Antongiulio Fornasiero

From September 2008 I am a postdoc at the University of Münster. I was a postdoc at the University of Freiburg and the University of Pisa.
I work mainly with Alessandro Berarducci; my interests are in logic and model theory, especially o-minimality and ordered structures.

Home address via Vivaldi 23 -- 35028 Piove di Sacco (PD), Italy Email antongiulio.fornasiero at googlemail.com Phone (in Germany!) Mobile (+49) 175 / 356 1704 Office (+49) 251 / 833 3766 Fax (+49) 251 / 833 3078

Teaching

Freiburg '08

O-minimality of Pfaffian functions

Münster, WS08

Exercises for Logik 3

Münster, SS09

Student seminar on Model Theory: Every Monday at 12:15, SR8.

Bio

I was born. Then, I graduted in mathematics at the University of Pisa. I got a Ph.D. in mathematics at the University of Edinburgh. I came back in Pisa, with a post-doc position, for 4 years, then I moved to Freiburg for 1 year. Now I am in Münster with a(nother) post-doc position.
CV (pdf file).

PhD thesis

Integration on Surreal Numbers (pdf, abstract)
Directed by A. Macintyre, at the University of Edinburgh,
Defended on January 2004 with the jury: A. Macintyre, A. Carbery, A. Maciocia, D. Richardson.

Articles

• Show abstracts. Hide abstracts.
• Dimension, matroids, and dense pairs of first-order structures (pdf file). A structure M is pregeometric if the algebraic closure is a pregeometry in all M' elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of omega and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique existential matroid.
  Generalising previous results by van den Dries, we define dense elementary pairs of structures expanding a field and with an existential matroid, and we show that the corresponding theories have natural completions, whose models also have a unique existential matroid. We extend the above result to dense tuples of structures.
• Baire's theorem and definably complete structures (with T. Servi) (pdf file), submitted. We consider definably complete and Baire expansions of ordered fields: every definalbe subset of the domain of the structure has a supremum and the domain can not be written as the union of a definable increasing family of nowhere dense sets. Every expansion of the real field is definably complete and Baire. So is every o-minimal expansion of a field. The converse is clearly not true. However, unlike the o-minimal case, the structures considered form an elementary class. In this context we prove a version of Kuratowski-Ulam's Theorem and some restricted version of Sard's Lemma. We also prove a relative version of Speissegger's Theorem on the Pfaffian closure of o-minimal structures.
• O-minimal cohomology: finiteness and invariance results (with A. Berarducci) (pdf file), submitted. We prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language. We also study the cohomology of the intersection of a definable decreasing family of definably compact sets, under the additional assumption that the o-minimal structure expands a field.
• O-minimal spectrum (pdf file), preliminary version. Let X be a definable sub-set of some o-minimal structure. We study the spectrum of X, in relation with the definability of types.
• Arithmetic of Dedekind cuts of ordered Abelian groups (with M. Mamino) (pdf file), published on Annals of Pure and Applied Logic, Volume 156, Issues 2-3, December 2008, pages 210--244; doi:10.1016/j.apal.2008.05.001. This paper fills a much needed (Dedekind) gap in the literature.
  We study the set of Dedekind cuts of a linearly ordered Abelian group as a structure over the language (0,<,+,-). Moreover, we obtain a simple set of axioms for the universal part of the theory of such structures. Finally, we prove that every structure satisfying the given axioms is a sub-structure of the set of cuts over a suitable group.
• Embedding Henselian fields into power series (pdf file), published in the Journal of Algebra, Volume 304, Issue 1, 1 October 2006, pages 112--156; doi:10.1016/j.jalgebra.2006.06.037. Every Henselian field of residue characteristic 0 admits a truncation-closed embedding in a field of generalised power series (possibly, with a factor set). As corollaries we obtain Ax-Kochen-Ershov theorem and an extension of Mourgues and Ressayre's theorem: every ordered field which is Henselian in its natural valuation has an integer part. We also give some results for the mixed and the finite characteristic cases. For a longer and better abstract, see also the MathSciNet review.
• Recursive definitions on surreal numbers (pdf file), submitted. Let No be Conway's class of surreal numbers. I will make explicit the notion of a function f on No recursively defined over some family of functions. Under some "tameness" and uniformity condition, f must satisfy some interesting properties; in particular, the supremum of the class of element x such that f(x) is greater or equal to a fixed d in No is actually an element of No. For similar reasons, the concatenation function x:y cannot be defined recursively in a uniform way over polynomial functions.
• O-minimality of the standard part (pdf file). Let M be an o-minimal structure expanding a field. Let R be the residue field of M, with the structure generated by the images of definable subsets of M under the residue map. Using a theorem by Baisalov and Poizat, we prove that R is weakly o-minimal.

