Singularities in the Midwest, V.
Schedule of Talks
All talks (and coffee breaks) will be in the 9th floor lounge, Room 911, in Van Vleck Hall.
Titles and Abstracts
- Pierre Albin:
The families index formula on stratified spaces
Thom-Mather stratified spaces arise naturally even when studying smooth objects, e.g.,
as algebraic varieties, orbit spaces of smooth group actions, and many moduli spaces.
There has recently been a lot of activity developing analysis on these spaces and studying
topological invariants such as the signature. I will report on joint work with J
esse Gell-Redman in which we study families of Dirac-type operators on stratified spaces
and establish a formula for the Chern character of their index bundle.
- Graham Denham:
Arrangements, matroids, and CSM classes
The maximal likelihood variety of a complex hyperplane arrangement
describes the set of critical points of all rational functions with
poles and zeros on the arrangement. Its bidegree encodes the
h-vector of the underlying matroid's broken circuit complex. I
will describe work-in-progress with Federico Ardila and June Huh that
constructs a tropical version of the maximal likelihood variety. This
sheds some new light on the tropical characteristic classes of López de
Medrano, Rincón, and Shaw, as well as on the h-vector of the broken
circuit complex of an arbitrary matroid.
- Eva Elduque:
On the signed Euler characteristic property for subvarieties of Abelian varieties
Franecki and Kapranov proved that the Euler characteristic of a perverse sheaf on a
semi-abelian variety is non-negative. This result has several purely topological
consequences regarding the sign of the (topological and intersection homology) Euler
characteristic of a subvariety of an abelian variety, and it is natural to attempt to
justify them by more elementary methods. In this talk, we'll explore the geometric tools
used recently in the proof of the signed Euler characteristic property.
Joint work with Christian Geske and Laurentiu Maxim.
- Christian Geske:
Algebraic Intersection Spaces
Associated to certain compact pseudomanifolds X are topological spaces IX called
intersection spaces. The reduced homology of IX with rational coefficients satisfies duality,
providing an alternative to the intersection homology of X, which, in the case that X is a
projective hypersurface with isolated singularities, better approximates the homology of a
smooth deformation of X. Intersection spaces have been constructed only for a small
subclass of pseudomanifolds. We describe a possible extension which applies to a wide
array of complex analytic spaces, including all projective varieties.
- Manuel Gonzalez Villa:
Motivic zeta functions: beyond the non-degenerate case
We will discuss how to generalize the formulas for the motivic and topological zeta functions
of hypersurfaces that are non-degenerate with respect to its Newton polyhedra.
Some of these ideas are based on previous joint work with P. Gonzalez-Perez about the quasi-ordinary hypersurface case.
- Alexandra Kjuchukova:
Branched covers of the four-sphere
We consider branched covering maps f: Y4 → S4, where Y is a manifold and the
branching set of f is a two-sphere embedded in the four-sphere with one singular point,
modeled on the cone of a slice knot K. I will explain how to obtain a ribbon obstruction
for K from this set-up. I will also discuss an interesting family of examples of
four-manifolds arising as 3-fold branched covers over the four-sphere.
Joint work with Patricia Cahn.
- Yongqiang Liu:
Jump loci of complex quasi-projective manifolds
Let X be a complex quasi-projective manifold. We would like to study the jump loci of
rank-1 local systems on X. Using the Albanese map, we reduce the problem to study the
jump loci of perverse sheaves on semi-abelain varieties. In particular, we show that the
jump loci of X have codimension bound and propagation property.
This is a joint work with Laurentiu Maxim and Botong Wang.
- Jörg Schürmann:
Hecke algebra and motivic Chern classes of Schubert cells
We explain in the context of complete flag varieties X=G/B the relations between motivic Chern classes of Schubert cells, duality, K-theoretical stable basis of Okounkov and convolution actions of Hecke-algebras as in the work of Ginzburg and Tanisaki .
This is joint work with P. Aluffi, L. Mihalcea and C. Su.
- Yuan Wang:
Characterization of abelian varieties for log pairs
Let X be a projective variety. A celebrated theorem of Kawamata says that if X is smooth,
the dimension of X is equal to that of its Albanese variety, and the Kodaira dimension of X is 0,
then X is birational to an abelian variety. Later it was shown by Chen and Hacon that instead of
requiring that the Kodaira dimension of X is 0, it suffices to require that the m-th plurigenera
of X is 1 for some m ≥ 2. In this talk I will discuss the case where X is not necessarily smooth.
In particular, I will present a result that generalizes Kawamata’s result beyond the
log canonical case and another one that generalizes the result of Chen and Hacon to klt pairs.
- Xiping Zhang:
Characteristic Classes of EIDS
Essential Isolated Determinantal Singularities (EIDS) are the rank loci of transverse maps
to the spaces of matrices. They are very interesting type of singularities, and it is
important to study their characteristic classes. In this talk I will introduce the total
Chern classes for singular varieties, (i.e., the Chern-Schwartz-MacPherson class),
and I will give formulas for such classes on EIDS and show some applications of them.