
October,
4th, 2006.
Jürg Kramer (HumboldtUniversität, Berlin)
A relation between Lseries of holomorphic modular forms and Lseries
of Maass forms
Let &Gamma ⊆ PSL_{2}(Z) be a noncompact
Fuchsian group of the first kind. In our talk we will establish a
relationship between symmetric square Lseries attached to holomorphic
cusp forms of weight 2 with respect to &Gamma and symmetric square
Lseries attached to Maass forms with respect to &Gamma. This
result leads to a relationship between the Fourier coefficients of the
holomorphic cusp forms and the Maass forms under consideration and provides
the key to attack the RamanujanPetersson conjecture for Maass forms using
the Deligne bounds for the Fourier coefficients of holomorphic cusp forms. The
results presented are obtained in joint work with Jay Jorgenson.

October,
18th, 2006.
Donatella Iacono (Università di Roma "La Sapienza")
Differential graded Lie algebras and deformations of holomorphic maps
We construct the deformation functor associated to a couple of
morphisms of differential graded Lie algebras, and use it to study the
infinitesimal deformations of an holomorphic map of compact complex
manifolds. In particular, in the case of Kähler manifolds we describe a
generalizzation of the semiregularity map.

October,
25th, 2006.
Gerard van der Geer (Universiteit van Amsterdam)
Siegel modular forms and the Harder conjecture
The talk aims to giving an introduction to Siegel's modular forms. We
will show how one can use genus two curves on finite fields to obtain
informations on degree two modular forms. Moreover, we will provide evidence
for the Harder conjecture on congruences between modular forms in one
variable and Siegel's modular forms. It
is a joint work with Carel Faber.

October,
31st, 2006.
Lucian M. Ionescu (Illinois State University)
Graph Complexes in Deformation Quantization and The Feynman Legacy
(Past, Present and Future)
The talk is centered on the joint work with Domenico Fiorenza on the
algebraic structures underlying Kontsevich solution to the deformation
quantization of Poisson manifolds. Deformation quantization is reviewed
(Flato, Kontsevich, CattaneoFelder) and the role of dgcoalgebra of graphs
is emphasized, in the light of renormalization and graph homology (ConnesKreimer,
Kontsevich). The underlying Feynman path integral interpretation is mentioned
as a leitmotif, and leading, in a quantum computing interpretation, to
Feynman processes as algebras over PROPs and quantum information flow (L. M.
I., B. Coecke, B. J. Hiley etc.).

November,
22th, 2006.
Angelo Vistoli (Scuola Normale Superiore, Pisa)
Transcendence degrees of fields of definition
I will discuss joint work with Patrick Brosnan and Zinovy Reichstein
on the notion of "essential dimension" for algebraic stacks. We
study the following problem: given a geometric object X on a
field K (e.g., an algebraic variety), what is the least
transcendence degree of a field of definition of X over the
prime field? In other words, how many independent parameters do we need to
define X? Our results give strict upper bounds in many cases,
for example, for smooth or stable curves of fixed genus.

November,
22nd, 2006.
Cristian Lenart (University at Albany (New York))
A combinatorial model in Lie theory
We present a simple combinatorial model for Kashiwara's crystals
corresponding to the irreducible representations of complex semisimple Lie
algebras, and, more generally, of complex symmetrizable KacMoody algebras.
Using the same model, we give an explicit Chevalleytype formula for the Tequivariant
Ktheory of generalized flag manifolds G/B. This model,
which was introduced in joint work with A. Postnikov, can be viewed as a
discrete counterpart to the Littelmann path model. While all the features of
Littelmann's model were recovered in ours, there are some additional features
too. One such example, developed in further solo work, is a combinatorial
realization of Lusztig's involution on irreducible crystals. This involution
exhibits a crystal as a selfdual poset, and corresponds to the action of the
longest Weyl group element on the corresponding representation. The talk will be largely selfcontained.

