相关论文: Model theory of valued fields

We work with semi-algebraic functions on arbitrary real closed fields. We generalize the notion of critical values and prove a Sard type theorem in our framework.

This note proposes a new notion of a gradient-like vector field and discusses its implications for the theory of Stein and Weinstein structures.

This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric…

The progress of the last decade in perturbative quantum field theory at high temperature and density made possible by the use of effective field theories and hard-thermal/dense-loop resummations in ultrarelativistic gauge theories is…

This article is the second part in the series of articles where we are developing theory of valuations on manifolds. Roughly speaking valuations could be thought as finitely additive measures on a class of nice subsets of a manifold which…

The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum $\mathrm{Spec}(\mathbb{Z})$ defined by Grothendieck serves this purpose. However, it is still not clear what…

The study of many problems in additive combinatorics, such as Szemer\'edi's theorem on arithmetic progressions, is made easier by first studying models for the problem in F_p^n for some fixed small prime p. We give a number of examples of…

We present closed forms for several functions that are fundamental in number theory and we explain the method used to obtain them. Concretely, we find formulas for the p-adic valuation, the number-of-divisors function, the sum-of-divisors…

Pseudo algebraically closed, pseudo real closed, and pseudo $p$-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this text, we propose a unified framework for…

We study completeness in partial differential varieties. We generalize many results from ordinary differential fields to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and…

We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.

New constructions in the theory of fields for multiple integrals are designed. Generalizations of the Legendre - Weyl - Caratheodory transforms and corresponding invariant integrals are introduced and explored. Connection and curvature of…

Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…

In this summary of my talk at Strings 2016, I explain how classical dynamics on an infinite tree graph can be dual to a conformal field theory defined over the $p$-adic numbers. An informal introduction to $p$-adic numbers is followed by a…

In this paper we develop a theory of class invariants associated to $p$-adic representations of absolute Galois groups of number fields. Our main tool for doing this involves a new way of describing certain Selmer groups attached to…

We introduce vectorial and topological continuities for functions defined on vector metric spaces and illustrate spaces of such functions. Also, we describe some fundamental classes of vector valued functions and extension theorems.

Machine learning methods based on statistical principles have proven highly successful in dealing with a wide variety of data analysis and analytics tasks. Traditional data models are mostly concerned with independent identically…

We study a Szemer\'edi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields.

I begin my discussion by summarizing the methodology proposed and new distributional results on multivariate log-Gamma derived in the paper. Then, I draw an interesting connection between their work with mean field variational Bayes.…

This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of a linear space. In this article we still work…