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 3-years assistant 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
A dichotomy for expansions of the real field (with P. Hieronymi and C. Miller)
(ArXiv); Proceedings of AMS 141 (2013), 697-698, doi:10.1090/S0002-9939-2012-11369-3.
A dichotomy for expansions of the real field is established: Either the set of integers is definable or every nonempty bounded nowhere dense definable subset of the real numbers has Minkowski dimension zero.
Expansions of the reals which do not define the natural numbers
(ArXiv); a version of this article has been submitted.
We study first-order expansions of the reals which do not define the set of natural numbers. We also show that several stronger notions of tameness are equivalent to each others.
Definably connected nonconnected sets
(pdf file); Math. Log. Quart. 58, No. 1-2, 125-126 (2012), doi: 10.1002/malq.201100062.
We give an example of a structure K on the real line, and a manifold M
definable in K, such that M is definably connected but is not connected.
Lovely pairs for independence relations
(pdf file); submitted.
In the literature there are two different notions of lovely pairs
of a theory T, according to whether T is simple or geometric.
We introduce a notion of lovely pairs for an independence relation,
which generalizes both the simple and the geometric case, and show
how the main theorems for those two cases extend to our general notion.
Hausdorff measure on o-minimal structures (with Elisa Vasquez Rifo)
(ArXiv); JSL, 77:2 (2012), 631-648,
doi:10.2178/jsl/1333566642.
We introduce the Hausdorff measure for definable sets in an o-minimal
structure, and prove the Cauchy-Crofton and co-area formulae for the
o-minimal Hausdorff measure. We also prove that every definable set can be
partitioned into "basic rectifiable sets", and that the Whitney arc property
holds for basic rectifiable sets.
Tame structures and open cores
(pdf file); some version of this article has been submitted.
We study various notions of "tameness" for definably complete expansions of
ordered fields.
We mainly study structures with locally o-minimal open core, d-minimal
structures, and dense pairs of d-minimal structures.
Part of this article appeared as:
Dimension, matroids, and dense pairs of first-order structures
(pdf file); APAL,
Volume 162, Issue 7, June-July 2011, pp. 514-543,
doi:10.1016/j.apal.2011.01.003.
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); some version of this article has been submitted.
The content of this article was published (in revised version) in three parts as:
Relative Pfaffian closure for definably complete Baire structures;
Illinois Journal of Mathematics, Volume 55, Number 3 (2011), 1203-1219; link.
Theorems of the complement; in: Lecture
Notes on O-Minimal Structures and Real Analytic Geometry, (C.
Miller, J.-P. Rolin, P. Speissegger Editors), Fields Institute Communications 62 (2012), 219-242; link to Google books.
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 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, expanded version);
Journal of Mathematical Logic, Volume 9, Issue 2 (2009) pp. 167--182, doi: 10.1142/S0219061309000859.
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.
Towers of complements to valuation rings and truncation closed embeddings of valued fields (with Franz-Viktor Kuhlmann and Salma Kuhlmann);
Journal of Algebra, Volume 323, Issue 3, 1 February 2010, pp. 574--600 doi:10.1016/j.jalgebra.2009.11.023.
We study truncation-closed embeddings of valued fields into fields of power series.
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);
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
substructure of the set of cuts over a suitable group.
Embedding Henselian fields into power series
(pdf file);
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), unpublished.
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.
Slides and notes
Notes on Hieronymi's proof of Baire property
(pdf file).
We give an exposition of P. Hieronymi's theorem that every definably complete expansion of an ordered field satisfies an analogue of Baire's category theorem.
There was a mistake in a previous version of these notes, which has now been corrected.
Lovely pairs for independence relations
(pdf file): commented slides of the talk given at the British Postgraduate Model Theory Conference, Leeds, 19-21 January 2011.
The topic is my article by the same name.
Notes on Hrushovski's article "Stable group theory and approximate subgroups"
(pdf file): notes for our seminar in Münster, WS2010-11.
They supplement van den Dries' notes on the same article.
Dimension, matroids, and dense pairs of structures
(pdf file): notes of the talk given at the workshop Model theory: around valued fields and dependent theories, Oberwolfach, January 2010.
The topic is my article by the same name.
Tame ordered structures
(pdf file): commented slides of the talk given at the Logic Colloquium 2009, Sofia, August 2009.
The topic is my article "Tame structures and open cores":
various notion of tameness for linearly ordered fields, generalyzing o-minimality. Addendum: I retract my claim in the talk that d-minimal non-o-minimal structures are not rosy (while I have no counterexamples, I found a mistake in the proof).
On the other hand, the conjecture that the open core of a dense pair of d-minimal structures (B,A) is B itself has been proved.
Snow White and the omega dwarves
(pdf file): notes of the seminar given at Freiburg, February 2008.
Some (surprising) variants of the Hat puzzle (thanks to all the guys in Pisa for the discussions on this puzzle).
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, 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, 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, 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, May 2006.
Let A be a semi-algebraic set, definable without using
parameters in some real closed field M. Let Ã
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
Ã 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.
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.
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 dot com
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 dot com
This page address: http://www.dm.unipi.it/~fornasiero/index.html
Last modified: Mon Sep 30 14:18:46 CEST 2013 by A. Fornasiero