versión en españolPlenary Talks
Abstracts
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Chern Classes of Singular Varieties, Graph Hypersurfaces, and Feynman Integrals (by Paolo Aluffi)
Numerical evidence indicates that the value of individual
contributions of graphs to Feynman integrals are multiple zeta values.
This fact can be interpreted as a statement concerning certain
hypersurfaces of projective space determined by graphs. I will
report on work aimed at understanding this phenomenon.
Characteristic classes of singular varieties play an important role
in this study, by quantifying the singularity of graph hypersurfaces,
and by giving an algebro-geometric construction of invariants
satisfying "Feynman rules". This is joint work with Matilde Marcolli.
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Homomorphisms Between Kähler Groups (by Donu Arapura)
A group is called Kähler if it is the fundamental group of a compact Kähler manifold, and in particular a
smooth projective variety. I want to talk about geometrically meaningful homomorphisms between such groups. I hope
the study of these maps sheds light on the original class of groups.
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Topological and Geometrical Aspects of Singularity Theory (by Enrique Artal)
Topology and geometry of quasi-projective varieties play an important role in local and global singularity theory.
When interested in fundamental groups, it is enough to study the surface case; complements of projective plane curves
are a particularly interesting case. One important invariant (for both the groups and the varieties) is given by
characteristic varieties, which can be seen as a generalization of Alexander polynomials. We review the state of the
art and present new results about the relationship of these invariants with mappings on 1-dimensional orbifolds
(following a joint work with J.I. Cogolludo).
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Characteristic classes of singular varieties and motivic theory (by Jean-Paul Brasselet)
This is a joint work with Jörg Schürmann (Münster) and Shoji Yokura
(Kagoshima).
The Hirzebruch theory unifies the three theories of characteristic classes:
Chern, Todd and Thom-Hirzebruch classes, whose degree zero components are respectively
Euler-Poincaré characteristic, arithmetic (or Todd) genus and the signature.
In the case of a singular algebraic complex variety, there are no characteristic
classes in cohomology anymore, but classes in homology. The three theories are
generalised by the (homology) transformations of Schwartz-MacPherson, Baum-Fulton-MacPherson
and Cappell-Shaneson respectively. The problem is that these transformations are defined
on different spaces. Using the motivic theory, more precisely the Grothendieck relative
group of algebraic varieties over the algebraic variety, one can unify the three transformations.
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Topology of Polynomial Maps at Infinity and Danielewski Surfaces (by Pierrette Cassou-Noguès)
In this talk we will speak about the relations between the topology at
infinity of polynomial maps and some singular surfaces on which those
maps are defined. Even if the subject is not completly understood it
has interesting applications. We will study some cases that we understand,
give some applications and ask questions.
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Some Examples of Contact Manifolds Associated to Non-Isolated Singularities (by Clément Caubel)
It is now well known that one can associate a well defined contact manifold to any isolated singular
point of a complex analytic variety. I will show how to do the same for some non-isolated singularities,
and give examples which cannot be obtained from isolated singularities.
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On the Milnor Fibers of Hyperplane Arrangements (by Alexandru Dimca)
We present new results on the monodromy action on the Milnor fiber of a
line arrangement in the complex projective plane. In particular we emphasize
the key role played by a certain class of nets in this question.
These results were obtained in joint work with N. Budur, S. Papadima and
M. Saito.
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Singularities of Pairs (by Lawrence Ein)
We will describe some of the invariants attached to singularities of pairs, such as log-canonical thresholds,
and log-discrepancies. We will also describe results on how to apply these invariants to study higher dimensonal
birational geometry.
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On the Nash Problem for Surface Singularities (by Javier Fernández de Bobadilla)
No abstract yet.
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On a Relation Between Algebraic and Geometric Properties of Braids (by Juan González Meneses)
This is a joint work with Volker Gebhardt and Bert Wiest.
We will introduce some algebraic operations, called cyclic slidings,
that can be used to solve the conjugacy problem in the braid groups, in a simple way.
