Generalized Differential Geometry

Generalized Functions play a central role in the understanding of differential equations containing singularities and nonlinearities. Introducing infinitesimals and infinities to deal with these obstructions leads to controversies concerning the existence, rigor and the amount of non-standard analysis needed to understand these theories. Milieus constructed over the generalized reals sidestep them all. A Riemannian manifold M embeds discretely into a generalized manifold $M^*$ on which singularities vanish and products of nonlinearities make sense. Linking this to an already existing global theory provides an algebra embedding $\kappa :\hat{{\cal{G}}}(M)\longrightarrow {\cal{C}}^{\infty}(M^*,\widetilde{\mathbb{R}}_f)$. Generalized Space-Time is constructed and its possible effects on Classical Space-Time are examined

The Hurewicz dichotomy for generalized Baire spaces

By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space $X$ is covered by a $K_\sigma$ subset of $X$ if and only if it does not contain a closed-in-$X$ subset homeomorphic to the Baire space ${}^\omega \omega$. We consider the analogous statement (which we call Hurewicz dichotomy) for $\Sigma^1_1$ subsets of the generalized Baire space ${}^\kappa \kappa$ for a given uncountable cardinal $\kappa$ with $\kappa=\kappa^{<\kappa}$, and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. Moreover, we show that if the Generalized Continuum Hypothesis (GCH) holds, then there is a cardinal preserving class-forcing extension in which the Hurewicz dichotomy for $\Sigma^1_1$ subsets of ${}^\kappa \kappa$ holds at all uncountable regular cardinals $\kappa$, while strongly unfoldable and supercompact cardinals are preserved. On the other hand, in the constructible universe L the dichotomy for $\Sigma^1_1$ sets fails at all uncountable regular cardinals, and the same happens in any generic extension obtained by adding a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the $\kappa$-perfect set property, the $\kappa$-Miller measurability, and the $\kappa$-Sacks measurability.Comment: 33 pages, final versio

Lebesgue's Density Theorem and definable selectors for ideals

We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold. In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which makes the ideal of countable sets satisfy an analogue to the density theorem. The proofs of the positive results use only elementary combinatorics of trees, while the negative results rely on forcing arguments.Comment: 28 pages; minor corrections and a new introductio

Perfect subsets of generalized Baire spaces and long games

We extend Solovay's theorem about definable subsets of the Baire space to the generalized Baire space ${}^\lambda\lambda$, where $\lambda$ is an uncountable cardinal with $\lambda^{<\lambda}=\lambda$. In the first main theorem, we show that that the perfect set property for all subsets of ${}^{\lambda}\lambda$ that are definable from elements of ${}^\lambda\mathrm{Ord}$ is consistent relative to the existence of an inaccessible cardinal above $\lambda$. In the second main theorem, we introduce a Banach-Mazur type game of length $\lambda$ and show that the determinacy of this game, for all subsets of ${}^\lambda\lambda$ that are definable from elements of ${}^\lambda\mathrm{Ord}$ as winning conditions, is consistent relative to the existence of an inaccessible cardinal above $\lambda$. We further obtain some related results about definable functions on ${}^\lambda\lambda$ and consequences of resurrection axioms for definable subsets of ${}^\lambda\lambda$

The Higher Cicho\'n Diagram

For a strongly inacessible cardinal $\kappa$, we investigate the relationships between the following ideals: - the ideal of meager sets in the ${<}\kappa$-box product topology - the ideal of "null" sets in the sense of [Sh:1004] (arXiv:1202.5799) - the ideal of nowhere stationary subsets of a (naturally defined) stationary set $S_{\rm pr}^\kappa \subseteq \kappa$. In particular, we analyse the provable inequalities between the cardinal characteristics for these ideals, and we give consistency results showing that certain inequalities are unprovable. While some results from the classical case ($\kappa=\omega$) can be easily generalized to our setting, some key results (such as a Fubini property for the ideal of null sets) do not hold; this leads to the surprising inequality cov(null)$\le$non(null). Also, concepts that did not exist in the classical case (in particular, the notion of stationary sets) will turn out to be relevant. We construct several models to distinguish the various cardinal characteristics; the main tools are iterations with $\mathord<\kappa$-support (and a strong "Knaster" version of $\kappa^+$-cc) and one iteration with ${\le}\kappa$-support (and a version of $\kappa$-properness).Comment: 84 page