Slides

• Show abstracts. Hide abstracts.
• Tame ordered structures (pdf file): commented slides of the talk given at the Logic Colloquium 2009, Sofia, July 31 - August 5, 2009. Various notion of tameness for linearly ordered structures, generalyzing o-minimality.
• Pairs of fields (pdf file): notes of the seminar given at Freiburg, June 2008. Exposition of some theorems on (dense) pairs of structures.
• Truncation-closed embeddings of Henselian fields into power series (gzipped postscript file). The topic is my article Embedding Henselian fields into generalised power series fields.
• O-minimal spectrum (pdf file): slides of the talk given at Regensburg, June 2007. The topic is my article O-minimal spectrum.
• Tagli di Dedekind di gruppi Abeliani ordinati (pdf file, in Italian): commented slides of the talk given at the Incontro Italiano Insiemi e Modelli, Torino, 2-4 April 2007.
  Dedekind cuts of ordered Abelian groups (pdf file): slides of a talk given at Freiburg, 22 Oct 2007. The topic is my article with M. Manino Arithmetic of Dedekind cuts of ordered Abelian groups.
• Hausdorff measure on o-minimal structures (pdf file): commented slides of the talk given at the Colloque autour de l'o-minimalité, Paris, 11-13 September 2006. Let K be an o-minimal structure expanding an ordered field. Expanding the work of Berarducci and Otero, we define the d-dimensional Hausdorff measure of definable subset of $K^n$, where d<n are natural numbers. We prove some of the properties for this measure, analogue of the ones for the Hausdorff measure on the reals.
• Initial Embeddings in the Surreal Numbers of Models of $T_{an}(\exp)$ (pdf file): commented slides on the talk given at the Oxford Workshop in Model Theory, 3-7 September 2006. Every model of $T_{an}$ or of $T_{an}(exp)$ admits an initial embedding in the field of surreal numbers No.
• Čech cohomology of definable sets in o-minimal structures (with A. Berarducci) (pdf file): commented slides of the talk given at the International Congress NonStandard Methods and Applications in Mathematics, Pisa, 25-31 May 2006. Let A be a semi-algebraic set, definable without using parameters in some real closed field M. Let à be its real spectrum, and A(R) be the realization of A on the reals. We will give a simple proof, based on the trasfer principle, of the fact that the Čech cohomology of à is isomorphic to the one of A(R) in a natural way. With a similar proof, we show an analogous result also for other o-minimal structures M.

Programs and files

Xmodmap Linux keymap for US windows keyboard: maps the Windows and Menu keys to something useful.
ansi-codes.el Elisp package for Emacs. An enhached version of A. Schroeder's package ansi-color.el.
How to add a random quotation to your signature in PINE.
My own Latex tips.

Links

See also: the (almost) complete list of my links at del.icio.us, and the list of Latex-related links on my latex page.

Entertainment

• The valuation theory home page
• MathSciNet
• Zentralblatt MATH
• arXiv
• RAAG

Pisa

• Biblioteca dipartimento di matematica
• Biblioteca SNS
• Dipartimento di matematica
• Mail

Words

• Wikipedia Free enciclopedia
• dictionary.com English dictionary
• Leo German dictionary
• Garzanti Italian dictionary
• Wordtheque Virtual library
• Biblioteca dei Classici Italiani Virtual library of Italian literature

Work

• XKCD
• Dresden Codak
• Get Fuzzy
• Calvin and Hobbes
• Sam and Fuzzy

Quotations

• On war...

  We make war that we may live in peace.
  — Aristotle, Nichomachean Ethics

• ...and peace

  You don't prevent anything by war except peace.
  — Harry Truman

• On working

  All paid jobs absorb and degrade the mind.
  — Aristotle

• On intellectual property

  It is not once nor twice but times without number that the same ideas make their appearance in the world.
  — Aristotle, On The Heavens

Please write me if you have suggestions, find mistakes and/or some links are not working.
My e-mail address is antongiulio dot fornasiero at googlemail.com

Last modified: Sat Aug 22 10:54:36 CEST 2009 by A. Fornasiero

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