|
Massimo Bertolini: On the Birch and Swinnerton-Dyer conjecture, I & II
|
|
Abstract: We will begin by formulating the Birch and
Swinnerton-Dyer conjecture,
which relates the analytic properties of the L-series of an elliptic
curve to certain arithmetic
invariants of the curve, notably the rank of its group of rational
points. We will then proceed
to illustrate the known results on the Birch and Swinnerton-Dyer
conjecture, and conclude
by touching upon analogues and generalisations of this conjecture.
|
|
|
Enrico Bombieri: The Classical Theory of Zeta and L-functions
|
Abstract:
Lecture I. Riemann's formula, zeta functions and L-functions.
Summary: Beginning with Riemann's "Explicit Formula", the zeta function,
Dirichlet L-functions, and the Dedekind zeta functions are
introduced and their main properties are discussed. If time allows,
zeta functions of modular forms and elliptic curves will be
briefly introduced.
Lecture II. The analytic theory of the zeta function, known
and conjectural results. Families and random matrix theory.
Summary: This lecture gives a survey of results about
the analytic behavior of the zeta function. The conjectural
correlations of zeros and the recent conjectures about moments
derived from random matrix theory will also be mentioned briefly.
Lecture III. The zeta function of curves over a finite field.
Summary: This lecture is dedicated to the zeta functions of
curves over a finite field and their analogy with the Dedekind zeta
function. A brief sketch of Weil's first proof of the analog of the
Riemann Hypothesis for them will be given. If time allows, Weil's
formulation of the "Explicit Formula" will also be given.
|
|
|
Mauro Carfora: Quantum fluctuations and Riemannian Geometry: The case of the Ricci flow
|
|
Abstract: The talk will be focalized on the interplay among Ricci flow and quantum
theory. After briefly introducing Ricci flow we discuss its connection
with renormalization group analysis in quantum field theory, (I will make
an effort to present this in a math. "reasonable" way). We will emphasize
the suggestive applications that such a relation has both in Riemannian
Geometry, and in improving our geometrical understanding of Quantum
Field Theories. This is an active field of investigation providing a nice
case of cross-fertilization between riemannian geometry and physics.
|
|
|
Jeff Cheeger: Curvature bounds and applications to 4-dimensional Einstein metrics
|
Abstract:  (39 KB)
|
|
|
Jean-Pierre Demailly: Singularities of currents and applications to algebraic geometry
|
Lecture 1. Fundamental approximation theorem for plurisubharmonic functions
Abstract: We will introduce basic invariants for plurisubharmonic singularities
(Lelong numbers, singularity exponents) and will discuss a fundamental
approximation result relying on the Ohsawa-Takegoshi extension theorem.
|
Lecture 2. Analytic Zariski decomposition and related results in algebraic geometry
Abstract: The goal of this second talk will be to outline an extension of
Zariski decomposition, as known for surfaces, to higher dimensions,
where it holds only approximately. This can be viewed as part of a new
intersection theory framework which has deep applications to algebraic
geometry - e.g. a characterization of the pseudoeffective cone and
a characterization of projective algebraic varieties possessing
a pseudoeffective canonical class.
|
Lecture 3. Semicontinuity of singularities, estimates for Monge-Ampére operators and existence of Kähler-Einstein metrics
Abstract: This third part will discuss a fundamental semicontinuity theorem for
psh singularities, which can be used to extend Nadel's criterion for the
existence of Kähler-Einstein metrics on Fano orbifolds. New
results concerning log canonical thresholds in relation with estimates
for Monge-Ampére operators will also be outlined, in connection with
applications to local algebra and birational geometry.
|
|
|
E. Kowalski: Some aspects and applications of the Riemann Hypothesis over finite fields
|
|
Abstract: Since Weil's proof of the Riemann Hypothesis for curves over finite
fields, and even more since Deligne's proof of his remarkable
generalizations of the general Weil conjectures, these results have had
numerous applications. We will present some of them, focusing
particularly on those in analytic number theory, and we will highlight
some of the important open questions still remaining.
|
|
|
Raghavan Narasimhan: Bernhard Riemann: Remarks on his life and some of his
mathematical work
|
|
Abstract: The mathematical work I will discuss will be mainly that on the zeta
function and his unpublished lectures on the hypergeometric series.
|
|
|
Alberto Perelli: Non-linear twists of L-functions
|
|
Abstract: (52 KB)
|
|
|
Mario Rasetti: Topological invariants of 3-manifolds and quantum computation
|
|
Abstract: After briefly reviewing the basic aspects of quantum
information
processing, based on topological quantum field theory methods, a
quantum
algorithm for approximating efficiently 3--manifold topological
invariants in
the framework of the SU(2) Chern--Simons--Witten topological
quantum
field theory at finite values of the coupling constant k will be
discussed.
