# Patrick SpeisseggerMcMaster University | McMaster · Department of Mathematics and Statistics

Patrick Speissegger

PhD

## About

45

Publications

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Introduction

Patrick Speissegger currently works at the Department of Mathematics and Statistics, McMaster University. Patrick does research in Analysis, Geometry and Model theory (a branch of mathematical Logic). His current projects are 'Quasianalytic Ilyashenko algebras' and 'Roussarie's conjecture for analytic families of planar vector fields with only hyperbolic singularities'.

Additional affiliations

August 2016 - July 2017

June 2003 - present

January 2000 - May 2003

Education

September 1992 - August 1996

October 1986 - January 1991

## Publications

Publications (45)

We construct a model complete and o-minimal expansion of the eld of real numbers in which each real function given on (0; 1) by a series P cnxn with 0 n !1 and P jcnjrn < 1 for some r> 1 is denable. This expansion is polynomially bounded.

We show that the expansion of the real field generated by the functions of a quasianalytic Denjoy-Carleman class is model complete and o-minimal, provided that the class satisfies certain closure conditions. Some of these structures do not admit analytic cell decomposition, and they show that there is no largest o-minimal expansion of the real fiel...

Every o-minimal expansion R-tilde of the real field has an o-minimal expansion P(R-tilde) in which the solutions to Pfaffian equations with definable C^1 coefficients are definable.

We construct a model complete and o-minimal expansion of the field of real numbers such that, for any planar analytic vector field X and any isolated, non-resonant hyperbolic singularity p of X, a transition map for X at p is definable in this structure. This structure also defines all convergent generalized power series with natural support and is...

Let R be an o-minimal expansion of the real field, and let L(R) be the language consisting of all nested Rolle leaves over R. We call a set nested subpfaffian over R if it is the projection of a boolean combination of definable sets and nested Rolle leaves over R. Assuming that R admits analytic cell decomposition, we prove that the complement of a...

We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both $\mathbb{R}_{\mathcal{G}}$ and the reduct of $\mathbb{R}_{\mathrm{an}^*}$ generated by all convergent gen...

We construct a Hardy field that contains Ilyashenko's class of germs at +∞ of almost regular functions found in [12] as well as all log-exp-analytic germs. This implies non-oscillatory behaviour of almost regular germs with respect to all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic e...

We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated with sequences such as $(-n^{s})_{n>0}$ (for $s>0$ ) and $(-s^{n})_{n>0}$ (for $s>1$ ), and also expansions by associated functions such as logarithmic derivatives. There are only...

We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated to sequences such as $(-n^s)_{n>0}$ (for $s>0$) and $(-s^n)_{n>0}$ (for $s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three p...

We construct a Hardy field that contains Ilyashenko's class of germs at infinity of almost regular functions as well as all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic expansion that is an LE-series as defined by van den Dries et al. As these series generally have support of order ty...

I discuss some recent work linking certain aspects of the second part of Hilbert's 16th problem to the theory of \hbox{o-minimality}. These notes are adapted from a lecture I gave in the Jour fixe seminar series at the Zukunfts\-kolleg of Universit\"at Konstanz in June 2017.

We describe maximal, in a sense made precise, analytic continuations of germs at infinity of unary functions definable in the o-minimal structure R_an,exp on the Riemann surface of the logarithm. As one application, we give an upper bound on the logarithmic-exponential complexity of the compositional inverse of an infinitely increasing such germ, i...

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the thirteenth publication in the Lecture Notes in Logic series, collects the proceedings o...

I construct a quasianalytic field $\mathcal F$ of germs at $0^+$ of real functions with logarithmic generalized power series as asymptotic expansions, such that $\mathcal F$ contains all transition maps of hyperbolic saddles of planar real analytic vector fields and is closed under differentiation and composition; in particular, $\mathcal F$ is a H...

Let X be an analytic vector field defined in a neighborhood of the origin of
R^3, and let I be an analytically non-oscillatory integral pencil of X; that
is, I is a maximal family of analytically non-oscillatory trajectories of X at
the origin all sharing the same iterated tangents. We prove that if I is
interlaced, then for any trajectory T in I,...

We consider a 2 2 -dimensional system of linear ordinary differential equations whose coefficients are definable in an o-minimal
structure R \mathcal {R} . We prove that either every pair of solutions at 0 of the system is interlaced or the expansion of R \mathcal {R} by all solutions at 0 of the system is o-minimal. We also show that if the coeffi...

Recent developments in the theory of pfaffian sets are presented from a model-theoretic point of view. In particular, the current state of affairs for Van den Dries's model-completeness conjecture is discussed in some detail. I prove the o-minimality of the pfaffian closure of an o-minimal structure, and I extend a weak model completeness result, o...

Given an o-minimal expansion R of the real field, we show that the structure obtained from R by iterating the operation of adding all total Pfaffian functions over R defines the same sets as the Pfaffian closure of R. 2000 Mathematics Subject Classification 14P10,03C64 (primary); 58A17 (sec-ondary) There are various possibilities for adding Pfaffia...

Preface.- Blowings-up of Vector Fields (F. Cano).- Basics of o-Minimality and Hardy Fields (C. Miller).- Construction of o-Minimal Structures from Quasianalytic Classes (J.-P. Rolin).- Course on Non-Oscillatory Trajectories.- F.S. Sanchez).- Pfaffian Sets and o-Minimality (P. Speissegger).- Theorems of the Complement (A. Fornasiero, T. Servi).

Let R be an o-minimal expansion of the real field. We introduce a class of
Hausdorff limits, the T-infinity limits over R, that do not in general fall
under the scope of Marker and Steinhorn's definability-of-types theorem. We
prove that if R admits analytic cell decomposition, then every T-infinity limit
over R is definable in the pfaffian closure...

