La Cristalera, Miraflores de la Sierra, Comunity of Madrid, Spain
GPS Location: 40 ° 49 '14 "N 3 ° 47' 1" O
From 9th to 27th September there will be a three weeks activity in Algebraic Geometry and Singularity Theory in Madrid, in honour of the 60th birthday of I. Luengo Velasco. In the first week a Conference will take place in the Conference Centre "La Cristalera" in Miraflores de la Sierra" a beautiful location in the mountais nearby Madrid. Its aim is to update the audience in the latest developments of Singuarity theory and neigbouring areas from Algebraic Geometry. In the last two weeks a Graduate School will take place at the Institute for Mathematical Sciences (ICMAT). Its aim is to present at a level comprehensible for graduate students and postdocs new techniques and aspects of Singularity Theory, and new connections with neighbouring areas as well. The teaches of the school will be N. A'Campo, L. Gottsche, J. Nicaise and D. van Straten, and topics like Deformation Theory, Berkovich Spaces, Enumerative Geometry and Vanishing Cycles in connection with Symplectic Geometry will be covered.
Singuarity month at ICMAT - Web page:
Algebra and Geometry and Topology of Singularities - Programme:
University of Vienna and Innsbruck
Title: Infinite Dimensional Algebraic Geometry
Artin's Approximation Theorem ensures the existence of analytic power series solutions of polynomial equations once formal or approximated solutions are known.
Instead of looking at specific solutions, one may study instead the entire set of all power series solutions. An example thereof are arc spaces and the Grinberg-Kazhdan-Drinfeld structure theorem, respectively the Denef-Loeser fibration theorem. We shall illustrate in the talk how Artin's theorem as well as these results embed into a much more general context of infinite dimensional algebraic subvarieties of power series spaces.
Université Pierre et Marie Curie
Title: Monodromy and Lefschetz fixed point formula
In 2002, Denef and Loeser computed the Lefschetz numbers of iterates of the monodromy in terms of space of arcs by direct computation on a resolution.
In this talk we shall present a new proof, relying on a Lefschetz fixed point formula and non-archimedean geometry. This is joint work with Ehud Hrushovski.
Title: Vanishing Cycles and Thom's a_f Condition
In this talk, we will describe the precise relationship between the vanishing cycles and Thom's a_f condition for an arbitrary complex analytic function on an arbitrary complex analytic space.
University of Wisconsin
Title: Characteristic classes of Hilbert schemes of points via symmetric products
I will explain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. The result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as a formula for the generating series of homology characteristic classes of symmetric products. This is joint work with Cappell, Schuermann, Ohmoto and Yokura.
Rényi Mathematical Institute of the Hungarian Academy of Sciences
Title: Lattice cohomology for superisolated and Newton nondegenerate singularities
I will review the definition of the lattice cohomology and relate with the geometric genus of hypersurface singularities.
We focus on the superisolated and Newton nondegenerate cases and discuss several new phenomenons (when the analytic structures do not choose the behavior predicted by the Seiberg Witten Invariant Conjecture).
University of Kaiserslautern
Title: Modular Computations
In this talk we present parallel modular algorithms to compute the radical of an ideal, a primary decomposition and a triangular decompositio. As an application we can compute all solutions (with multiplicities) of a given zero-dimensional polynomial system of equations over the rational numbers.
Title: Hypersurface singularities with normal crossings in codimension one
By a conjecture of Zariski, proved by Deligne and Fulton, the complement of a plane projective nodal curve has abelian fundamental group. This result was generalized by L^e and Saito in the case of complex hypersurfaces: Here the complement has abelian local fundamental groups if and only if the hypersurface has only normal crossings in codimension one. It was conjectured by Kyoji Saito and confirmed by Granger and Schulze that this latter property is also equivalent to the condition that all residues of logarithmic 1-forms are weaky holomorphic functions. Another related condition, studied by Eleonore Faber with the aim of characterizing normal crossing divisors, is the reducedness of the Jacobian scheme. We discuss the above mentioned and further related conditions and their mutual implications, putting special emphasis on the case of free divisors. We also indicate a generalization beyond the hypersurface case.
Title: Higher discriminants and the topology of algebraic maps
We define 'higher discriminants' of a proper map between complex algebraic varieties, and show (1) any component of the characteristic cycle of the pushforward of the constant function is a conormal variety to a component of a higher discriminant and (2) the support of any summand of the pushforward of the IC sheaf is a component of a higher discriminant. This is joint work with Luca Migliorini.