November,
29th, 2006.
Domenico Fiorenza (Università di Roma "La Sapienza")
The period map as a morphism of deformation theories
We prove that, for every compact Kähler manifold, its universal period
map is induced by a natural L_{∞}morphism. This implies, by
standard theory of L_{∞}structures, that the universal period
map is a "morphism of deformation theories" and then commutes with
all deformation theoretic constructions (e.g. obstruction theories). As a
corollary, one obtains a proof of Kodaira's Principle that ambient cohomology
annihilates obstruction to deforming complex structures on compact Kähler
manifolds. These results were obtained in a joint work with Marco Manetti.

December,
6th, 2006.
MoulayTahar Benameur (Université de Metz)
Application of AtiyahSingerConnes index theory to the gaplabel
conjecture
The gaplabel conjecture has origins in solid state physics, in the
study of quasicrystals. It was first stated by Bellissard in the late 80's
and has been solved recently. One way to solve this conjecture is to use
Connes' index theory for foliations and the BaumConnes isomorphism for the
free abelian group. We shall explain in this talk how to reinterprete the
gaplabel question in terms of a pairing between a holonomy invariant measure
on some lamination and the leafwise Chern character of idempotents on this
lamination. The solution to the gaplabel problem is then obtained by using a
topological integrality result.

December,
13th, 2006.
Carlo Petronio (Università di Pisa)
Complexity of knots, graphs and 3orbifolds
Matveev's definition of the complexity c(M) of a
(closed, irreducible) 3manifold M gives a very natural measure
of how complicated M is. In addition, c has very
nice properties, including additivity under connected sum. This talk will
describe variations of the definition of c which apply to the
objects listed in the title. Along with the discussion of some properties of
this extended notion of c, the talk will mention a subtlety
which arises when trying to reproduce for 3orbifolds the proof of Haken,
Kneser and Milnor of existence and uniqueness of the splitting along spheres
of a 3manifold into irreducible ones. If time permits an account will also
be given of work in progress with Hodgson and Pervova on computer tabulation
of objects as listed in the title in increasing order of complexity.

December,
20th, 2006.
Davide Gaiotto (Harvard University)
GopakumarVafa invariants and modular forms
The OSV conjecture suggests a relation between topological string
amplitudes and the number of microstates of certain supersymmetric black
holes. Mtheory packages those integer numbers into vectors of holomorphic
modular forms. We show through some basic examples how to build such modular
forms from enumerative geometry data of the CalabiYau. Modularity implies unexpected constraints on
GopakumarVafa invariants.

January,
10th, 2007.
Gabriele Mondello (M.I.T.)
Triangulated Riemann surfaces and the WeilPetersson Poisson structure
Given a Riemann surface with boundary S, the lengths of a
maximal system of disjoint simple geodesic arcs on S that start and
end at ∂S perpendicularly are coordinates on the Teichmüller space &tau(S).
We compute the WeilPetersson Poisson structure on &tau(S) in this
system of coordinates and we prove that it limits pointwise to the
piecewiselinear Poisson structure defined by Kontsevich on the arc complex of
S. As a byproduct of the proof, we obtain a formula for the
firstorder variation of the distance between two closed geodesic under
Fenchel Nielsen deformation.

January,
17th, 2007.
Gabriele Mondello (M.I.T.)
Triangulated Riemann surfaces and the WeilPetersson Poisson
structure,II
I will describe two global coordinate systems (the Acoordinates
and the Wcoordinates) for the Teichmüller space of hyperbolic
surfaces with geodesic boundary and I will write an explicit expression for
the WeilPetersson Poisson structure in the Acoordinates. In the
limit for boundary lengths going to zero, the (normalized) Acoordinates
tend to the Penner coordinates and the cellularization of the Teichmüller
space becomes the Penner and BowditchEpstein's one. In the same limit, the
explicit expression we found for the WeilPetersson Poisson structure in the Acoordinates
reduces to Penner's formula for the WeilPetersson 2form. In the (not so
classically studied) limit for boundary lengths going to infinity, the
(normalized) Wcoordinates tend to the natural affine coordinates on
the arc complex used by Kontsevich in his proof of the Witten conjecture, and
the Teichmüller space cellularization becomes the one described by
HarerMumfordThurston by means of JenkinsStrebel quadratic differentials.
In the same limit, the WeilPetersson symplectic form converges pointwise to
the Kontsevich's one (Thurston's one one measured laminations); the limit of
the WeilPetersson Riemannian volum will instead differ by the symplectic
volume by an exp[(p_1+...+p_n)/2] factor, which is precisely the
factor appearing in Kontsevich's proof.

