Let &Gamma ⊆ PSL₂(Z) be a non-compact Fuchsian group of the first kind. In our talk we will establish a relationship between symmetric square L-series attached to holomorphic cusp forms of weight 2 with respect to &Gamma and symmetric square L-series 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 Ramanujan-Petersson 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.

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.

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.

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, Cattaneo-Felder) and the role of dg-coalgebra of graphs is emphasized, in the light of renormalization and graph homology (Connes-Kreimer, 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.).

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.

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 Kac-Moody algebras. Using the same model, we give an explicit Chevalley-type formula for the T-equivariant K-theory 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 self-dual poset, and corresponds to the action of the longest Weyl group element on the corresponding representation. The talk will be largely self-contained.

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.

The gap-label conjecture has origins in solid state physics, in the study of quasi-crystals. 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 Baum-Connes isomorphism for the free abelian group. We shall explain in this talk how to reinterprete the gap-label 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 gap-label problem is then obtained by using a topological integrality result.

Matveev's definition of the complexity c(M) of a (closed, irreducible) 3-manifold 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 3-orbifolds the proof of Haken, Kneser and Milnor of existence and uniqueness of the splitting along spheres of a 3-manifold 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.

The OSV conjecture suggests a relation between topological string amplitudes and the number of microstates of certain supersymmetric black holes. M-theory 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 Calabi-Yau. Modularity implies unexpected constraints on Gopakumar-Vafa invariants.

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 Weil-Petersson 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 first-order variation of the distance between two closed geodesic under Fenchel- Nielsen deformation.

I will describe two global coordinate systems (the A-coordinates and the W-coordinates) for the Teichmüller space of hyperbolic surfaces with geodesic boundary and I will write an explicit expression for the Weil-Petersson Poisson structure in the A-coordinates. In the limit for boundary lengths going to zero, the (normalized) A-coordinates tend to the Penner coordinates and the cellularization of the Teichmüller space becomes the Penner and Bowditch-Epstein's one. In the same limit, the explicit expression we found for the Weil-Petersson Poisson structure in the A-coordinates reduces to Penner's formula for the Weil-Petersson 2-form. In the (not so classically studied) limit for boundary lengths going to infinity, the (normalized) W-coordinates 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 Harer-Mumford-Thurston by means of Jenkins-Strebel quadratic differentials. In the same limit, the Weil-Petersson symplectic form converges pointwise to the Kontsevich's one (Thurston's one one measured laminations); the limit of the Weil-Petersson 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.

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 Lefshetz-type theorems.

We extend the results of Cellini-Papi 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.

An affine Hecke algebra is the q-analogue of an affine Weyl group, which plays major roles in representation theory of Chevalley groups and several other fields. The Deligne-Langlands-Lusztig conjecture (proved by Kazhdan-Lusztig, 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 two-parameter deformation H (and this is best possible in some sense). In this talk, we realize H as the equivariant K-group of a certain variety, which we refer as the exotic Steinberg variety. This enables us to present a Deligne-Langlands type classification of simple H-modules when the values and ratios of deformation parameters are not too bad.

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.

As we know that theta functions provide a section of a non-trivial line bundle (theta bundle) on elliptic curves, in this talk we will give the geometric realization of elliptic Gamma functions whose highly non-trivial 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 CP²-RP²/SL(3,Z)× Z³. It relates to the classical theta functions in the following way: the restriction of this gerbe on a substack C/Z³, 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.

We define a universal version of the Knizhnik-Zamolodchikov-Bernard (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.

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.

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_qU(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_qU(d) among abstract tensor categories. For tensor categories containing Rep S_qU(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.

The lecture is devoted to a description of the so-called Hirzebruch formula in different aspects which form a basic list of problems related to noncommutative geometry and topology. They include 1. Finite-dimensional unitary representations. 2. Continuous family of finite-dimensional representations. 3. Functional version of the Hirzebruch formula. Infinite-dimensional representations. 4. Smooth version of the Hirzebruch formula. 5. Combinatorial local Hirzebruch formula.

W-algebra is a "non-linear" 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 1-6(p-q)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 W-algebras.

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^{1/2-1/9+&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.

The "first-order approximation" functor, associating to every Lie group its Lie algebra, can be described in a way that admits a far-reaching 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 Chevalley-Eilenberg 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 non-abelian gerbes. On the infinitesimal side we have higher homotopy Lie algebras and algebroids. It is well-known that, with only a minor restriction on the topology of a Lie group (1-connectedness), the first-order approximation is actually an equivalence of categories: the quasi-inverse -- integration functor -- assigns to every Lie algebra what in the present set-up should be viewed as its "fundamental group". However, proving smoothness of the resulting object is nontrivial even in this case, whereas constructing the quasi-inverse and proving its smoothness for arbitrary differential graded manifolds remains an open problem. In my talk I will describe the general set-up, 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.

We give a solution to the geometric smallness problem raised by Nakajima : we characterize the Drinfeld polynomials of the corresponding small Kirillov-Reshetikhin modules. From the point of view of algebraic geometry, these results are equivalent to the smallness in the sense of Borho-MacPherson of certain projective morphisms involving quiver varieties (analog to the Springer resolution). We also discuss the representation theoritical consequences of the small property.

Dimostreremo un teorema di classificazione di intorni infinitesimali analogo al teorema di classificazione delle deformazioni di una varietà su un anello Artiniano.

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 log-concavity of Littlewood-Richardson coefficients.

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 Harish-Chandra 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.

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.

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.

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.

Let H_N be the degenerate affine Hecke algebra corresponding to the group GL_N over a p-adic 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 $\mathfrak{so}_n\,$.

Le algebre cluster sono state introdotte da S. Fomin e A. Zelevinsky con l'idea di creare una teoria algebrico-combinatoria 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 Z-base 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 A₂^{(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 self-contained. In particolare nessuna conoscenza della teoria delle algebre cluster verrà assunta.