Singularity Theory Conference
Schedule of Talks
All talks (and coffee breaks) will be in Room 504, Block A, Science Building (on campus).
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July 10 |
July 11 |
July 12 |
July 13 |
July 14 |
| 9-9:50 |
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| 9:50-10:10 |
Coffee Break |
Coffee Break |
Coffee Break |
Coffee Break |
Coffee Break |
| 10:10-11:00 |
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| 11:10-12:00 |
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| LUNCH |
Lunch |
Lunch |
Lunch |
Lunch |
Lunch |
| 14:00-14:50 |
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| 14:50-15:10 |
Coffee Break |
Coffee Break |
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Coffee Break |
Coffee Break |
| 15:10-16:00 |
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Titles and Abstracts
- Patrick Brosnan:
Hessenberg varieties, monodromy and the Shareshian-Wachs conjecture
I will explain joint work with Tim Chow which proves a conjecture of Shareshian and Wachs relating a combinatorial object,
the chromatic quasi-symmetric function, to an action of the symmetric group on the cohomology of certain smooth
subvarieties of a flag manifold, the regular semi-simple Hessenberg varieities. The proof reinterprets the
symmetric group action, originally defined by Tymoczko in terms of equivariant cohomology, as a monodromy action.
We then show that this action can be computed in terms of the cohomology of certain degenerations of the Hessenberg
varieties which, although singular, happen to have palindromic cohomology groups (i.e., groups that numerically satisfy
Poincare duality). There is an interesting independent proof of the conjecture by Guay-Paquet and I will mention
this as well.
- Nero Budur:
Absolute sets and the Decomposition Theorem
The celebrated Monodromy Theorem states that the eigenvalues
of the monodromy of a polynomial are roots of unity. In this talk we
give on overview of recent results on local systems achieving a vast
generalization of the Monodromy Theorem. We end up with a conjecture
of André-Oort type for special loci of local systems. The conjecture
is true in rank one, and if true in general, it would provide a simple
proof in all generality of the DecompositionTheorem of
Beilinson-Bernstein-Deligne-Gabber. Joint work with Botong Wang.
- Dan Burghelea:
Towards a computational MORSE-NOVIKOV theory
In this talk I will sketch a computer friendly alternative to Morse-Novikov theory and explain what it can do for the
topology of complements of complex hypersurfaces in Cn, in particular about the calculations of basic topological
invariants of such varieties.
- José Ignacio Cogolludo Agustín:
Quasi-projective Artin groups and the QP K(π,1)-conjecture
A group is called quasi-projective if it can be realized as the fundamental
group of a quasi-projective space. Even quasi-projective Artin groups are
fully characterized in terms of their graph in a joint work with R.Blasco.
We will discuss recent results on a potential characterization of
quasi-projective groups as well as the QP K(π,1)-conjecture, that is,
whether or not such groups can be realized by Eilenberg-MacLane quasi-
projective spaces.
- Lawrence Ein:
- James Fullwood:
A canonical Chern class for embeddable schemes
For a scheme which is embeddable in a smooth variety (over an
arbitrary algebraically closed field), we define a canonical
characteristic class in its group
of algebraic cycles modulo rational equivalence, which coincides with its
total homological Chern class when the scheme is smooth. The definition is
very similar to the definition of Fulton's canonical class, but with a
modification according to the singularities of the scheme. If the
underlying field is algebraically closed of characteristic zero and the
scheme is embeddable as a hypersurface in a smooth variety, then it
immediately follows via a formula of Aluffi that the class we define
coincides with its Chern-Schwartz-MacPherson class, and we conjecture that
this is in fact true even when the scheme is not a hypersurface in a
smooth variety. If true, the definition of the class we define would yield
a very simple formula for the Chern-Schwartz-MacPherson class of arbitrary
embeddable schemes, and since our class is defined for fields of arbitrary
charcteristic, the
class we define may lead to a functorial theory of singular Chern classes
in positive characteristic.
- Manuel González Villa:
Multiplier ideals associated to plane curve singularities
Multiplier ideals and their jumping numbers are a powerful tool to study the singularities of an ideal
on complex algebraic variety with mild singularities. Jumping numbers of (analytically) irreducible plane
curve singularities have been independently computed and studied by Jarvilehto, Naie and Tucker.
We will report in a work in progress to describe the multiplier ideals associated to irreducible plane
curve singularities. This is a joint work with Carlos Rodrigo Guzman (CIMAT).