January,
24th, 2007.
Mario Salvetti (Università di Pisa)
Combinatorial Morse theory and Hyperplane Arrangements
We show how, using (almost) purely combinatorial methods, it is
possible to deduce the minimality of the complementary of an hyperplane
arrangement without referring to Lefshetztype theorems.

February,
7th, 2007.
Céline Righi (Université de Poitiers)
Adnilpotent ideals of a parabolic subalgebra
We extend the results of CelliniPapi on the characterizations of
nilpotent and abelian ideals of a Borel subalgebra to parabolic subalgebras
of a simple Lie algebra. These characterizations are given in terms of
elements of the affine Weyl group and faces of alcoves. In the case of a
parabolic subalgebra of a classical Lie algebra, we give formulas for the
number of these ideals.

February,
14th, 2007.
Kato Syu (University of Tokyo)
An exotic DeligneLanglands correspondence for symplectic groups
An affine Hecke algebra is the qanalogue of an affine
Weyl group, which plays major roles in representation theory of Chevalley
groups and several other fields. The DeligneLanglandsLusztig conjecture
(proved by KazhdanLusztig, Ginzburg) asserts that each simple module of an
affine Hecke algebra corresponds to some geometric datum. An affine Hecke
algebra of type C admits a natural twoparameter deformation H
(and this is best possible in some sense). In this talk, we realize H
as the equivariant Kgroup of a certain variety, which we refer
as the exotic Steinberg variety. This enables us to present a
DeligneLanglands type classification of simple Hmodules when
the values and ratios of deformation parameters are not too bad.

February,
20st, 2007.
Alessandro Ruzzi (Université de Grenoble)
Smooth projective symmetric varieties with Picard number equal to one
We classify the smooth projective symmetric varieties with Picard
number equal to one. Moreover we prove a criterion for the smoothness of the
simple (normal) symmetric varieties whose closed orbit is complete. In
particular we prove that a such variety X is smooth if and only if an
appropriate toric variety contained in X is smooth.

February,
21st, 2007.
Chenchang Zhu (Université de Grenoble)
A gerbe of Gamma functions
As we know that theta functions provide a section of a nontrivial
line bundle (theta bundle) on elliptic curves, in this talk we will give the
geometric realization of elliptic Gamma functions whose highly nontrivial
identities are developed by Felder and Varchenko. They can also be regarded
as the difference of theta functions. It turns out these identities can be
geometrically inteperated as the fact that Gamma functions give a meremorphic
section of a holomorphic gerbe over the stack CP^{2}RP^{2}/SL(3,Z)×
Z^{3}. It relates to the classical theta functions in the
following way: the restriction of this gerbe on a substack C/Z^{3},
viewed as a central extension of groupoid, is exactly a union of products of
the theta bundles; the restricted meremorphic section is provided by theta
functions.

February,
28th, 2007.
Benjamin Enriquez (Université de Strasbourg)
Universal KZB equations
We define a universal version of the KnizhnikZamolodchikovBernard
(KZB) connection in genus 1. This is a flat connection over a principal
bundle on the moduli space M_{1,[n]} of elliptic curves with
marked points. We show that this connection can be used for reproving the
formality of the pure braid groups on genus 1 surfaces. We study its
monodromy and show that it gives rise to a relation between the KZ associator
and a generating series for iterated integrals of Eisenstein forms. Going
from the "universal" to the "concrete" setup, we
introduce the notion of elliptic structure on a (quasi)bialgebra, which gives
rise to representations of the braid groups in genus 1.