These operations allow to find conjugates of a given braid, which are as simple
as possible from the algebraic point of view (related to left greedy normal forms).
We will then show that cyclic slidings also simplify braids from the geometric point of view,
untangling the reduction curves. Hence they provide a simple algorithm to determine
whether a braid is pseudo-Anosov, periodic or reducible, and in the latter case
to find its reduction system of curves.
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Graphical Bracket and Jones Polynomials for Knots and Links in Thickened Surfaces (by Louis Kauffman)
If one examines the proof of the invariance of the bracket polynomial using
oriented knot and link diagrams it becomes apparent that much of the local oriented structure can be
preserved in the state summation. A closer look reveals that none of this extra structure contributes to
the evaluation of link diagrams that are drawn in the plane or on the surface of the two-sphere.
However, if these diagrams originate on a higher genus surface, then there is indeed much extra information
in the original bracket state sum model. We exploit this extra information in this talk
to formulate new invariants of knots and links in thickened surfaces and new invariants
of virtual knots and links. (A virtual knot is
an equivalence class of knots in thickened (orientable) surfaces taken up to handle
stabilization and surface homeomorphism. This talk will construct these invariants and
discuss their relationship with classical and virtual knot theory. We will discuss joint
work with Heather Dye and Vassily Manturov on the categorification of these invariants.
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Hirzebruch Invariants of Symmetric Products of Quasi-Projective Varieties (by Laurentiu Maxim)
Hirzebruch characteristic classes for singular varieties have been recently defined by Brasellet-Schuermann-Yokura
as an attempt to unify previously known characteristic class theories for singular spaces
(e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes and resp. Cappell-Shaneson L-classes).
In this talk, I will discuss equivariant analogues of these classes, and show how these can be used to calculate
generating series for Hirzebruch classes of symmetric products of possibly singular quasi-projective varieties.
This is joint work with Cappell, Schuermann and Shaneson.
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The Ascending Chain Condition for Log Canonical Thresholds (by Mircea Mustață)
Shokurov conjectured that the set of all log canonical thresholds of pairs of bounded dimension satisfies the Ascending Chain Condition.
It is known that this conjecture has strong consequences for the Minimal Model Program. I will discuss a proof of this conjecture in
the case when the ambient variety is smooth (or more generally, a locally complete intersection).
This is based on joint work with Tommaso de Fernex and Lawrence Ein.
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Some Results and Open Problems on Surface Singularities (by András Némethi)
Recently some amazing constructions, connections and results were
found and developed about the invariants of normal complex surface
singularities. Here I just mention the construction of splice-quotient
singularities by Neumann and Wahl, and the computation of some of
their invariants (by Neumann, Wahl, Okuma and myself), the connection
described by the Campillo, Delgado and Guzein-Zade formula (targeting by the
identification of two series the topological description
of the Poincaré series associated with the multivariable divisorial
filtration), or the connection around the `Casson Invariant Conjecture'
(of Neumann-Wahl) or its generalization, the `Seiberg-Witten Invariant
Conjecture' (Némethi-Nicolaescu), which lead to a deep interplay with
Heegaard-Floer homology of the link, having the outcome the construction
of the `lattice homology of the link'.
We list some key results and some guiding open problems.
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Separating Sets in Isolated Complex Singularities (by Walter Neumann)
Joint work with Lev Birbrair and Alexandre Fernandes. A "separating set" (or "Cheeger cycle")
in the germ of a point x in an algebraic set X of real dimension n is a
codimension 1 set which has negligible (n-1)-dimensional density at x, but which
separates X locally into pieces of positive n-dimensional density. The talk will describe
the ubiquity of such sets in germs of isolated complex singularities.
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On the Milnor Fibration of the Mixed Functions (by Mutsuo Oka)
Mixed functions are analytic functions in variables z1,..., zn and their conjugates
z1,..., zn
We introduce the notion of Newton non-degeneracy for mixed functions and develop a basic tool for the study of mixed
hypersurface singularities. We show the existence of a canonical resolution of the singularity, and the existence of
the Milnor fibration under the strong non-degeneracy condition.