- Javier Fernandez de Bobadilla:
Representation of surface homeomorphisms by tête-à-tête graphs
We use tête-à-tête graphs as defined by N. A'Campo and extended versions to codify all periodic mapping classes
of an orientable surface with non-empty boundary, improving work of N. A'Campo and C. Graf. We also introduce the notion
of mixed tête-à-tête graphs to model some pseudo-periodic homeomorphisms. In particular we are able to codify the
monodromy of any irreducible plane curve singularity. The work ends with an appendix that studies all the possible
combinatorial structures that make a given filtered metric ribbon graph with some regularity conditions into a mixed
tête-à-tête graph.
- Guangfeng Jiang:
Falk Invariants of signed Graphic arrangements
The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological
invariant. The third rank of successive quotients in the lower central series of the fundamental group was called
Falk invariant of the arrangement since M. Falk gave the first combinatorial formula and asked to give a combinatorial
interpretation. We proved that the Falk invariant of an arrangement associated with signed graph G without loops
is double of the sum of k3, k4 and k3±, where kl is the number of subgraphs of G
that are switching equivalent to the cliques with l vertices, k3± is that of subgraphs which have 3
vertices and each two vertices are connected by double edges with different signs.
This formula modifies the one given by H. Schenck and A. Suciu, and answers partially Falk's question in the case
of signed graphic arrangements.
- Jun Li:
Enumerating curves with prescribed singularities, and beyond
I will report the approach to enumerating nodal curves via cobordism theory, initiated by
Tzeng; on the later joint work with Tzeng on enumerating curves with arbitrary prescribed singularities.
Some later development will also be discussed.
- Yongqiang Liu:
Topology of smooth closed subvarieties of complex semi-abelian varieties
We use the non-proper Morse theory of Palais-Smale to investigate the topology of smooth closed subvarieties
of complex semi-abelian varieties, and that of their infinite cyclic covers. As main applications, we obtain the
finite generation (except in the middle degree) of the corresponding integral Alexander modules,
as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties.
Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic
property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result
for the L^2 Betti numbers of very affine manifolds. Our method also reproves and generalizes a theorem of June Huh
about the master function on very affine manifolds to smooth closed sub-varieties of semi-abelian varieties.
This is a joint work with Laurentiu Maxim and Botong Wang.
- Ignacio Luengo:
Bernstein polynomial of 2-Puiseux pairs irreducible plane curve singularities
I will report on a joint work with E. Artal, P. Cassou-Nogues and A. Melle (arxiv 1611.01091 ).
In 1982, Yano proposed a conjecture predicting the b-exponents of an irreducible plane curve singularity
which is generic in its equisingularity class. In a previous work we proved the conjecture for the case
in which the germ has two Puiseux pairs and its algebraic monodromy has distinct eigenvalues.
In this work we aim to study the Bernstein polynomial for any function with the hypotheses above.
In particular the set of all common roots of those Bernstein polynomials is given.
- Alejandro Melle Hernández:
Power structure over the Grothendieck ring of maps
A power structure over a ring is a method to give sense to expressions of the form
(1+a1t+a2t2+...)m, where ai (i=1, 2,...) and m are elements of the ring.
The (natural) power structure over the Grothendieck ring of complex quasi-projective varieties
appeared to be useful for a number of applications. In this talk we meanly focus on
the Grothendieck ring of maps of complex quasi-projective varieties. I describe two natural
λ-structures on it which lead to the same power structure.
We show that this power structure is effective. In the terms of this power structure we write
some equations containing classes of Hilbert-Chow morphisms. We describe some generalizations of
this construction for maps of varieties with some additional structures.
This is a joint work with S.M. Gusein-Zade and I. Luengo.
- Toru Ohmoto:
C1-triangulation and semialgebraic de Rham homotopy theory
I will talk about a solution to a fundamental question on the C1-regularity of triangulations
for semialgebraic sets (over any real closed field). It would be useful for e.g. the semialgbraic de Rham homotopy theory
introduced by Kontsevich-Soibelman. This is a joint work with Masahiro Shiota.
- Mutsuo Oka:
Łojasiewicz exponents of non-degenerate holomorohic and mixed functions
We consider Łojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity
at the origin. We give an explicit estimation of the Łojasiewicz exponent in a slightly weaker form than the
assertion in Fukui. For a weighted homogeneous polynomial, we give a better estimation in the form which is conjectured
by S. Brzostowski, T. Krasiński and C. Oleksik under under some condition (the Łojasiewicz non-degeneracy).
