Schedule for: 24w5224 - Exponential Fields

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

"In terms of real fields with exponential, in particular models of the theories of $R_{exp}$ or $R_{an,exp}$, there are two important classes of objects in parallel: $\mathbf{Hardy\, fields}$ on the functional side, certain subfields of $\mathbf{generalized\, power\, series}$ on the formal side, namely exp-log-series, log-exp-series, transseries. In particular, both are encompassed by Conway's field of $\mathbf{surreal\, numbers}$, which is itself a monster model of $R_{an,exp}$. \[\] During this workshop, there will be several talks connected to these topics. The first purpose of my talk is to provide a quick introduction and overview of the basic definitions and classical properties of these objects, in particular connecting the exponential structure with the ordering, valuation and derivation. The other purpose will be to quote some of the very last achievements, as much as time permits.

I will explain the notions of quasiminimality and exponential algebraic closedness, and survey some progress towards proving that they hold in the complex exponential field.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

This talk provides a survey of exponentiation on the surreals together with an overview of forthcoming work by Ovidiu Costin and myself on integration on the surreals, work in which exponentiation on the surreals plays an important role.

Gareth Jones and I have been suspecting for some time that the commonly used notion of pfaffian function is not equivalent to Khovanskii’s original definition. Jim Freitag’s recent proof that Klein’s $j$-function is not pfaffian (in the “common” sense) confirms this suspicion, and it implies also that the germs of pfaffian functions at 0, say, are not closed under taking implicit functions. Upon close inspection of Khovanskii’s pfaffian functions, we obtain what we call “nested pfaffian functions”. The germs at 0 of the latter are closed under taking implicit functions, which implies that the $j$-function is nested pfaffian. Together with Siegfried van Hille, we are currently exploring other possible closure properties of this class of germs.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

More than three decades after first being considered, the question is still open as to whether there exist o-minimal expansions of the real field that are not exponentially bounded. This is particularly striking as there have always been some (seemingly) obvious candidates for examples. I will discuss this issue and several other related matters, in particular, whether there exists an o-minimal expansion of the ordered additive group of rational numbers that defines a unary function from a bounded set onto an unbounded set.

I will trace some developments in the model-theoretic approach to analytic/algebraic geometry and discuss further prospects.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we will talk about the prospects for characterizing which complex analytic functions are Pfaffian. The perspective will be mostly differential algebraic, but will also touch on o-minimality.

We prove a version of a Nullstellensatz for partial exponential fields even though the ring of exponential polynomials is not a Hilbert ring. We show that under certain natural conditions, one can embed an ideal of the ring of exponential polynomials into an exponential ideal. In case the ideal consists of exponential polynomials with one iteration of the exponential function, we show that these conditions can be met. We apply our results to the case of ordered exponential fields. This is a joint work with Nathalie Regnault.

Zilber's exponential-algebraic closedness conjecture predicts that polynomial-exponential equations must have solutions, unless they have a very good (geometric) reason not to. The same can be asked of other functions having some symmetries, such as abelian exponentials, modular functions, and similar. I'll discuss what V. Aslanyan, S. Eterović and I have done for equations in j, j', j'' in one variable. Even in just one variable, this raises questions that were not addressed in the literature; for instance, we show that the equation j''=0 has "Zariski dense" many solutions, in a suitable sense. I'll discuss the reasoning behind Zilber's conjecture and the ingredients we used to study j and its derivatives.

Hahn groups endowed with the canonical valuation play a fundamental role in the classification of valued abelian groups. In this talk I report on a joint work with Salma Kuhlmann [1], where we study the group of valuation (respectively order) preserving automorphisms of a Hahn group G. Under the assumption that G satisfies some lifting property, we prove a structure theorem decomposing the automorphism group into a semidirect product of two notable subgroups. We characterise a class of Hahn groups satisfying the aforementioned lifting property and, for some special cases, we provide a matrix description of the automorphism group. \[\] [1] S. Kuhlmann and M. Serra, Automorphisms of valued Hahn groups, 2023, arXiv:2302.06290

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Let K be an algebraically bounded structure expanding a field of characteristic 0; for instance, K may be algebraically closed, real closed, Henselian, large and model complete, or the expansion of an algebraically closed field by a generic predicate. If T has elimination of quantifiers, then the expansion of T by one or more derivations has a model completion Tδg. This theory Tδg inherits many of the model-theoretic properties of T: if T is stable or NIP or simple, Tδg is also so. Joint work with G. Terzo.

