A polyfold proof of the Arnold conjecture

We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin–Salamon–Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from $${{\mathbb {C}}{\mathbb {P}}}^1\times M$$ C P 1 × M to $${{\mathbb {C}}}^+ \times M$$ C + × M and $${{\mathbb {C}}}^-\times M$$ C - × M , as developed by Fish–Hofer–Wysocki–Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.

Random fractals and their intersection with winning sets

We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$ ), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$ , $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$ , where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$ . We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$ . We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.

Extreme Points and Majorization: Economic Applications

We characterize the set of extreme points of monotonic functions that are either majorized by a given function f or themselves majorize f and show that these extreme points play a crucial role in many economic design problems. Our main results show that each extreme point is uniquely characterized by a countable collection of intervals. Outside these intervals the extreme point equals the original function f and inside the function is constant. Further consistency conditions need to be satisfied pinning down the value of an extreme point in each interval where it is constant. We apply these insights to a varied set of economic problems: equivalence and optimality of mechanisms for auctions and (matching) contests, Bayesian persuasion, optimal delegation, and decision making under uncertainty.

On Fermat-Torricelli Problem in Frechet Spaces

We study the Fermat-Torricelli problem (FTP) for Frechet space X, where X is considered as an inverse limit of projective system of Banach spaces. The FTP is defined by using fixed countable collection of continuous seminorms that defines the topology of X as gauges. For a finite set A in X consisting of n distinct and fixed points, the set of minimizers for the sum of distances from the points in A to a variable point is considered. In particular, for the case of collinear points in X, we prove the existence of the set of minimizers for FTP in X and for the case of non collinear points, existence and uniqueness of the set of minimizers are shown for reflexive space X as a result of strict convexity of the space.

On the Hilbert boundary-value problem for Beltrami equations with singularities

We investigate the Hilbert boundary-value problem for Beltrami equations $\overline\partial f=\mu\partial f$ with singularities in generalized quasidisks $D$ whose Jordan boundary $\partial D$ consists of a countable collection of open quasiconformal arcs and, maybe, a countable collection of points. Such generalized quasicircles can be nowhere even locally rectifiable but include, for instance, all piecewise smooth curves, as well as all piecewise Lipschitz Jordan curves. Generally speaking, generalized quasidisks do not satisfy the standard $(A)-$condition in PDE by Ladyzhenskaya-Ural'tseva, in particular, the outer cone touching condition, as well as the quasihyperbolic boundary condition by Gehring-Martio that we assumed in our last paper for the uniformly elliptic Beltrami equations. In essence, here, we admit any countable collection of singularities of the Beltrami equations on the boundary and arbitrary singularities inside the domain $D$ of a general nature. As usual, a point in $\overline D$ is called a singularity of the Beltrami equation, if the dilatation quotient $K_{\mu}:=(1+|\mu|)/(1-|\mu|)$ is not essentially bounded in all its neighborhoods. Presupposing that the coefficients of the problem are arbitrary functions of countable bounded variation and the boundary data are arbitrary measurable with respect to the logarithmic capacity, we prove the existence of regular solutions of the Hilbert boundary-value problem. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann, and Poincar\'{e} boundary-value problems for equations of mathematical physics with singularities in anisotropic and inhomogeneous media.

Presentations of left-c.e. reals

This chapter examines presentations of left–c.e. reals, proving Theorem 1.4. One of the main ideas of this book is unifying the combinatorics of constructions in various subareas of computability theory. The chapter looks at one such subarea: algorithmic randomness. It provides a brief account of the basics of algorithmic randomness, and includes the basic definitions required in the chapter. While algorithmic randomness has a history going back to the early work of Borel on normal numbers, von Mises, and even Turing, the key concept in the modern incarnation of algorithmic information theory is Martin-Löf randomness. A notion of randomness is determined by a countable collection of null sets, with each null set considered a statistical test. Elements of the null sets are those which have failed the test; they are atypical, in the sense of measure. One of the reasons the notion of ML-randomness is central is that it is robust.

Hypergraphs and proper forcing

Given a Polish space [Formula: see text] and a countable collection of analytic hypergraphs on [Formula: see text], I consider the [Formula: see text]-ideal generated by Borel anticliques for the hypergraphs in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies.

Properties of domain representations of spaces through dyadic subbases

A dyadic subbase S of a topological space X is a subbase consisting of a countable collection of pairs of open subsets that are exteriors of each other. If a dyadic subbase S is proper, then we can construct a dcpo DS in which X is embedded. We study properties of S with respect to two aspects. (i) Whether the dcpo DS is consistently complete depends on not only S itself but also the enumeration of S. We give a characterization of S that induces the consistent completeness of DS regardless of its enumeration. (ii) If the space X is regular Hausdorff, then X is embedded in the minimal limit set of DS. We construct an example of a Hausdorff but non-regular space with a dyadic subbase S such that the minimal limit set of DS is empty.

The unions of dense metrizable subspaces with certain local properties

Many important examples of topological spaces can be represented as a union of a finite or countable collection of metrizable subspaces. However, it is far from clear which spaces in general can be obtained in this way. Especially interesting is the case when the subspaces are dense in the union. We present below several results in this direction. In particular, we show that if a Tychonoff space X is the union of a countable family of dense metrizable locally compact subspaces, then X itself is metrizable and locally compact. We also prove a similar result for metrizable locally separable spaces. Notice in this connection that the union of two dense metrizable subspaces needn?t be metrizable. Indeed, this is witnessed by a well-known space constructed by R.W. Heath.

STRATIFIED TRANSVERSALITY OF HOLOMORPHIC MAPS

We discuss genericity and stability of transversality of holomorphic maps to complex analytic stratifications. We prove that the set of maps between Stein manifolds and Oka manifolds transverse to a countable collection of submanifolds in the target is dense in the space of holomorphic maps with the weak topology. This greatly generalizes earlier results on the genericity of transverse maps by Forstnerič and by Kaliman and Zaidenberg. As an application we show that the Whitney (a)-regularity of a complex analytic stratification is necessary and sufficient for the stability of transverse holomorphic maps between a Stein manifold and an Oka manifold. This gives an analogue of a theorem in the real case due to Trotman.