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We survey discrete and continuous model-theoretic notions which have important connections to general topology. We present a self-contained exposition of several interactions between continuous logic and $C_p$-theory which have applications to a classification problem involving Banach spaces not including $c_0$ or $l^p$, following recent results obtained by P. Casazza and J. Iovino for compact continuous logics. Using $C_p$-theoretic results involving Grothendieck spaces and double limit conditions, we extend their results to a broader family of logics, namely those with a first countable weakly Grothendieck space of types. We pose $C_p$-theoretic problems which have model-theoretic implications.
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I summarize Density Functional Theory (DFT) in a language familiar to quantum field theorists, and introduce several apparently novel ideas for constructing {it systematic} approximations for the density functional. I also note that, at least within the large $K$ approximation ($K$ is the number of electron spin components), it is easier to compute the quantum effective action of the Coulomb photon field, which is related to the density functional by algebraic manipulations in momentum space.
This is a largely expository paper about how groups arise or are of interest in model theory. Included are the following topics: classifying groups definable in specific structures or theories and the relation to algebraic groups, groups definable in stable, simple and NIP theories, definable compactifications of groups, definable Galois theory (including differential Galois theory), connections with topological dynamics, model theory of the free group.
Lectures given at the Theoretical Advanced Study Institute (TASI 2020), 1-26 June 2020. The topics covered include quantum circuits, entanglement, quantum teleportation, Bell inequalities, quantum entropy and decoherence, classical versus quantum measurement, the area law for entanglement entropy in quantum field theory, and simulating quantum field theory on a quantum computer. Along the way we confront the fundamental sloppiness of how we all learned (and some of us taught) quantum mechanics in college. Links to a Python notebook and Mathematica notebooks will allow the reader to reproduce and extend the calculations, as well as perform five experiments on a quantum simulator.
We prove that uniform metastability is equivalent to all closed subspaces being pseudocompact and use this to provide a topological proof of the metatheorem introduced by Caicedo, Duenez and Iovino on uniform metastability and countable compactness for logics.
The Grothendieck property has become important in research on the definability of pathological Banach spaces [CI], [HT], and especially [HT20]. We here answer a question of Arhangelskiu{i} by proving it undecidable whether countably tight spaces with Lindelof finite powers are Grothendieck. We answer another of his questions by proving that $mathrm{PFA}$ implies Lindelof countably tight spaces are Grothendieck. We also prove that various other consequences of $mathrm{MA}_{omega_1}$ and $mathrm{PFA}$ considered by Arhangelskiu{i}, Okunev, and Reznichenko are not theorems of $mathrm{ZFC}$.
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