SINGULARITY THEORY CONFERENCE
Schedule of Talks
Titles and Abstracts
- Alexander Aleksandrov: Residue theory and logarithmic de Rham complex
The purpose of the talk is to give an elementary introduction to the
theory of residue of logarithmic differential
forms, and to describe some of the less known
applications of this theory, developed by the author in the past few
years. In particular, we briefly discuss the notion of residue due
to H. Poincare, J. de Rham, J. Leray and K. Saito, and then
obtain a nice description of regular meromorphic
differential forms in terms of residues of logarithmic differential
forms. We also discuss a new method for computing
the topological index of complex vector fields on hypersurfaces with
arbitrary singularities, some applications to the theory of
holonomic D-modules of Fuchsian and logarithmic types,
to the theory of Hodge structures on singular varieties, etc.
- Paolo Aluffi: Chern classes of hyperplane arrangements
We relate invariants such as the Chern-Schwartz-MacPherson class of the class in the Grothendieck group of varieties to combinatorial invariants, for the case of hyperplane arrangements in projective space. We will explain the connection between these algebro-geometric invariants and results of Orlik-Solomon, Mustata-Schenck, and Wakefield-Yoshinaga.
- Donu Arapura: Hodge theory and the fundamental group
I want to discuss some thoughts about Hodge
and motivic structures associated to the fundamental group of an algebraic variety
using Tannakian methods.
I will compare this to other approaches due
to Hain, Morgan, Simpson and others.
- Markus Banagl: Singularities and Intersection Spaces: Theory and Application.
In many situations, it is homotopy theoretically possible to associate to a
singular space in a natural way a generalized geometric Poincare complex,
called its intersection space, whose cohomology turns out to be a new
cohomology theory for singular spaces, not isomorphic in general to
intersection cohomology or Cheeger-L2-cohomology. An alternative description
of the new theory by a de Rham complex of global differential forms is
available. The talk will discuss the properties of the new theory, stressing
differences to intersection cohomology, as well as indicate applications of
these methods, even outside singularity theory.
We will consider the K-theory of intersection spaces. We will see how, as a
by-product, one obtains results on equivariant cohomology and flat bundles.
The theory also addresses questions in type II string theory and mirror
symmetry. While intersection cohomology is stable under small resolutions,
the new theory is often stable under deformations of singularities. The
latter result is partly joint work with L. Maxim. An analytic description
remains to be found, but we shall indicate a partial result based on results
of Melrose and Hausel-Hunsicker-Mazzeo.
- Jean-Paul Brasselet: Some insights on the Euler local obstruction
The lecture concerns a joint work with N. Grulha and M. Ruas.
The local Euler obstruction was introduced by R. MacPherson as a key ingredient
for his construction of characteristic classes of singular complex algebraic varieties.
Then, an equivalent definition was given by J.-P. Brasselet and M.-H. Schwartz using vector fields
and many other definitions and interpretations have been provided.
The notion of local Euler obstruction developed mainly in two directions: the first one
comes back to MacPherson's definition and concerns with differential forms.
That is developed by W. Ebeling and S. Gusein-Zade in a series of papers.
The second one relates local Euler obstruction with functions and maps defined on the variety.
That approach is useful to relate local Euler obstruction with other indices.
Aim of the lecture is to present these new features on the subject.
- Sylvain Cappell: Symmetries and rigidity of aspherical manifolds
Classical work of Borel had shown that an action of the
circle on a manifold with contractible universal cover yields non-trivial
center in the manifold's fundamental group. In the 60's, Conner and
Raymond made further deep investigations which led them to conjecture
a converse to Borel's result. We construct counter-examples to this
conjecture, i.e., we exhibit aspherical manifolds (in all dimensions greater
than or equal to 6) which have non-trivial center in their fundamental groups
but no circle actions. The constructions and invariants involve synthesizing
disparate ideas of geometric topology of singular spaces, geometric group
theory and hyperbolic geometry.
(This is joint work with Shmuel Weinberger and Min Yan.)
- Zhi Chen: KZ connections and Brauer type algebras
We introduce a Brauer type algebra and a BMW type algebra for every Coxeter group and every pseudo reflection group. This is an application of KZ connections on the complementary spaces of hyperplane arrangements.
- Manuel Gonzalez Villa : Motivic zeta functions for quasi-ordinary hypersurface singularities
Denef and Loeser applied the theory of motivic integration to define new invariants of hypersurface singularities. The motivic Milnor fibre, introduced as a formal limit of the motivic zeta function, encodes much of the geometry of the Milnor fibration. We discuss using toric methods how to compute these invariants for the class of irreducible hypersurface quasi-ordinary singularities. We prove that these invariants are determined by the embedded topological type of the singularity, which is known to be completely characterized by a finite number of characteristic monomials of some fractional power series associated to the hypersurface This is a joint work with P.D. Gonzalez-Perez.