An open U ⊆ R is produced such that (R, +, ·, U) defines a Borel isomorph of (R, +, ·, N) but does not define N. It follows that (R, +, ·, U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expa...

Let A be an o-minimal expansion of the real feld, M a submanifold of R n and ω a differentiable 1-form on M. We assume that M and ω are definable in A and ω defines a foliation on M of codimension one. Then there are definable, open subsets M i of M, for i = 1, ..., r, such that every C 1 loop contained in M i is tangent to ker(ω) at some point.

Soit M une sous-variété définissable dans une structure o-minimale A et soit omega une 1-forme différentielle A-définissable et qui définit un feuilletage de codimension un sur M. Nous montrons qu'il existe un recouvrement fini de M par des ouverts A-définissables M1 , ..., Mr qui vérifient la propriété suivante : pour chaque i, tout lacet C1 inclu...

For a vector field F on the Euclidean plane we construct, under certain assumptions on F, an ordered model-theoretic structure associated to the flow of F. We do this in such a way that the set of all limit cycles of F is represented by a definable set. This allows us to give two restatements of Dulac's Problem for F--that is, the question whether...

We are interested in expanding o-minimal structures on the real field by trajectories 1 of definable vector fields. This note 2 is a preliminary report on some progress. We do not attempt to state and prove results as efficiently as possible or in the greatest generality. The reader is assumed to be familiar with o-minimal expansions of the real fi...

Let R be an o-minimal expansion of the real field, and let P(R) be its Pfaffian closure. Let L be the language consisting of all Rolle leaves added to R to obtain P(R). We prove that P(R) is model complete in the language L, provided that R admits analytic cell decomposition. We do this by proving a somewhat stronger statement, the theorem of the c...

We give a geometric proof of the following well-established theorem for o-minimal expansions of the real field: the Hausdorff
limits of a compact, definable family of sets are definable. While previous proofs of this fact relied on the model-theoretic
compactness theorem, our proof explicitly describes the family of all Hausdorff limits in terms of...

I survey two methods of constructing o-minimal expansions of the real field: those generated by certain quasianalytic classes of functions, and those obtained by closing under Pfaffian functions. I also discuss several open questions related to these constructions and examples.

We investigate the asymptotic behavior at +∞ of non-oscillatory solutions to differential equations y′ = G(t, y), t > a, where G: ℝ 1+l → ℝ l is definable in a polynomially bounded o-minimal structure. In particular, we show that the Pfaffian closure of a polynomially bounded o-minimal structure on the real field is levelled.

In this paper, we continue investigations into the asymptotic behavior of solutions of differential equations over o-minimal structures.
Let ℜ be an expansion of the real field (ℝ, +, ·).
A differentiable map F = ( F 1 ,…, F 1 ): ( a, b ) → ℝ i is ℜ-Pfaffian if there exists G : ℝ 1+ l → ℝ l definable in ℜ such that F ′( t ) = G ( t, F ( t )) for al...

We show that the field of real numbers with multisummable real power series is model complete, o-minimal and polynomially
bounded. Further expansion by the exponential function yields again a model complete and o-minimal structure which is exponentially
bounded, and in which the Gamma function on the positive real line is definable. 2000 Mathematic...

. Let e R be an o-minimal expansion of the field of real numbers. We show that if e R has analytic cell decomposition, then its Pfaffian closure P Gamma e R Delta also has analytic cell decomposition. In particular, if e R has analytic Whitney stratification, then so does P Gamma e R Delta . Introduction Let U ` R n be open and ! = a 1 dx 1 + Delta...

We extend results about the asymptotic behaviour of non-oscillatory solutions to semialgebraic ODEs y 0 = G(t; y), with y = (y 1 ; : : : ; y l ) and t > a, by allowing G : R 1+l to be definable in a polynomially bounded o-minimal structure on the real field, and considering only solutions that are non-oscillatory with respect to the structure. As a...

. The open core of a structure R := (R; !; : : : ) is defined to be the reduct (in the sense of definability) of R generated by all of its definable open sets. If the open core of R is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of R is finite or uncount...

. For any o-minimal expansion e IR of the ordered additive group of real numbers, the expansion of e IR by the usual multiplication is o-minimal. Recent developments concerning o-minimal expansions of the field of real numbers show that such structures can be further expanded by solutions to Pfaffian equations (relative to the structure in question...

Introduction During the academic year of 1996--97, the Fields Institute hosted a program in Algebraic Model Theory. As a part of that program, a course on topics from the subject of o-minimality was taught jointly by A. Macintyre, M. Spivakovsky and myself during the fall semester. The main theme of the course was the study of the expansion of the...

We examine how in any o-minimal expansion of a dense linear order, fiberwise open implies pecewise open for sets definable
with parameters, and fiberwise continuous implies piecewise continuous for functions definable with parameters.

Let e R be an o-minimal expansion of the eld of real numbers. We show that if e R has analytic cell decomposition, then its Pfaaan closure P ? e R also has analytic cell decomposition. In particular, if e R has analytic Whitney stratiication, then so does P ? e R .

vating the abstract developments. Applied model theory is using ideas and methods from other parts of mathematics, ranging from homology theory to complex analytic geometry. These two strands of research were exhibited at the BIRS workshop. The workshop was used as an opportunity to exhibit and elucidate two large pieces of technical work which hav...

Printout. Thesis (Ph. D.)--University of Illinois at Urbana-Champaign, 1996. Vita. Includes bibliographical references (leaves 117-118).

## Projects

Projects (8)