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Variations on the Skein Relation (by Luis Paris)
The HOMFLYPT polynomial is an invariant of knots and links discovered
simultaneously by many people in the early 80s. Its definition is
simple, it is easy to calculate, and it has many applications. So, this
polynomial has played an important role in the theory of knots since its
discovery. The starting point of the lecture is the following question:
what can be the theory of HOMFLYPT polynomials for other knot-type
objects such as singular knots and links, or virtual knots and links, or
links in 3-manifolds? The approach I propose to develop is through the
theory of Markov traces on generalized Hecke algebras. This is the
approach of Jones for the Jones polynomials as well as for the HOMFLYPT
polynomials.
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The Kite of a Constellation (by Patrick Popescu Pampu)
A constellation is a set of infinitely near points obtained by successive blowing-ups
starting from a smooth point of a surface. Two graphs are usually used to encode the topology of
finite constellations : the Enriques diagram and the dual graph of the divisor obtained by totally
blowing-up the constellation. I will explain the construction of a 2-dimensional simplicial complex,
the kite of the constellation, in which both graphs embed naturally. This allows to understand
geometrically their relation, as well as the algorithms of passage from one to the other using
continued fractions.
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The Topology of Hyperplane Arrangements (by Richard Randell)
This talk surveys the topology of hyperplane arrangements,
particularly those aspects where the work of Libgober heavily impacts
the subject. Thus there will be a focus on covers and homotopy groups.
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On the Morse Complex of Complexified Arrangements (by Mario Salvetti)
In a previous paper we constructed a discrete field over the complement of a
complexified hyperplane arrangement, which in particular produces a Morse complex
computing local cohomology. In a subsequent paper we improved the two dimensional case.
Here we give some further reductions in the higher dimensional case.
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Real Analytic Singularities in Complex Geometry (by José Seade)
During the 1960s there was a rich interplay between manifolds theory and
the topology of complex singularities, leading to a deeper understanding of
both of these areas of mathematics. A keystone for this was Milnor's
fibration theorem. This result gives important information about the
topology of holomorphic map-germs, and it also plays a fundamental role for
studying the topology of the link, as well as for applications to foliations
theory, open-book decompositions, knots theory and contact geometry. In this
talk we make a quick review of Milnor's theorem in the classical setting of
holomorphic mappings, with a new viewpoint that lends itself to
generalizations for real singularities, and then move forward to explaining
some analogous questions and known results, for families of real analytic
map-germs that arise in complex geometry. We also discuss how these ideas
may yield to new insights into the geometry and topology of manifolds.
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Characteristic Varieties and Homological Finiteness Properties (by Alex Suciu)
The characteristic varieties of a space
are the support loci for homology with coefficients
in rank one local systems. Much of the original
motivation for the study of these varieties
(present in the pioneering work of Anatoly
Libgober on the subject) comes from the very
precise information they give about the homology
of finite abelian covers of the given space.
In this talk, I will explain how the characteristic
varieties also hold information about the homological
finiteness properties of free abelian covers. This
information leads to computable upper bounds for the
Bieri-Neumann-Strebel-Renz invariants of the space,
and its fundamental group. Under suitable hypothesis,
these bounds can be expressed in terms of even simpler
data, namely, the resonance varieties associated to
the cohomology ring.
This is joint work with Stefan Papadima.
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Degenerations and Regenerations - and Applications to K3-Surfaces (by Mina Teicher)
In the talk we shall show two different K3-surfaces with equisingular branch curves and different
related fundamental groups. We shall use degenerations of the two surfaces to a union of planes,
one called "The Pillow" and the other one called "The Magician" and the respective regeneration of the braid monodromy
of the branch curves of the degeneration (a union of lines) to the braid monodromy of the singular branch
curves of the original surfaces.
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Resonance Varieties of Arrangement Complements (by Sergey Yuzvinsky)
The talk is a survey of the resonance varieties of the Orlik-Solomon algebras and their
relations with topology and algebraic geometry. In particular we recall Libgober's results
in this area and discuss further developments.
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