We also introduce Łojasiewicz inequality for strongly non-degenerate mixed functions and generalize this
estimation for mixed functions.
- Tomohiro Okuma:
Cohomology of ideals in normal surface singularities
We study normal surface singularities in terms of cohomology of ideal
sheaves on resolution spaces. We introduce an invariant q for
integrally closed ideals in the local ring of normal surface
singularities. We show that the ideals with maximal q satisfy some
nice properties. The invariant q induces the normalized reduction
number r of the singularities. Rational singularities are
characterized by r=1. We show that (weakly) elliptic singularities
satisfy r=2. This is based on joint works with Kei-ichi Watanabe and
Ken-ichi Yoshida.
- Markus Pflaum:
Stratified groupoids and inertia spaces
The inertia space of a compact Lie group action or more
generally of a proper Lie groupoid has an interesting singularity
structure. Unlike the quotient space of the group action, respectively
the groupoid, the inertia space can not be stratified by orbit types, in
general. In the talk we explain this phenomenon and provide a
stratification and local description of the inertia space. Moreover, we
show that that leads naturally to the concept of a stratified groupoid
which lies in between the one of a Lie groupoid and the one of a
topological groupoid. Finally we show that a de Rham theorem holds for
inertia spaces and explain the connection of the inertia space with the
non-commutative geometry of the underlying groupoid.
- Jörg Schürmann:
On Ginzburg's bivariant Chern classes for singular spaces
We explain an extension of the (equivariant) MacPherson Chern class transformation to a bivariant
theory with respect to suitable convolutions as conjectured by Ginzburg.
This is based on the Lagrangian approach via characteristic cycles. As an application we show,
that for flag manifolds G/B the two cohomological Weyl group actions constructed by Ginzburg and
Aluffi-Mihalcea coincide. These Weyl group actions permute the (equivariant) Chern classes of
the corresponding Schubert cells. This is joint work with P. Aluffi, L. Mihalcea and C. Su.
- José Seade:
- Kiyoshi Takeuchi:
On the monodromies and the limit mixed Hodge structures
of families of algebraic varieties
We study the monodromies and the limit mixed Hodge structures
of families of complete intersection varieties over a punctured
disk in the complex plane. For this purpose, we express their motivic
nearby fibers in terms of the geometric data of some Newton polyhedra.
In particular, the limit mixed Hodge numbers and some part
of the Jordan normal forms of the monodromies of such a family will
be described explicitly. This is a joint work with Takahiro Saito.
- Botong Wang:
Algebraic geometry approach to enumerative combinatorics
We will discuss two applications of algebraic geometry in enumerative combinatorics.
The first one is the proof of the "top-heavy" conjecture of Dowling and Wilson in 1975 using
hard Lefschetz theorem for intersection cohomology groups. The conjecture is a higher dimensional
generalization of the following theorem due to de Bruijn and Erdos: n points in the plane determine
at least n lines unless all the points lie on a line. The second one is the proof of a log-concave
conjecture on the number of independent sets using Hodge index theorem. This is joint work with June Huh.
- Shoji Yokura:
On poset-stratified spaces and related topics
I will discuss some thoughts about poset-stratified spaces from a naive general topological viewpoint and talk
about some applications such as hyperplane arrangements and a poset-stratified space structure of the set of
morphisms of a locally small category.
- Huaiqing Zuo:
Non-existence of negative weight derivations of isolated
singularities and new Lie algebras
Let R=C[x1,x2,..., xn]/(f1,..., fm) be a
positively graded Artinian algebra. A long-standing conjecture in
algebraic geometry, commutative algebra and rational homotopy theory is
the non-existence of negative weight derivations on R. Alexsandrov
conjectured that there are no negative weight derivations when R is a
complete intersection algebra and Yau conjectured there are no negative
weight derivations on R when R is the moduli algebra of a weighted
homogeneous hypersurface singularity. This problem is also important in
rational homotopy theory and differential geometry. Wahl conjectured that
non-existence of negative weight derivations is still true for positive
dimensional positively graded R. In this talk we present our recent
progress on these problems. Joint work with Stephen Yau, Hao Chen and Bingyi Chen.