d$, subject to a sliding boundary constraint where competitors for $E$ are obtained by deforming $E$ by a one-parameter family of functions $\varphi_t$ such that $\varphi_t(x) \in L$ when $x\in E$ lies on the boundary $L$. In the simple case when $L$ is an affine subspace of dimension $d-1$, the monotone or almost monotone functional is given by $F(r) = r^{-d} \cal{H}^d(E \cap B(x,r)) + r^{-d} \cal{H}^d(S \cap B(x,r))$, where $x$ is any point of $E$ (not necessarily on $L$) and $S$ is the shade of $L$ with a light at $x$. We then use this, the description of the case when $F$ is constant, and a limiting argument, to give a rough description of $E$ near $L$ in two simple cases. ----- On donne une formule de monotonie pour des ensembles minimaux ou presque minimaux pour la mesure de Hausdorff $\cal{H}^d$, avec une condition de bord o\`u les comp\'etiteurs de $E$ sont obtenus en d\'eformant $E$ par une famille \`a un param\`etre de fonctions $\varphi_t$ telles que $\varphi_t(x)\in L$ quand $x\in E$ se trouve sur la fronti\`ere $L$. Dans le cas simple o\`u $L$ est un sous-espace affine de dimension $d-1$, la fonctionelle monotone ou presque monotone est donn\'ee par $F(r) = r^{-d} \cal{H}^d(E \cap B(x,r)) + r^{-d} \cal{H}^d(S \cap B(x,r))$, o\`u $x$ est un point de $E$, pas forc\'ement dans $L$, et $S$ est l'ombre de $L$, \'eclair\'ee depuis $x$. On utilise ceci, la description des cas o\`u $F$ est constante, et un argument de limite, pour donner une description de $E$ pr\`es de $L$ dans deux cas simples.Comment: 100 page

Counter-example in 3D and homogenization of geometric motions in 2D

In this paper we give a counter-example to the homogenization of the forced mean curvature motion in a periodic setting in dimension $N\ge 3$ when the forcing is positive. We also prove a general homogenization result for geometric motions in dimension $N=2$ under the assumption that there exists a constant $\delta>0$ such that every straight line moving with a normal velocity equal to $\delta$ is a subsolution for the motion. We also present a generalization in dimension $2$, where we allow sign changing normal velocity and still construct bounded correctors, when there exists a subsolution with compact support expanding in all directions

Haag's theorem in renormalised quantum field theories

We review a package of no-go results in axiomatic quantum field theory with Haag's theorem at its centre. Since the concept of operator-valued distributions in this framework comes very close to what we believe canonical quantum fields are about, these results are of consequence to quantum field theory: they suggest the seeming absurdity that this highly victorious theory is incapable of describing interactions. We single out unitarity of the interaction picture's intertwiner as the most salient provision of Haag's theorem and critique canonical perturbation theory to argue that renormalisation bypasses Haag's theorem by violating this very assumption.Comment: 80 pages, 8 Feynman diagrams, 3 appendix section

Coherent sheaves, superconnections, and RRG

Given a compact complex manifold, the purpose of this paper is to construct the Chern character for coherent sheaves with values in Bott-Chern cohomology, and to prove a corresponding Riemann-Roch-Grothendieck formula. Our paper is based on a fundamental construction of Block.Comment: 161 pages, 1 figure. In version 2, references to earlier work have been adde

Introduction to Nonsmooth Analysis and Optimization

This book aims to give an introduction to generalized derivative concepts useful in deriving necessary optimality conditions and numerical algorithms for infinite-dimensional nondifferentiable optimization problems that arise in inverse problems, imaging, and PDE-constrained optimization. They cover convex subdifferentials, Fenchel duality, monotone operators and resolvents, Moreau--Yosida regularization as well as Clarke and (briefly) limiting subdifferentials. Both first-order (proximal point and splitting) methods and second-order (semismooth Newton) methods are treated. In addition, differentiation of set-valued mapping is discussed and used for deriving second-order optimality conditions for as well as Lipschitz stability properties of minimizers. The required background from functional analysis and calculus of variations is also briefly summarized.Comment: arXiv admin note: substantial text overlap with arXiv:1708.0418

Basic introduction to higher-spin theories

This is a collection of my lecture notes on the higher-spin theory course given for students at the Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University. The goal of these lectures is to give an introduction to higher-spin theories accessible to master level students which would enable them to read the higher-spin literature. I start by introducing basic relevant notions of representation theory and the associated field-theoretic descriptions. Focusing on massless symmetric fields I review different approaches to interactions as well as the no-go results. I end the lectures by reviewing some of the currently available positive results on interactions of massless higher-spin fields, namely, holographic, Chern-Simons and chiral higher-spin theories.Comment: 159 pages, minor corrections, references adde

Land Perspectives: People, Tenure, Planning, Tools, Space, and Health

Good land administration and spatial enablement help to improve people’s living conditions in urban, peri-urban, and rural areas. They protect people’s land rights (including of individuals, communities, and the state) through good governance principles and practices. This makes research concerning land administration practices and geographic (spatial) sciences—whether in developed or developing countries—essential to developing tools or methods for securing natural resource rights for people. In the time of COVID-19, understanding the land and health or wellbeing nexus is also crucial for adequate living conditions for people in living urban, peri-urban, and rural areas. This Special Issue comprises 15 articles (including the editorial) that present insights on theories and practices on land administration and geographic (spatial) sciences in the context of land/water/forest–people–health–wellbeing nexus

A monotonicity formula for minimal sets with a sliding boundary condition

We prove a monotonicity formula for minimal or almost minimal sets for the Hausdorff measure Hd, subject to a sliding boundary constraint where competitors for E are obtained by deforming E by a one-parameter family of functions yt such that yt(x) ∈ L when x ∈ E lies on the boundary L. In the simple case when L is an affine subspace of dimension d-1, the monotone or almost monotone functional is given by F(r) = r-d Hd (E∩B(x, r)) + r-d Hd (S∩B(x,r)) where x is any point of E (not necessarily on L) and S is the shade of L with a light at x. We then use this, the description of the case when F is constant, and a limiting argument, to give a rough description of E near L in two simple cases