20191209 ~ 20191213 4092
Dates: 913 December 2019
This conference intends to provide a good exchange platform on exciting new results on singularities and promote further the development of algebraic and geometric aspects of singularity theory.
Organizers
Name  University 

Igor Burban  University of Paderborn 
Stanislaw Janeczko  Banach Institute 
Gerhard Pfister  University of Kaiserslautern 
Stephen Yau  Tsinghua University 
Huaiqing Zuo  Tsinghua University 
NO. 
English Name 
Chinese Name 
Employer's Name in English 
Employer's Name in Chinese 
1 
Igor Burban 
University of Paderborn 

2 
Stanisaw Janeczko 
Banach Center and Warsaw University of Technology 

3 
Gerhard Pfister 
University of Kaiserslautern 

4 
Stephen ShingToung Yau 

5 
Huaiqing Zuo 
左怀青 
Tsinghua University 

6 
Anne Frühbis Krüger 
Leibniz Universität Hannover 

7 
Duco van Straten 
Hohannes Gutenberg Universität Mainz 
美因茨大学 

8 
Mutsuo Oka 
Tokyo University of Science 
东京理科大学 

9 
Viktor Kulikov 
Steklov Mathematical Instit 

10 
Ngo Viet Trung 
Vietnam Academy of Science and Technology 
越南科学院 

11 
Shihoko Ishii 
TokyoWoman's Chiristian University 

12 
Antonio Campillo 
University of Valladolid, Spain 
西班牙瓦拉多利德大学 

13 
Claus Hertling 
University of Mannheim 
缅因大学 

14 
Wolfram Decker 
Technische Universität Kaiserslautern 

15 
Victor Goryunov 
University of Livepool 
利物浦大学 

16 
Sabir GuzeinZade 
Moscow State University 
莫斯科国立大学 

17 
Kyoji Saito 
Kavli IPMU, University of Tokyo 
京都大学数理解析研究所 

18 
Morihiko Saito 
Kyoto University 

19 
Lê Dũng Tráng 
The Abdus Salam, International Centre for Theoretical Physics 

20 
Mihai Tibar 
University of Lille, FRANCE 
法国里尔大学 

21 
Wolfgang Ebeling 
University of Hannover 
汉诺威大学 

22 
Eleonore Faber 
the University of Leeds 
利兹大学 

23 
Thomas Reichelt 
Universität Heidelberg 

24 
Thomas Markwig 
the University of Tübinge 
图宾根大学 

25 
Nguyen Duc Hong 
BCAM – Basque Center for Applied Mathematics Mazarredo, 14, 48009 Bilbao, Basque Country – Spain 

26 
Pham Thuy Huong 
Quy Nhon University 
昆恩大学 

27 
Xiaotao Sun 
孙笑涛 
AMSS 
中国科学院数学与系统科学研究院 
28 
Guangfeng Jiang 
姜广峰 
Beijing University of Chemical Technology 
北京化工大学 
29 
Fanning Meng 
孟凡宁 
Guangzhou University 
广州大学 
30 
Chuangqiang Hu 
胡创强 
Tsinghua University 
清华大学 
31 
Xiankui Meng 
孟宪奎 
Tsinghua University 
清华大学 
32 
Naveed Hussian 
哈那维 
Huashang College Guangdong University of Fiance and Economics 
广东财经大学 
33 
Shuanghe Fan 
Tsinghua University 
清华大学 