The model of computation adopted is the q--deformed spin network
model
viewed as a quantum recognizer automaton, where each basic unitary
transition function can be efficiently processed by a standard
quantum
circuit. The algorithm is an extension of that constructed for the
(additive approximation) evaluation of polynomial invariants (Jones) of
colored
oriented links. All the significant quantities --partition functions
as well as
observables-- of the quantum field theory can be processed
efficiently on
a quantum computer, reflecting the intrinsic, field--theoretic
solvability of
such theory at finite k. The talks aims as well to providing a
critical overview
of some basic conceptual tools underlying the construction of the
quantum
invariants of knots, links and 3--manifolds and their connections
with
algorithmic questions that arise in geometry and quantum gravity
models.
|
|
|
Harold Rosenberg: Entire minimal graphs in H x R and the
construction of surjective harmonic diffeomorphisms from the complex plane C to the hyperbolic plane H.
|
|
Abstract: In the 1950's, Heinz proved there is no harmonic
diffeomorphism from H onto C. He then used this to prove Bernsteins' theorem:
An entire minimal graph over the euclidean plane is linear.
It was conjectured by R. Schoen that there are no surjective harmonic
diffeomorphisms from C onto H. We will construct such
diffeomorphisms by constructing entire minimal graphs in H x R that
are conformally C.
This is joint work with Pascal Collin.
|
|
|
Peter Sarnak: L-functions of modular forms and some applications
|
Lecture 1: GL(2)
Abstract: We review the classical theory of Hecke
and Maass which generalizes Riemann's work on the zeta function to
L-functions associated with modular forms in the upper half plane. In
particular we pay attention to the converse theorem of Weil, its
philosophical implications and higher symmetric power L-functions.
Lecture 2: GL(n)
Abstract: We give a quick introduction the theory of modular forms on GL(n).
Via Riemann and Hecke's methods they produce "standard L-functions"
which are expected to contain all such functions that arise in "nature". We
discuss the basic conjectures associated with these forms and
their L-functions that is the Grand Riemann Hypothesis and the
Generalized Ramanujan Conjectures and we report on what is known towards them.
Lecture 3: Subconvexity and some applications
Abstract: After explaining the subcovexity problem for general
L-functions and some methods that have been used to resolve
it (mainly in GL(2)), we give applications via period formulae to
local to global problems in the theory of integral quadratic forms and to the
Quantum Unique Ergodicity Conjecture.
|
|
|
Nina Snaith: A short history of Riemann and random matrices.
|
|
Abstract: It has been almost 40 years since Montgomery and Dyson
uncovered the similarity between statistics of the Riemann zeros and
the eigenvalues of random matrices. In the past dozen or so years we
have seen random matrix theory have a significant impact in number
theory as we start to understand how to use it to attack some of the
biggest questions concerning the Riemann zeta function and other
L-functions. This talk will chart the rise of random matrix theory as
a reputable means of studying Riemann's famous zeta function.
|
|
|
Gudlaugur Thorbergsson: Singular Riemannian foliations and isoparametric submanifolds
|
|
Abstract: In the talk I will explain both old and new results
due to myself and others on singular Riemannian foliations that
generalize or are analogous to isoparametric foliations in spheres
and Euclidean spaces. I will also discuss the question when such
foliations are homogeneous, i.e., when they are orbit foliations
of isometric actions.
|
|
|
Claire Voisin: Algebraic geometry versus Kaehler geometry.
|
|
Abstract: Hodge theory is a common tool to study the topology of
compact Kaehler or complex projective manifolds.
We will explain in the first lecture how it can be used however to
exhibit topological obstructions for
a compact Kaehler manifold to admit a projective complex structure.
Another very important difference between compact Kaehler manifolds
and complex projective
varieties comes from the absence of "motive" in the first case,
possibly explaining the failure of the Hodge conjecture
in the Kaehler setting. Here the motive is understood in the simplest
sense, and provides a way of computing the
cohomology of a projective variety by purely algebraic means. This
will be the object of the second lecture.
|
