Abstracts
| Benedetti |
On the periodic motions of a charged particle in a magnetic field.
Following seminal ideas of Novikov, we employ variational methods to find
periodic orbits of a charged particle moving under the influence of a magnetic
field on a closed Riemannian manifold. In particular, we generalize to our
setting a classical result of Lyusternik and Fet in the theory of periodic
geodesics. Namely, we prove that if the ambient manifold is not a K(G,1), then
for almost every positive real number E, there exists a contractible periodic
orbit with kinetic energy E.
This result was obtained in joint work with Luca Asselle.
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| Bolognese |
Strange Duality and Verlinde numbers on abelian surfaces.
With the purpose of examining some relevant geometric properties of the moduli space of sheaves over an algebraic surface, Le Potier conjectured some unexpected duality between the complete linear series of certain natural divisors, called Theta divisors, on the moduli space. Such conjecture is widely known as Strange Duality conjecture. After having motivated the problem by looking at certain instances of quantization in physics, we will work in the setting of surfaces. We will then sketch the proof in the case of abelian surfaces (joint work with Marian, Oprea and Yoshioka), giving an idea of the techniques that are used. In particular, we will show how the theory of discrete Heisenberg groups and fiber wise Fourier-Mukai transforms, which might be applied to other cases of interest, enter the picture.
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| Bruno |
On hyperplane sections of K3 surfaces.
I will talk about joint work with E. Arbarello and E. Sernesi and with
E. Arbarello, G. Farkas and G. Saccà.
Which canonically embedded curves are hyperplane sections of K3
surfaces? In the first work, following conjectures of Wahl, Mukai and
Voisin,
we give a complete characterization of curves which are hyperplane
sections of K3 surfaces (or of limits of such). This involves proving
two conjectures stated by Wahl in 1997.
In the second work we we find an explicit family of curves, of any
given genus,
satisfying the Brill-Noether-Petri condition.
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| Carocci |
Homological projective duality and blow ups.
Kuznetsov's homological projective duality is a powerful tool for investigating semiorthogonal decompositions of algebraic varieties, which in turn are interesting as they seem to contain a lot of information about the geometry of the variety in question. I will recall the notion of homological projective duals and present a new example of geometric HP duals. Its construction is a special case of a more general story coming from blowing up base loci of linear systems. The example also highlights an interesting phenomenon: starting with a noncommutative HP dual pair one can obtain a commutative HP dual via the blowing up process. This example is a generalisation of other people's work on rationality of cubic fourfolds. This is joint work with Zak Turcinovic, Imperial College.
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| Cavazzani |
Complete homogeneous varieties.
Compactifying moduli spaces is a problem that goes all the way back to the middle of 19th century. Many times in history, changing the compactification lead to new results in enumerative geometry, and more; one example of that is the space of complete conics, that solved the problem of how many smooth conics are tangent to 5 conics in general position (3264). In many cases, the moduli space to compactify is of the form G/H, where G is the projective linear group of P^N (in the case of conics, for instance, it is PGL_3/PO_3). In my thesis, I studied what compactifications of such G/H are obtained compactifying G to the space X of complete collineations, and then taking the G.I.T quotient X//H by H. In this way, it is possible to use many representation theoretic tools to study the geometry of these spaces; in particular, intersection theory on X//H just becomes a matter of counting H-invariant vectors in irreducible representations of G. As an example, I’ll show how this process creates substantially new moduli spaces for twisted cubics in P^3, and new enumerative answers about them.
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| Fanelli |
Effective Matsusaka for Surfaces in Positive Characteristic.
The problem of determining an effective bound on the multiple which makes an ample divisor D on a smooth variety X very ample is natural and many results are known in characteristic zero. In this talk, based on a joint paper with Gabriele Di Cerbo, I will discuss this problem on surfaces in positive characteristic, giving a complete solution in this setting.
Our strategy requires an ad hoc study of pathological surfaces, on which Kodaira-type theorems can fail. A Fujita-type theorem and a vanishing result for big and nef divisors on pathological surfaces will also be discussed.
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| Faonte |
Nerve construction, A-infinity functors and homotopy theory of dg-categories.
In this talk we show how to obtained Toen's derived enrichment of the model category of dg-categories defined by Tabuada using by the dg-category of A-infinity functors. This approach was suggested by Kontsevich. We further put this construction into the framework of (infinity,2)-categories. Namely, we show that the categories of dg and A-infinity categories can be enhanced to (infinity,2)-categories. The enhancement is defined using the nerve construction for A-infinity categories, which generalizes the dg-nerve of Lurie. We prove that the (infinity,1)-truncation of to the (infinity,2)-category of dg-categories is a model for the simplicial localization at the model structure of Tabuada. As an application, we prove that the homotopy groups of the mapping space of endomorphisms at the identity functor in the (infinity,2)-category of A-infinity categories compute the Hochschild cohomology.
