This is a survey on recent advances in classical decidability issues for local and global fields and for some canonical infinite extensions of those.

# Past Logic Seminar

Abstract: The universal algorithm is a Turing machine program that can in principle enumerate any finite sequence of numbers, if run in the right model of PA, and furthermore, can always enumerate any desired extension of that sequence in a suitable end-extension of that model. The universal finite set is a set-theoretic analogue, a locally verifiable definition that can in principle define any finite set, in the right model of set theory, and can always define any desired finite extension of that set in a suitable top-extension of that model. Recent work has uncovered a $\Sigma_1$-definable version that works with respect to end-extensions. I shall give an account of all three results, which have a parallel form, and describe applications to the model theory of arithmetic and set theory. Post questions and commentary on my blog at http://jdh.hamkins.org/parallels-in-universality-oxford-math-logic-semin...

Both in the real and in the p-adic case, I will talk about recent results about C^r-parameterizations and their diophantine applications. In both cases, the dependence on r of the number of parameterizing C^r maps plays a role. In the non-archimedean case, we get as an application new bounds for rational points of bounded height lying on algebraic varieties defined over finite fields, sharpening the bounds by Sedunova, and making them uniform in the finite field. In the real case, some results from joint work with Pila and Wilkie, and also beyond this work, will be presented,

in relation to several questions raised by Yomdin. The non-archimedean case is joint work with Forey and Loeser. The real case is joint work with Pila and Wilkie, continued by my PhD student S. Van Hille. Some work with Binyamini and Novikov in the non-archimedean context will also be mentioned. The relations with questions by Yomdin is joint work with Friedland and Yomdin.

In joint work with Gareths Boxall and Jones we prove a poly-logarithmic bound for the number of rational points on the graph of functions on the disc that exhibit a certain decay. I will present an application of this counting theorem to the arithmetic of dynamical systems. It concerns fields generated by the solutions of equations of the form $P^{\circ n}(z) = P^{\circ n}(y)$ for a polynomial $P$ of degree $D \geq 2$ where $y$ is a fixed algebraic number. The general goal is to show that the degree of such fields grows like a power of $D^n$.

This lecture will give a brief review of the theory of non-abelian reciprocity maps and their applications to Diophantine geometry, and pose some questions for model-theorists.

Several works (by Kontsevich, Soibelman, Berkovich, Nicaise, Boucksom, Jonsson...) have shown that the limit behavior of a one-parameter family $(X_t)$ of complex algebraic varieties can often be described using the associated Berkovich t-adic analytic space $X^b$. In a work in progress with E. Hrushovski and F. Loeser, we provide a new instance of this general phenomenon. Suppose we are given for every t an $(n,n)$-form $ω_t$ on $X_t$ (for n= dim X). Then under some assumptions on the formula that describes $ω_t$, the family $(ω_t)$ has a "limit" ω, which is a real valued (n,n)-form in the sense of Chambert-Loir and myself on the Berkovich space $X^b$, and the integral of $ω_t$ on $X_t$ tends to the integral of ω on $X^b$.

In this talk I will first make some reminders about Berkovich spaces and (n,n)-forms in this setting, and then discuss the above result.

In fact, as I will explain, it is more convenient to formulate it with $(X_t)$ seen as a single algebraic variety over a non-standard model *C of C and (ω_t) as a (n,n) differential form on this variety. The field *C also carries a t-adic real valuation which makes it a model of ACVF (and enables to do Berkovich geometry on it), and our proof uses repeatedly RCF and ACVF theories.

I will report on two recent papers with D. Ghioca and U. Zannier (joined by P. Corvaja and F. Hu, respectively) in which we consider variants of the Mordell-Lang conjecture. In the first of these, we study the dynamical Mordell-Lang conjecture in positive characteristic, proving some instances, but also showing that in general the problem is at least as hard as a difficult diophantine problem over the integers. In the second paper, we study the Mordell-Lang problem for extensions of abelian varieties by the additive group. Here we have positive results in the function field case obtained by using the socle theorem in the form offered as an aside in Hrushovski's 1996 paper and in the number field case we relate this problem to the Bombieri-Lang conjecture.

I will present a variation of *positive model theory* which addresses the issues of approximations of conventional geometric structures by sequences of Zariski structures as well as approximation by sequences of finite structures. In particular I am interested in applications to quantum mechanics.

I will report on a progress in defining and calculating oscillating in- tegrals of importance in quantum physics. This is based on calculating Gauss sums of order higher or equal to 2 over rings **Z**/*m***Z **for very specific *m*.

We reflect on the set-theoretic ineffability of the Cantorian Absolute of all sets. If this is done in the style of Levy and Montague in a first order manner, or Bernays using second or higher order methods this has only resulted in principles that can justify large cardinals that are `intra-constructible', that is they do not contradict the assumption that V, the universe of sets of mathematical discourse, is Gödel's universe of constructible sets, namely L. Peter Koellner has advanced reasons that this style of reflection will only have this rather limited strength. However set theorists would dearly like to have much stronger axioms of infinity. We propose a widened structural `Global Reflection Principle' that is based on a view of sets and Cantorian absolute infinities that delivers a proper class of Woodin cardinals (and more). A mereological view of classes is used to differentiate between sets and classes. Once allied to a wider view of structural reflection, stronger conclusions are thus possible.

Obtaining Woodin's Cardinals

P. D. Welch, in ``Logic in Harvard: Conference celebrating the birthday of Hugh Woodin''

Eds. A. Caicedo, J. Cummings, P.Koellner & P. Larson, AMS Series, Contemporary Mathematics, vol. 690, 161-176,May 2017.

Global Reflection principles,

P. D. Welch, currently in the Isaac Newton Institute pre-print series, No. NI12051-SAS,

to appear as part of the Harvard ``Exploring the Frontiers of Incompleteness'' Series volume, 201?, Ed. P. Koellner, pp28.