March,
7th, 2007.
Tomoyoshi Ibukiyama (Osaka University)
Dimensions of Siegel Modular Forms
Siegel modular forms naturally appear when one considers the moduli
spaces of abelian varieties of various complex dimensions. They form a graded
ring which, for each weight, is finite dimensional. I would like to give a
general survey of what is presently known on these dimensions. I will talk on
history, general conjectures, as well as very new results.

March,
14th, 2007.
Claudia Pinzari (Università di Roma "La Sapienza")
The embedding problem for tensor C*categories
Motivated by the operator algebraic approach to low dimensional
quantum field theory, we discuss the problem of embedding an abstract tensor
category into the category of Hilbert spaces. We survey Woronowicz notion of
compact quantum groups with its main example: the quantum deformation S_{q}U(d)
of the group of unitary matrices with determinant 1. We discuss, in the
operator algebraic approach, a characterization of the representation
category of S_{q}U(d) among abstract tensor categories. For
tensor categories containing Rep S_{q}U(d), we outline a
construction (obtained in a joint work with S. Doplicher and J.E. Roberts) of
an embedding into the category of bimodules over a noncommutative C*algebra.

March,
21st, 2007.
Aleksandr S. Mischenko (Moskow State University)
The Hirzebruch formula and the signature of manifolds
The lecture is devoted to a description of the socalled Hirzebruch
formula in different aspects which form a basic list of problems related to
noncommutative geometry and topology. They include 1. Finitedimensional
unitary representations. 2. Continuous family of finitedimensional
representations. 3. Functional version of the Hirzebruch formula.
Infinitedimensional representations. 4. Smooth version of the Hirzebruch
formula. 5.
Combinatorial local Hirzebruch formula.

March,
27th, 2007.
Victor Kac (MIT)
On semisimplicity of Walgebras
Walgebra is a "nonlinear" generalisation
of the Virasoro algebra, attached to a nilpotent orbit of a simple Lie
algebra. It is known that any positive energy representation of the Virasoro
vertex algebra is completely reducible iff the central charge is of the form 16(pq)^{2}/pq,
where p and q are relatively prime integers greater than 1.
In my talk I will discuss how to solve the semisimplicity problem for general
Walgebras.

March, 27th, 2007.
Enrico Bombieri (IAS)
Una costruzione dei polinomi di Kahane (in collaborazione con Jean
Bourgain)
J.P. Kahane ha dimostrato,
con un metodo probabilistico, l'esistenza di polinomi trigonometrici di grado
n con coefficienti di valore assoluto 1, con modulo uguale a n^{1/2}
+ O(n^{1/2  1/17}(log n)^{1/2}) in ogni punto. Questa
conferenza dà due costruzioni, una probabilistica e l'altra costruttiva, di
tali polinomi con una approssimazione n^{1/2}+O(n^{1/21/9+&epsilon}).
La costruzione effettiva di questi polinomi utilizza in maniera essenziale
l'ipotesi di Riemann per funzioni L di varietà in caratteristica p,
per varietà di dimensione arbitraria, insieme ad altri risultati di natura
aritmetica.