The abelian exponential-algebraic closedness conjecture, stated by Bays and Kirby, predicts sufficient conditions for solvability of systems of equations involving algebraic operations and the exponential map of a complex abelian variety. This is phrased geometrically, interpreting existence of solutions to these systems as existence of points in the intersection between the graph of the exponential and an algebraic subvariety of the tangent bundle of the abelian variety, and it is motivated by model-theoretic questions on the structure of subsets of the complex numbers that are definable using polynomials and the exponential. In this talk I will report on work in progress concerning the case of the conjecture in which the subvariety of the tangent bundle splits as a product of subvarieties; I will discuss how a method developed by Peterzil and Starchenko to approximate algebraic varieties and a previous result of mine on a simpler case of the conjecture can be used to tackle this problem.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Pfaffian functions are solutions to triangular, polynomial, first-order systems of differential equations -- the exponential function is an example. They were introduced by Khovanskii, who proved effective bounds on the number of non-degenerate roots of polynomial systems of such functions, given in terms of a natural measure of complexity. The Pila--Wilkie Theorem is a seminal result in o-minimality, which provides an upper bound on the number of rational points of bounded numerator and denominator lying on (the transcendental parts of) sets definable in o-minimal expansions of the real field. Following critical insights of Pila and Zannier, there are by now a great many applications of the Pila--Wilkie Theorem (and its variants for algebraic and semi-rational points) to diophantine geometry. However, neither the proof of this theorem (nor that of any of its variants) gives a bound that is effective, which limits the precision of applications. In this talk, aimed at non-experts, I will discuss some joint work with Gal Binyamini, Gareth O. Jones and Harry Schmidt in which we obtain effective, uniform versions of the Pila--Wilkie Theorem and its variants for sets definable using Pfaffian functions (including an effective, uniform Yomdin--Gromov parameterization result in the restricted case), an approach which has subsequently inspired a new perspective on point counting and improved bounds due to Binyamini, Novikov and Zack. I will also discuss how our effective, uniform Pila--Wilkie bounds allow us to derive several effective diophantine applications, including an effective, uniform Manin--Mumford statement for products of elliptic curves with complex multiplication.

Studying the growth properties of definable functions in o-minimal settings, Miller established the following remarkable growth dichotomy: an o-minimal expansion of an ordered field is either power bounded or admits a definable exponential function (see [2]). Going one step further in the hierarchy of growth, Miller’s dichotomy result naturally led to the question whether there exist o-minimal expansions of ordered fields that are not exponentially bounded. Recent research activity in this area is therefore motivated by the search for either an o-minimal expansion of an ordered exponential field by a transexponential function that eventually exceeds any iterate of the exponential or, contrarily, for a proof that any o-minimal expansion of an ordered field is already exponentially bounded. \[\] In this talk, I will firstly present our axiomatic and valuation theoretic approach towards the study of ordered fields equipped with a transexponential function from [1]. Namely, denoting by e an exponential and by T a compatible transexponential, we establish a first-order theory of ordered transexponential fields in which the functional equation $T(x + 1) = e(T(x))$ holds for any positive $x$. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural (i.e. the finest non-trivial convex) valuation. \[\] My talk will then focus on the description and construction of countable non-archimedean ordered transexponential fields, as these can be fully characterised in terms of algebraic featurs of their residue fields and value groups. All relevant valuation theoretic background will be introduced. \[\] [1] L. S. Krapp and S. Kuhlmann, ‘Ordered transexponential fields’, preprint, 2023, arXiv:2305.04607v2. \[\] [2] C. Miller, ‘A Growth Dichotomy for O-minimal Expansions of Ordered Fields’, Logic: from Foundations to Applications (eds W. Hodges, M. Hyland, C. Steinhorn and J. Truss; Oxford Sci. Publ., Oxford Univ. Press, New York, 1996) 385–399.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The exponential is a formal power series that can be evaluated in various contexts, including non-commutative ones. For certain algebras involved in Lie group theory, it induces a bijective correspondence between derivations $\partial$ and automorphisms $\sigma$ of the algebra. Relying on an axiomatic notion of sum of infinite families in certain algebras, we establish a correspondence between derivations and automorphisms in said algebras. This applies in particular to the Lie algebra of contracting derivations which commute with infinite sums defined on fields of generalised power series. This gives a description of one of the groups in Serra and Kuhlmann's decomposition of valuation preserving automorphisms on such fields. This is joint work with Sebastian Krapp, Salma Kuhlmann, Michele Serre, and Daniel Panazzolo.

