Singularities in the Midwest
FRIDAY, March 19 (Room: B231 Van Vleck):
Schedule of talks
- 1:20-2:10 PM : Sylvain Cappell
- 2:30-3:20 PM : Andrei Caldarau
- 4:00-4:50 PM : David Massey
SATURDAY, March 20 (Room: B231 Van Vleck):
- 9:00-9:50 : Jörg Schürmann
- 10:20-11:10 : Donu Arapura
- 11:30-12:20 : Anatoly Libgober
- 2:30-3:20 : Mircea Mustata
- 3:50-4:40 : Nero Budur
TITLES and ABSTRACTS:
- Sylvain: The role of singularities in transformation groups
The talk explores classifications of topological group actions
using their singularities. The methods, developed in joint work with Min Yan
and Shmuel Weinberger, yield functorialities for such classifications
- Andrei: A conjecture of Duflo and the Ext algebra of branes
The Duflo theorem is a statement in Lie theory which allows us to compute the ring structure of the center of the universal enveloping algebra of a finite-dimensional Lie algebra. A categorical version of it was used by Maxim Kontsevich to give a spectacular proof of the so-called "Theorem on complex manifolds," which computes the multiplicative structure of Hochschild cohomology of a complex manifold in terms of the algebra of polyvector fields. In Lie theory there are also more general Duflo-type statements (mostly conjectural), which study the case of a pair (Lie algebra, Lie subalgebra). I will explain how these translate into conjectures about the
multiplicative structure of the Ext-algebra of the structure sheaf of a complex submanifold of a complex manifold, and how from this interaction we can hope to gain new insights into both algebraic geometry and Lie theory. (Joint with Damien Calaque and Junwu Tu.)
- David: Intersection cohomology, nearby monodromy, and vanishing cycles
A characteristic zero version of Gabber's result on the nilpotent/weight filtration on the
nearby cycles of intersection cohomology (with, possibly, twisted coefficients) would, in particular,
imply that, if the monodromy T on the nearby cycles has finite order, then the nearby cycles are a direct
sum of intersection cohomology complexes.
In the most trivial case, where T is the identity or, equivalently, where the vanishing cycles are zero,
one would hope that the nearby cycles are, in fact, an intersection cohomology complex. However,
this is false, which means that having no vanishing cycles is not a "good" cohomological analog of
having a normally nonsingular slice.
In this talk, I will discuss examples/counterexamples and results on when shifted hypersurface
restrictions of intersection cohomology yield intersection cohomology.
- Jörg: Nearby cycles and characteristic classes for mixed Hodge modules.
The Hodge-Chern class transformation of a mixed Hodge module
is a K-theoretical characteristic class capturing information about the
graded pieces of the corresponding filtered de Rham complex.
Application of a suitable Todd class transformation gives the Hirzebruch
classes in homology. We explain that these transformations commute with
specialization, which for Hodge modules means the corresponding nearby
cycles defined in terms of the V-filtration of Malgrange-Kashiwara.
This generalizes a corresponding specialization result of Verdier about
MacPherson's Chern class transformation.
- Donu: A category of motivic sheaves
I want to discuss some ideas for building a category of motivic "sheaves" in
characteristic 0 using a method due Nori. Not everything works yet, but many things do:
The category is abelian, there are pullbacks, and etale Betti and (with some restrictions)
- Anatoly: Characteristic varieties
of arrangements with isolated non normal crossings.
I will discuss the problem
of describing the structure of the families of rank one local systems
with non-vanishing cohomology on the complements to arrangements of hyperplanes and new examples of such families.
- Mircea: Invariants of graded systems of ideals and psh functions.
To an ideal on a singular variety one can associate invariants via the sequence of multiplier ideals: these are the jumping numbers. More generally, one can attach such invariants to graded systems of ideals, and to plurisubharmonic functions. I will describe these invariants, and I will discuss
a conjecture relating them to quasimonomial valuations. This is joint work with
Mattias Jonsson and Rob Lazarsfeld.
- Nero: Local systems and singularities.
The local systems on the complement of an arbitrary divisor
in a smooth complex projective variety contain a wealth of information
about the singularities of the divisor. We describe various
filtrations on local systems (Hodge, polar), their induced
stratifications on the space of unitary rank one local systems, and
their relation with the singularities of the divisor. Applications
include: some results about Hodge numbers of abelian covers, and some
results about singularity invariants (Hodge spectra, b-functions, and
topological zeta functions) for hyperplane arrangements. This surveys
joint work with various people.