March,
28th, 2007.
Dmitry Roytenberg (MPI Bonn)
Simplicial manifolds and higher Lie theory
The "firstorder approximation" functor, associating to
every Lie group its Lie algebra, can be described in a way that admits a
farreaching generalization: to every simplicial manifold satisfying an
appropriate smooth version of Kan's extension condition it associates a
"differential graded manifold"  a sheaf of differential graded
commutative algebras over the base manifold which are locally free as graded
algebras. For the nerve of a Lie group this yields the ChevalleyEilenberg
complex of its Lie algebra, whereas for the nerve of the pair (or
fundamental) groupoid of a manifold the result is the de Rham complex of the
manifold. Other examples of Kan simplicial manifolds include (nerves of)
higher Lie groups and groupoids, such as Lie crossed modules and their
weakened versions that appear as symmetries of TFT's and in differential
geometry of nonabelian gerbes. On the infinitesimal side we have higher
homotopy Lie algebras and algebroids. It is wellknown that, with only a
minor restriction on the topology of a Lie group (1connectedness), the
firstorder approximation is actually an equivalence of categories: the
quasiinverse  integration functor  assigns to every Lie algebra what in
the present setup should be viewed as its "fundamental group".
However, proving smoothness of the resulting object is nontrivial even in
this case, whereas constructing the quasiinverse and proving its smoothness
for arbitrary differential graded manifolds remains an open problem. In my
talk I will describe the general setup, give examples, describe a solution
of the integration problem in the cases where one exists and explain
analytical and combinatorial obstructions to solving the general case.

April,
4th, 2007.
David Hernandez (CNRS  Université de Versailles)
Geometric smallness property of quiver varieties and applications
We give a solution to the geometric smallness problem raised by
Nakajima : we characterize the Drinfeld polynomials of the corresponding
small KirillovReshetikhin modules. From the point of view of algebraic
geometry, these results are equivalent to the smallness in the sense of
BorhoMacPherson of certain projective morphisms involving quiver varieties
(analog to the Springer resolution). We also discuss the representation
theoritical consequences of the small property.

April, 11th, 2007.
Catriona MacLean (Université de Grenoble)
Costruzione di intorni infinitesimali
Dimostreremo un teorema di
classificazione di intorni infinitesimali analogo al teorema di
classificazione delle deformazioni di una varietà su un anello Artiniano.

May, 9th,
2007.
Jerzy Weyman (Northeastern University)
Counterexamples to Okounkov logconcavity conjecture
This is joint work with Calin Chindris and Harm Derksen. I will
explain how one can use quiver representations to find counterexamples to the
conjecture of Andrei Okounkov on the logconcavity of LittlewoodRichardson
coefficients.

May, 16th, 2007.
Paolo Papi (Università di Roma La Sapienza)
Multiplets of representations and Dirac operators in affine setting
La prima parte del
seminario sara' dedicata ad una estesa panoramica sulla teoria dei multiplets
di Kostant nel caso finito dimensionale e alla congettura di Vogan sul
carattere infinitesimale di un modulo di HarishChandra in termini di
coomologia di Dirac. Illustreremo poi le versioni affini di alcuni degli
enunciati. Questi ultimi risultati sono stati ottenuti in collaborazione con
V. Kac e P. Moseneder Frajria.

May, 23th, 2007.
Lidia Stoppino (Università di Roma Tre)
Gonalità di superficie fibrate
Data una superficie fibrata
su una curva, chiamiamo "slope" il rapporto tra l'autointersezione
del canonico relativo e la caratteristica di Eulero relativa. Parlerò
dell'influenza sulla slope della gonalità delle fibre generali: è stata
variamente congetturata l'esistenza di una limitazione dal basso della slope
dipendente in modo crescente dalla gonalità, ma il problema è sostanzialmente
ancora aperto. Discuterò questo problema e in particolare illustrerò un
risultato riguardante le fibrazioni trigonali ottenuto in collaborazione con
Miguel Angel Barja.

May, 29th,
2007.
Ernesto Mistretta (Université Paris VII)
Line bundle transforms on curves and theta divisors
We show some constructions of tranforms of vector bundles, i.e.
kernels of evaluation maps on subspaces of global sections. We consider the
cases where these are stable, and explain why they were studied in the
literature. In the cases where the slope of the transform is integer, we
analyse the existence and properties of its theta divisor.

