Cover

Title page

Contents

Preface

Program of the conference

Some results on global real analytic geometry

1. Emergence of real analytic spaces

2. Inequalities and the global approach

3. Amenable 𝐶-semianalytic sets and irreducible components

4. Nullstellensätze

References

Dimension in the realm of transseries

Introduction

1. Differential-algebraic dimension

2. Dimension and codimension

3. The case of \T

4. Dimension 0 = discrete

5. Parametrizability by constants

6. Dimension 0 = co-analyzable relative to the constant field

References

Stability index of algebras with involution

1. Introduction

2. Notation

3. Signatures revisited

4. Stability index of algebras with involution

References

Efficient algorithms for computing the Euler-Poincaré characteristic of symmetric semi-algebraic sets

1. Introduction

2. Mathematical preliminaries

3. Equivariant deformation

4. Algorithms and the proofs of the main theorems

5. Appendix

References

Gram spectrahedra

Introduction

1. Gram matrices and sums of squares

2. Connections to toric geometry

3. Ranks on Gram spectrahedra

4. Examples: Binary forms and ternary quartics

5. Hermitian Gram spectrahedra

6. Sums of squares on products of simplices

References

Constructions in the category of real semigroups

1. Fundamentals

2. Real semigroups of continuous functions

3. Inductive limits

4. RS-sums

5. The space of RS-characters of products of real semigroups

6. The Functor from Preordered Rings to Real Semigroups

References

Recent progress on definability of Henselian valuations

1. Definability of a given henselian valuation

2. Definable henselian valuations on a given henselian field

3. Quantifier complexity and questions of uniformity

Acknowledgements

References

Polynomial, regular and Nash images of Euclidean spaces

1. First examples and obstructions

2. One dimensional polynomial and regular images of \Rⁿ

3. Examples and obstructions for dimension two

4. Obstructions for the boundaries of polynomial images

5. Piecewise linear semialgebraic sets as polynomial and regular images

6. Nash images of \Rⁿ

References

Witt equivalence of fields: A survey with a special emphasis on applications of hyperfields

1. Introduction

2. Hyperfields

3. Quadratic hyperfields

4. Witt equivalence

5. Henselian valued fields

6. Local fields

7. Behaviour of orderings and valuations under Witt equivalence

8. Global fields

9. Function fields

10. Function fields over local and global fields

References

Infinite dimensional moment problem: open questions and applications

Introduction

1. Preliminaries

2. The moment problem on lmc topological \RR-algebras

3. The moment problem on symmetric algebras of lc real spaces

Acknowledgments

References

A new proof for the existence of degree bounds for Putinar’s Positivstellensatz

1. Introduction, definitions and historical outline of the problem

2. The purpose of the main theorem and its requirements

3. Proof of the theorem

References

Coincidence point theorems for ball spaces and their applications

1. Introduction

2. Proofs of the main theorems

3. Applications to ultrametric spaces

4. An application to complete metric spaces

5. An application to ordered abelian groups and fields

6. Fixed point theorems

References

Valuation theory of exponential Hardy fields II: Principal parts of germs in the Hardy field of o-minimal exponential expansions of the reals

1. Introduction

2. Some preliminaries

3. A general structure theorem for 𝐿𝐸_{\cal𝐹}(𝑥)

4. An intrinsic version of “truncation at 0”

5. Applications

References

Théories géométriques pour l’algèbre des nombres réels

Introduction

Introduction

1. Théories \gmqsz

2. Théories dynamiques pour les corps ordonnés

3. Corps ordonnés réels clos non discrets

Conclusion

Conclusion

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Surreal numbers with derivation, Hardy fields and transseries: a survey

1. Introduction

2. Surreal numbers

3. From Hardy fields to transseries and 𝐻-fields

4. Surreal derivations

References

Polynomials nonnegative on the cylinder

Introduction

1. Proof of Theorem 1

2. Proof of Theorem 2

References

Positive semifields and their ideals

1. Introduction

2. Definition of semifields, and examples

3. Homomorphisms

4. Ideals and factor semifields

5. The ideal spectrum of a semifield

6. Cancellation modulo ideals and the cancellation radical

7. The 𝑙-kernel

8. Ideals under semifield extensions

References

On the strength of some topological lattices

1. Introduction

2. The weak monadic structure of a first order structure

3. Finite unions of closed intervals

4. Defining closed and bounded intervals

5. Linear spaces and convex sets

6. Lattices of closed definable sets in the ordered context

7. Zariski closed sets and p-adic sets

8. Grzegorczyk’s paper

References

On the local geometry of definably stratified sets

1. Introduction

2. Definitions

3. Normal pseudo-flatness in the non-polynomially bounded case

4. Pawłucki’s example

References

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