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| Fatighenti |
Hodge Theory and Deformations of Affine Cones of Subcanonical Projective Varieties.
Hodge Theory and Deformation Theory are known to be closely related. Amongst the many avatars of this friendship we have Griffiths's Residues calculus for hypersurfaces or the Calabi-Yau case, were the first order deformations of a smooth algebraic variety are identified with a special piece of its Hodge structure. In this talk we show how in the more general case of a smooth projective subcanonical variety X we can reconstruct part of its Hodge Theory by looking at a distinguished graded component of the first order deformations module of its affine cone A. In order to get a global reconstruction theorem we then move to the study of the Derived deformations of A (à la Kontsevich), showing how to find amongst them the missing Hodge spaces. This is a joint work with Carmelo Di Natale and Domenico Fiorenza.
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| Foscolo |
Exotic nearly Kähler structures on the 6-sphere and the product of two 3-spheres.
Compact 6-dimensional nearly Kähler manifolds are the cross-sections of Riemannian cones with G2 holonomy. Viewing Euclidean 7-space as the cone over the round 6-sphere endows the latter with a nearly Kähler structure which coincides with the standard G2-invariant almost complex structure induced by octonionic multiplication. A long-standing problem has been the question of existence of complete nearly Kähler 6-manifolds besides the four known homogeneous ones. We resolve this problem by proving the existence of an exotic (inhomogeneous) nearly Kähler structure on the 6-sphere and on the product of two 3-spheres. This is joint work with Mark Haskins, Imperial College London.
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| Gandini |
Abelian subalgebras of theta representations and sphericity.
Let theta be an automorphism of a semisimple Lie algebra g of finite order m and let G be the semisimple simply connected algebraic group with Lie algebra g. Fix a primitve m-th root of unity zeta and denote by g_i the weight space of theta of eigenvalue zeta^i, then g_0 is a reductive subalgebra of g, and the corresponding connected subgroup G_0 of G acts on g_i for all i=0, ... , m-1. Following Vinberg's terminology the representation of G_0 on g_1 is called a theta-representation. In the talk I will speak about two ongoing projects in collaboration with Moseneder-Papi and with Bravi-Chirivì-Maffei, about subalgebras of g contained in g_1 which are stable under a Borel subalgebra of g_0 and spherical G_0 orbits in g_1. Generalizing recent work of Panyushev, given such a subalgebra of g, I will explain how it decomposes into finitely many orbits under the action of a Borel subgroup of G_0, and I will explain how to parametrize these orbits.
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| Ortega |
The Prym map of degree-7 cyclic coverings.
Given a finite morphism between smooth projective curves one can associate
to it a Prym variety (an abelian variety, not necessarily principal). The
corresponding Prym map is the map between the moduli space of coverings
and the moduli space of abelian varieties with some fixed polarization
type. By dimension reasons, only in very few cases one can expect the Prym
map to be generically finite over its image.
In this talk we will explain the case of étale cyclic coverings of degree
7 over a genus 2-curve. We show that the Prym map is generically finite
over a special subvariety of the moduli space of 6-dimensional abelian
varieties with polarization type (1,1,1,1,1,7). By extending the map to a
proper map on a partial compactification of the space of coverings and
performing a local analysis we compute that the degree of this Prym map is
10.
This a joint work with Herbert Lange.
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| Saccà |
Geometry of O'Grady's 6 dimensional example.
There are not many known examples of compact irreducible hyperkähler
manifolds. Two series of examples appear in dimension 2n, for every
n>1, and are related to the Hilbert scheme of points on a K3 or an
abelian surface; and in dimension 6 and 10 there is one extra, or
exceptional, deformation class, each of which was found by O'Grady.
While considerable work has been devoted to studying hyperkähler
manifolds belonging to the first two deformation classes, not much is
known for the exceptional deformation classes. In this talk I will
present joint work with Giovanni Mongardi and Antonio Rapagnetta, regarding the
geometry of O'Grady's six dimensional example. By realizing these
examples as "quotients" of another hyperkähler manifold by a
birational involution, we are able to compute all the Hodge numbers
and, in work in progress, also study properties of their moduli
spaces/deformation class.
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| Valeri |
Adler type pseudodifferential operators and integrable systems.
In this talk I will introduce the notion of an Adler type pseudodifferential operator and show its application to get integrable Hamiltonian equations.
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