- Jose Ignacio Cogolludo: Mordell-Weil groups of elliptic threefolds over P2, Alexander polynomials, and quasi-toric relations of curves
Given C a (possibly reducible and non-reduced) projective plane curve, we define the Alexander polynomial A(t) of C
with respect to the multiplicities of its components. To each root of A(t) one associates a cyclic covering of P2
ramified along C and an elliptic threefold W with a natural automorphism.
The purpose of this talk will be to establish a connection between the multiplicity of roots of A(t),
the rank of the Mordell-Weil group of W, and the existence of enough "independent" maps onto elliptic orbifold curves
(a generalization of pencils) containing C as fibers. Curves for which such pencils exist are a generalization
of torus-type curves and are called here quasi-toric curves.
This connection allows one to use the theory of elliptic curves over a function field to find properties of
quasi-projective groups and Alexander polynomials. In particular one can find bounds for the degree of Alexander
polynomials of curves with only nodes and cusps as singularities.
This is a joint work with Anatoly Libgober.
- Guangfeng Jiang: The Global Invariant of Hyperplane Arrangements
- Ludmil Katzarkov: Degenerations, singularities and wall crossings
In this talk we will introduce new categorical structures
from classical prospective of degnerations.
- Toshitake Kohno: Monodromy groups of conformal field theory
There is an action of the mapping class groups on the space of the
conformal blocks for Riemann surfaces defined by monodromy.
We give a qualitative estimate for the images of such representations
of mapping class groups. In particular, we show that the image of
any Johnson subgroup contains a non-abelian free group.
In the case of braid groups we describe the monodromy group
in relation with triangle groups. Based on the estimate of the
monodromy groups we give an answer to conjectures by Squier
on Burau representations of braid groups.
It was shown by P. Gilmer and G. Masbaum that the
monodromy groups of conformal field theory for Riemann surfaces
are defined over cyclotomic integers. We show that,
in general, the monodromy groups are not isomorphic to a higher
rank irreducible lattices in semi-simple Lie groups.
This is a joint work with Louis Funar.
- Zhi Lu: A differential operator and tom Dieck-Kosniowski-Stong localization theorem
In this talk, we define a differential operator on the "dual"
algebra of the unoriented G-representation algebra introduced by
Conner and Floyd, where G is a mod 2 torus group of rank n.
With the help of G-colored graphs (or mod 2 GKM graphs),
we may use this differential operator to give a very simply
equivalent description of tom Dieck--Kosniowski--Stong
localization theorem in the setting of smooth closed n-manifolds
with effective smooth G-actions (also called n-dimensional 2-torus
manifolds). Some applications will be considered.
This is a joint work with Qiangbo Tan.
- Teresa Monteiro Fernandes: Propagation for solutions with moderate growth of D-Modules
We obtain a Cauchy Theorem for holomorphic solutions of D-modules with moderate growth conditions in a space of parameters and obtain an estimate for the obstruction to the propagation (microsupport).
- Mutsuo Oka: Mixed polynomials and mixed varieties
Mixed varieties is a complex analytic technique to study real algebraic variety of codimension 2.
I will explain basic properties of mixed varieties and
- Piotr Pragacz: On positivity of Thom polynomials
The pioneering papers of Griffiths and Fulton and Lazarsfeld
investigated numerical positivity related to ample vector
bundles in differential and algebraic geometry.
Their various variants are nowadays widely investigated
in algebraic geometry. Among main objects of global
singularity theory are the Thom polynomials of singularity
classes. We shall consider Thom polynomials of singularities
of mappings and Lagrangian and Legendrian Thom polynomials.
We shall show that in some bases coming from representation
theory, they admit positive expansions. (This is a report on joint work with M. Mikosz and A. Weber.)
- Richard Rimanyi: Thom series via equivariant localization and iterated residues
Thom polynomials measure how topology forces singularities.
Natural infinite sequences of Thom polynomials can be arranged in formal
power series, the Thom series. In the talk we will explore different
interpretations and computational strategies of Thom series,
as well as present some open problems.
This is a joint work with L. Feher, and we will also report on
recent results of Berczi-Szenes and Kazarian.
- Christian Schnell: Hodge modules on abelian varieties
I will explain some results on Hodge modules on abelian varieties,
and applications to the study of irregular varieties. (Joint work with Mihnea Popa.)
- Jörg Schürmann: Characteristic classes of Hilbert schemes of points via symmetric products.