34 
Klaus Altmann 
Freie Universität Berlin, Fachbereich Mathematik und Informatik, Mathematisches Institut 
柏林自由大学 

35 
GertMartin Wilhelm Greuel 
technische universitt kaiserslautern 
德国凯撒斯劳滕工业大学 

36 
Hua Zheng 
华诤 
University of Hong Kong 
香港大学 
37 
Alexander Elashvili 

38 
Guorui Ma 
马国瑞 
Tsinghua University 
清华大学 
39 
Xiping Zhang 

40 
Qiwei Zhu 
朱其蔚 
Tsinghua University 
清华大学 
=============================== Monday ===================================
9:0010:00
Claus Hertling
Marked singularities, their moduli spaces, distinguished bases and Stokes regions
10:0010:30 Coffee break
10:3011:30
Zheng Hua
Noncommutative MatherYau theorem and its applications
11:3014:30 Lunch break
14:3015:30
Thomas Reichelt
Hodge theory of GKZ systems
15:3016:00 Coffee break
16:0016:40
Mutsuo Oka
On the Milnor fibration of $f(z)\bar g(z)$
16:5017:30
Wolfgang Ebeling
Dual invertible polynomials with permutation symmetries and the orbifold Euler characteristic
=============================== Tuesday ===================================
9:0010:00
Javier Fernandez de Bobadilla
Classification of Reflexive Modules on Gorenstein Surface Singularities and a conjecture of Drodz, Greuel and Kashuba
10:0010:30 Coffee break
10:3011:30
Eleonore Faber
Countable CohenMacaulay type and the infinitygon
11:3014:00 Lunch break
14:0014:40
MihaiMarius Tibar
Polar degree conjectures
14:5015:30
Ngyuen Duc Hong
Cohomology of contact loci
15:3016:00 Coffee break
16:0016:40
Goryunov, Victor
Vanishing cycles of matrix singularities
16:5017:30
Victor Kulikov
On rigid germs of finite morphisms of smooth surfaces
=============================== Wednesday =================================
9:0010:00
Kyoji Saito
Primitive forms without metric structure and integrable hierarchy
10:0010:30 Coffee break
10:3011:30
GertMartin Greuel
On Semicontinuity of Singularity Invariants in Families of Formal Power Series
11:3014:00 Lunch break
=============================== Thursday ===================================
9:0010:00
Shihoko Ishii
A new bridge between positive characteristic and characteristic 0
10:0010:30 Coffee break
10:3011:30
Klaus Altmann
Universal extensions of semigroups
11:3014:00 Lunch break
14:0014:40
Stanislaw Janeczko
Geometric and algebraic restrictions of differential forms
14:5015:30
Guangfeng Jiang
Free subarrangements of SHI and ISH arrangements
15:3016:00 Coffee break
16:0016:40
Ngo Viet Trung
Depth functions of homogeneous ideals
16:5017:30
Antonio Campillo
Poincare series of matroids
=============================== Friday ====================================
9:0010:00
Xiaotao Sun
A MiyaokaYau type inequality of surfaces in characteristic $p>0$
10:0010:30 Coffee break
10:30 11:30
Sabir GuseinZade
Poincaré polynomials of filtrations and algebraic links in the Poincaré sphere and in the links of simple surface singularities.
11:3014:00 Lunch break
==========================Monday ========================================
Claus Hertling
Marked singularities, their moduli spaces, distinguished bases and Stokes regions
Abstract: One part of the talk is on a global study of μconstant
families of holomorphic function germs with isolated singularities.
Some new data are defined and discussed, μconstant monodromy groups,
marked singularities, their global moduli spaces, and a global Torelli
type conjecture for their Brieskorn lattices. Another part of the talk
is on universal unfoldings, Brieskorn lattices at semisimple points,
their Stokes data and distinguished bases, and a global
LyashkoLooijenga map. For the simple singularities (by work of
Looijenga and Deligne 73/74) and the simple elliptic singularities
(joint work with C. Roucairol 18) this leads to an understanding of a
certain global base space as an atlas of Stokes data.
===================================================================================
Zheng Hua
Noncommutative MatherYau theorem and its applications
Abstract: I will survey some recent progress in noncommutative differential calculus of formal and analytic potentials and their applications to algebraic geometry and cluster algebra. This is based on joint work with Guisong Zhou and with Bernhard Keller.
===================================================================================
Thomas Reichelt
Hodge theory of GKZ systems
Abstract: GKZ hypergeometric systems were introduced by Gelfand, Kapranov and Zelevinsky as a generalization of Gauss
hypergeometric differential equation. It can be shown that for certain parameters the GKZsystems carry the structure of
a mixed Hodge module in the sense of Morihiko Saito. We will discuss the Hodge and weight filtration of these Dmodules.
===================================================================================
Mutsuo Oka
On the Milnor fibration of $f(z)\bar g(z)$