May, 30th,
2007.
Marco Zambon (Universität Zürich)
Reduction of branes in generalized complex geometry
We show that a generalized complex submanifold ("brane") of
a generalized complex manifold is endowed with a natural foliation and that
the quotient, when smooth, is again a generalized complex manifold. This can
be seen as a generalized complex analog of quotienting a coisotropic
submanifold in symplectic geometry. Along the way we will consider exact
Courant algebroids as well.

May, 30th,
2007.
Maxim Nazarov (University of York)
Drinfeld functor for twisted Yangians
Let H_{N} be the degenerate affine Hecke algebra corresponding
to the group GL_{N} over a padic field. There are two well
known functors in the representation theory of H_{N}. One of them,
first introduced by Cherednik and then studied by Arakawa, Suzuki and
Tsuchiya, is a functor from the category of modules over the Lie algebra
$\mathfrak{gl}_m$ to the category of H_{N}modules. The other,
introduced by Drinfeld, is a functor from the latter category to the category
of modules over the Yangian $\operatorname{Y}(\mathfrak{gl}_n)$ of the Lie
algebra $\mathfrak{gl}_n\,$. This Yangian is a deformation of the universal
enveloping algebra of the polynomial current Lie algebra $\mathfrak{gl}_n[t]$
in the class of Hopf algebras. The composition of two functors is
particularly important. It provides a corres\pondence between the \lq\lq
extremal cocycle\rq\rq\ on the Weyl group of $\mathfrak{gl}_m$ defined by
Zhelobenko, and intertwining operators on the tensor products of
$\operatorname{Y}(\mathfrak{gl}_n)\,$modules. This correspondence involves
the classical dual pair of Lie groups $(GL_m,GL_n)\,$. The aim of the talk is
to introduce an analogue of the composition of two functors when the pair
$(GL_m,GL_n)$ is replaced by the dual pair $(Sp_{2m},O_n)$ introduced by
Howe. The role of the Hopf algebra $\operatorname{Y}(\mathfrak{gl}_n)$ is
then played by the twisted Yangian $\operatorname{Y}(\mathfrak{so}_n)$. This
twisted Yangian is a coideal subalgebra of
$\operatorname{Y}(\mathfrak{gl}_n)$, and a deformation of the universal
enveloping algebra of the twisted polynomial current Lie algebra $$
\{\,X(t)\in\mathfrak{gl}_n[t]\ \ \sigma(X(t))=X(t)\,\} $$ where $\sigma$ is
the involutive automorphism of $\mathfrak{gl}_n$ with the fixed point
subalgebra $\mathfrak{so}_n\,$.

June,
6th, 2007.
Giovanni Cerulli Irelli (Università di Padova)
Positività e base canonica in un'algebra cluster di tipo A_{2}^{(1)}:un
approccio geometrico
Le
algebre cluster sono state introdotte da S. Fomin e A. Zelevinsky con l'idea
di creare una teoria algebricocombinatoria nella quale fosse possibile
studiare esplicitamente le basi canoniche duali alle basi canoniche di
Lusztig. In un'algebra cluster A esiste infatti una naturale nozione
di positività: il problema è vedere se l'insieme degli elementi positivi e
indecomponibili formino una Zbase B di A. Nel caso
questo sia vero, una tale base è detta canonica. Questo problema è stato
risolto in algebre cluster di rango due utilizzando tecniche elementari. In
questo seminario verrà presentata una generalizzazione all'unico caso
antisimmetrico di rango tre, ovvero il caso in cui A sia di tipo A_{2}^{{(1)}}.
In tale generalizzazione si fara' uso di tecniche geometriche introdotte da
P. Caldero e B. Keller. Queste tecniche hanno il vantaggio di produrre
espressioni esplicite per gli elementi della base canonica. Il seminario sarà
il più possibile selfcontained. In particolare nessuna conoscenza della
teoria delle algebre cluster verrà assunta.

June,
13th, 2007.
Giovanni Cerulli Irelli (Università di Padova)
Positività e base canonica in un'algebra cluster di tipo A_{2}^{(1)}
 seconda parte