We explain a new formula for the generating series of
the Hirzebruch characteristic homology classes of the Hilbert schemes of
points for a smooth quasi-projective variety, push-forward to the
corresponding symmetric products. This result is based on two facts:
(i) a nice interplay between the geometric definition of a motivic power
structure and a motivic Pontrjagin ring of the symmetric products,
(ii) a formula for the generating series of the Hirzebruch characteristic
homology classes of the symmetric products.
This is joint work in progress with L. Maxim, S. Cappell,
T. Ohmoto and S. Yokura.
- Kiyoshi Takeuchi: Motivic Milnor fibers and Newton polyhedra
By computing the equivariant mixed Hodge numbers
of motivic Milnor fibers introduced by Denef-Loeser
etc., we obtain various formulae for the Jordan normal forms
of the local and global monodromies of polynomials. Especially
we focus our attention on the global ones, i.e., the monodromies at infinity.
For polynomials over affine complete intersection
varieties the results will be described by the mixed
volumes of the faces of their Newton polyhedra.
This is a joint work with Y. Matsui and A. Esterov.
- Andrzej Weber: Local equivariant Chern classes of Singular varieties
Equivariant cohomology is a powerful tool to study of complex manifolds equipped with a torus action. The localization theorem of Atiyah and Bott and the resulting formula of Berline-Vergne allow to compute global invariants of singular subsets in terms of the fixed points of the action. We will concentrate on the Chern(-Schwartz-MacPherson) classes. The global class is determined by the local contributions coming from the fixed points. The local contributions are in a form of a quotients with the local Chern classes in the numarators. On the other hand, as in the "residue theorem" for meromorphic functions, the sum of local Chern classes is equal to zero. Especially for Grassmanians we obtain interesting calculations with nontrivial formulas involving rational functions. We will discuss the issue of positivity: the local Chern class may be presented in a various ways, depending on some choice of a certain graphs. For some choices we find that the coefficients of the presentation are nonnegative. Also the coefficients in an appropriate Schur basis are nonnegative in many examples.
- Jaroslaw Wlodarczyk: Weights on the cohomology and invariants of singularities
We study the weight filtration on the cohomology of a proper
complex algebraic variety and obtain natural upper bounds on its size,
when it is the exceptional divisor of a singularity. The invariants of
singularities introduced here gives rather
strong information about the topology of rational and related singularities. (based on joint paper with D.Arapura and P.Bakhtary)
- Jie Wu: Obtaining Higher Homotopy Groups from the Fundamental Groups
The braid groups and link groups are the fundamental groups of configurations and link complements, respectively. In this talk, we give an illustration that the general higher homotopy groups of spheres can be obtained as the quotient groups of the intersections of certain canonical subgroups of these groups. The talk will give a philosophical view that the higher homotopy groups may be discovered by systematically studying the subgroups of the fundamental groups of canonical objects.
- Min Yan: Homotopy Classification of Multiaxial Actions
A multiaxial action by U(n) is locally modeled on kC^n + C^p, where U(n) acts canonically on C^n and trivially on C^p. There are similar multiaxial actions by O(n) and Sp(2n). Mike Davis gave diffeomorphic classification of multiaxial smooth manifolds in case k \le n. We describe how to homotopically classify multiaxial topological manifolds without assuming k \le n. Moreover, we compute the classification in case of the multiaxial sphere. Such classification is a vast generalization of the classical result on fake complex projective spaces. In fact, up until now, not much is known about the homotopy classifications of actions by positive dimensional Lie groups.
This is a joint work with S. Cappell and S. Weinberger.
- Shoji Yokura: Fiberwise bordism groups
We introduce a notion of fiberwise bordism for certain families of smooth manifolds. We explain motivations for considering this bordism, some connections with other known results, etc. This is a joint work with
M. Banagl and J. Schuermann.
- Masahiko Yoshinaga: Minimal Stratifications for Line Arrangements
The homotopy type of complements of complex hyperplane arrangements have a
special property, so called minimality (Dimca-Papadima and Randell,
around 2000). In this talk, we introduce the "dual" object to the minimal CW complex for
two dimensional real line arrangements, which we call minimal stratification.
It is a real semialgebraic stratification which induces a partition into
contractible manifolds. We also see associated presentation of the
fundamental group. This talk is based on arXiv:1105.1857.
- Sergey Yuzvinsky: Topological complexity of arrangement complements
Topological complexity of (motion planning on) a topological space has been defined by M.Farber as a
specialization of Schwartz`s genus.
Its calculation for hyperplane arrangement complements is important for topological robotics and relates to interesting problems involving the Orlik-Solomon algebras. We will survey old results on the topic and discuss some new ones.
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