When we work with exponential polynomial rings over an exponential field (E-field) some classical results fail, as Hilbert’s Basis Theorem and Nullstellensatz. We investigate E-ideals in the E-polynomial ring over an E-ring. Among E-ideals three categories stand out: prime E-ideals; E-ideals which are maximal as ideals, which we will call “strongly maximal”; E-ideals which are maximals among E-ideals, which we will call “E-maximal”. We prove that these three notions are independent, unlike in the classical case. We introduce also the notion of radical E-ideal and we try to give a characterization of it. (Joint work with Paola D’Aquino and Antongiulio Fornasiero)

The model theory of the expansion of the ordered real field by the real exponential function has been studied for several decades. A result of Bianconi states that no restriction of the complex exponential function to a disc is definable in this structure. It turns out that we can consider corresponding results for other transcendental functions such as the p-function and j-function. The association of the p-function to a complex lattice means that one can consider definability questions for structures involving several p-functions, which was done by Jones, Kirby and Servi. This leads naturally to definability questions involving the exponential maps of abelian varieties. This latter work is joint with Jones and Kirby.

The elementary theory of the differential field of (logarithmic-exponential) transseries has been completely axiomatized by Aschenbrenner--van den Dries--van der Hoeven, and this theory is model complete. This talk concerns pairs of models of this theory such that one is a tame substructure of the other. Tameness here is meant in the sense of a tame pair of real closed fields, which goes back to van den Dries--Lewenberg and, ultimately, Macintyre and Cherlin--Dickmann. I will describe work in progress on the model theory of such transserial tame pairs, including a model completeness result for them. An example comes from the differential field of hyperseries, constructed by Bagayoko--Kaplan--van der Hoeven and shown to be an elementary extension of the differential field of transseries by Bagayoko, equipped with an exponentially bounded subfield.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The group of transseries of Dulac type is the natural framework to study the asymptotic expansions of germs of first return maps near hyperbolic polycycles in analytic planar vector fields. Motivated by a problem in dynamical systems, we delve into the classification of Dulac type transseries up to conjugation by non-ramified power series. In a joint work with M. Resman, we uncover a quite surprising rigidity phenomenon, demonstrating that formal conjugacy of formal series implies analytic conjugacy of the underlying analytic germs.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

Conway's proper class-sized field of surreal numbers are constructed from the empty set by an elegant recursive process. This process endows the surreals with a well-founded partial order, and the downward closed subclasses with respect to this partial order are called initial. The surreals also admit an exponential function defined by Gonshor, which makes them an elementary extension of the real exponential field. We consider the following question: which ordered exponential fields are isomorphic to initial exponential subfields of the surreals? We give an answer in terms of Schmeling's conception of a transseries field. As a corollary, we recover Fornasiero's result that any elementary extension of the real field with restricted analytic functions and the unrestricted exponential function admits an initial elementary embedding into the surreal numbers. We also prove some new results on embeddings of transseries fields into the surreals, related to work of Berarducci and Mantova. This is joint work with Philip Ehrlich.