curriculum vitae et studiorum personal data present position studies

CURRICULUM VITAE ET STUDIORUM
Simone Calamai
PERSONAL DATA
Given name: SIMONE
Family name: CALAMAI
Place of birth: FLORENCE
Date of birth: 11/23/1982
Home Address: Via Ugo Foscolo 3, 50041 Calenzano, Florence, Italy
Nationality: ITALIAN
E-mail Address: [email protected]
PRESENT POSITION
Ph.D. in Mathematics.
Since January 2007 member of GNSAGA national research group in
Geometry of INDAM.
STUDIES AND HONORS
High School certificate in 2001.
Laurea (B.S.) in mathematics at University of Florence on September 2004
(advisor: Professor Giorgio Patrizio).
Laurea Specialistica (M.Sc.) in mathematics at University of Florence on
October 16th, 2006 (advisor: Professor Giorgio Patrizio).
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Admitted after competitive examination, from 2007 to 2009 student in the
Ph.D. program in Mathematics of the University of Florence.
On May 2007 he attended congress Complex Analysis and Geometry - XVIII
organized by CIRM.
For three years (2001-2004) holder of fellowship for undergraduate students in
Mathematics promoted by Istituto Nazionale di Alta Matematica (INdAM).
STUDIES ABROAD
From January to July, 2008, Honorary Fellow at the University of Wisconsin at Madison, USA, for a scientific collaboration with Professor Xiu Xiong
Chen.
From September to December, 2008, Honorary Fellow at the University of
Wisconsin at Madison, USA, for a scientific collaboration with Professor Xiu
Xiong Chen.
From May to August, 2009, Visiting Student at the University of Science
and Technology of China, in Hefei, China, for a scientific collaboration with
Professor Xiu Xiong Chen.
TEACHING ACTIVITIES AND SEMINARS
Teaching assistant for the course ’Funzioni di variable complessa’ at the Dipartimento di Matematica ’Ulisse Dini’ in Florence, academic year 2009-2010,
fall semester. March-April 2008: Eight seminars on the Calabi’s conjecture
at the University of Wisconsin at Madison.
July 2009: Seminar on the Calabi’s metric at the University of Science and
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Technology of China.
ACADEMIC THESIS
M.Sc. Thesis: ’Funzioni definenti di Monge-Ampère per domini con bordo
analitico’.
Ph.D. Thesis: ’The Calabi’s metric for the space of Kähler metrics’.
PREPRINTS
[1] ’Monge-Ampère Defining Functions’, (2009)
[2] ’The Calabi’s Metric for the Space of Kähler Metrics’, (2009)
RESEARCH INTERESTS
Kähler geometry, Geometry of Kähler class, extremal metrics, several complex variables, in particular geometric properties of complex degenerate MongeAmpère equations, complex differential geometry.
RESEARCH’S ACTIVITY DESCRIPTION
For a given domain D of the space Cn , a defining function f is a real function
defined in a neighborhood of partialD such that f = 0 precisely on ∂D and
df 6= 0 on ∂D. A particular class of defining function is given by those which
satisfy the Homogeneous Complex Monge-Ampère equation. In [1] I prove
in various situations the existence of Monge-Ampère defining functions for
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pseudoconvex domains. Moreover I try to characterize the case in which the
foliation associated to the Monge-Ampère defining function is holomorphic.
In my Ph.D. thesis and in [2] is proposed a new geometry for any Kähler
class of a closed Kähler manifold of any dimension. This geometry arises
from a pointwise smoothly varying positive scalar product which I called
the Calabi’s metric. In fact the intuition for this scalar product comes from
Eugenio Calabi in a private communication to Xiu Xiong Chen; I learned it
from Xiu Xiong Chen himself. Classically, the Riemannian metric considered
on a Kähler class is the Mabuchi’s metric, rediscovered later by Semmes and
Donaldson. The papers of these three authors showed that the Mabuchi’s
metric generates is own Levi Civita covariant derivative, that it induces a
structure of non positively curved locally symmetric space on the Kähler class
and that the geodesic equation is a Homogeneous Complex Monge-Ampère
equation. Donaldson conjectured that the Kähler class is a connected by
smooth geodesics and that the space is a genuine metric space. Xiu Xiong
Chen proved partially the first conjecture and completely the second one;
moreover Chen was able to prove the uniqueness of constant scalar curvature
metrics when the first Chern class is zero or negative. In spite of this success
of the theory, there are some problems. For example, Chen’s geodesic are
of class C 1,1 and it is not known if they degenerate. Moreover Donaldson
showed that the Cauchy problem for the geodesic does not always admit a
solution. I my paper I prove that the Calabi’s metric induces a structure of
locally symmetric space with constant positive sectional curvature, namely
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,
4V ol
where V ol is the volume of the given Kähler manifold in the fixed Kähler
class. The geodesic equation in the Calabi’s case is an harmonic ordinary
differential equation. Thus I proved that the Cauchy problem admits an
unique and analytic solution, of explicit and nice expression. In spite of the
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Mabuchi’s case, I defined a notion of exponential map and Jacobi fields. I
also proved the uniqueness of analytic and explicit solutions for the Dirichlet
problem; this let to the computation of the diameter of the space, which is
√
π V ol. As for the Mabuchi’s case, I proved the Calabi’s metric to induce
a genuine distance function on the Kähler class. The Calabi’s metric enjoys
better properties than the Mabuchi’s metric and motivates to continue the
study of its application which can embrace extremal metrics, Kähler-Ricci
solitons, constant scalar curvature metrics for Riemann surface with conical
singularities.
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