Abstract: We consider a mixed function of type $H(\mathbf z,\bar{\mathbf z})=f(\mathbf z)\bar g({\mathbf z})$ where $f$ and $g$ are which are nondegenerate with respect to the Newton boundary.We assume also that the variety $f=g=0$ is a nondegenerate complete intersection variety.Under the convenience assumption of the Newton boundary and the multiplicity condition,we will show that $H$ has both a tubular and a spherical Milnor fibrations. For nonconvenience case, we assume the local tame nondegeneracy, we can also prove the existence of Milnor fibration.
===================================================================================
Wolfgang Ebeling
Dual invertible polynomials with permutation symmetries and the orbifold Euler characteristic
Abstract:
P.Berglund, T.H\"ubsch, and M.Henningson proposed a method to construct mirror symmetric CalabiYau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A.Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal symmetries and some permutations of variables. It turns out that in order to get certain mirror symmetric properties in this case, one needs a special condition on the permutation group called parity condition (PC). We prove that, if the permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign. This is joint work with Sabir GuseinZade.
====================================================================================
======================== Tuesday==========================================
Javier Fernandez de Bobadilla
Classification of Reflexive Modules on Gorenstein Surface Singularities and a conjecture of Drodz, Greuel and Kashuba
Abstract: In this paper we generalize ArtinVerdier, Esnault and Wunram construction of McKay correspondence to arbitrary Gorenstein surface singularities. The key idea is the definition and a systematic use of a degeneracy module, which is an enhancement of the first Chern class construction via a degeneracy locus. We study also deformation and moduli questions. Among our main result we quote: a full classification of special reflexive MCM modules on normal Gorenstein surface singularities in terms of divisorial valuations centered at the singularity, a first Chern class determination at an adequate resolution of singularities, construction of moduli spaces of special reflexive modules, a complete classification of Gorenstein normal surface singularities in representation types, and a study on the deformation theory of MCM modules and its interaction with their pullbacks at resolutions. For the proof of these theorems we prove several isomorphisms between different deformation functors that we expect that will be useful in further work.
====================================================================================
Eleonore Faber
Countable CohenMacaulay type and the infinitygon
Abstract: By a result of BuchweitzGreuelSchreyer, a hypersurface is of countable CohenMacaulay representation type, if and only if it is isomorphic to a singularity of type $A_{\infty}$ or $D_{\infty}$. In this talk, we show how the category CM(R) of maximal CohenMacaulay modules over the coordinate ring $R$ for the $A_{\infty}$curve gives a categorical model for arcs in an ``$\infty$gon''. This allows us to construct triangulations of the $\infty$gon, making use of the language of cluster categories.
This is joint work with J. August, M. Cheung, S. Gratz, and S. Schroll.
====================================================================================
MihaiMarius Tibar
Polar degree conjectures
My talk is based on joint work with Dirk Siersma and Joseph Steenbrink.
Dolgachev (Michigan Math J, 2000) has initiated the study of Cremona polar transformations i.e. birational maps gradf:ℙn⇢ℙn defined by the gradient map of a homogeneous polynomial. He conjectured that the topological degree of gradf depends only on the projective zero locus V of f,so that it can be called polar degree of V, denoted pol(V). The hypersurfaces with pol(V)=1 are called homaloidal; Dolgachev classified the homaloidal plane curves. I'll discuss the proof of Dolgachev's conjecture found by Dimca and Papadima (Annals of Math 2003),their conjecture on the classification of homaloidal hypersurfaces with isolated singularities proved by Huh (Duke Math J, 2014), and Huh's conjecture on the classification of hypersurfaces with isolated singularities and pol(V)=2 with its recent proof.
====================================================================================
Nguyen Duc
Cohomology of contact loci
Abstract: We construct a spectral sequence converging to the
cohomology with compact support of the $m$th contact locus of a
complex polynomial. The first page is explicitly described in terms
of a log resolution and coincides with the first page of McLean's
spectral sequence converging to the Floer cohomology of the $m$th
iterate of the monodromy, when the polynomial has an isolated
singularity. Inspired by this connection, we conjecture that if two
germs of holomorphic functions are embedded topologically equivalent,
then the Milnor fibers of the their tangent cones are homotopy
equivalent. (Joint work with J. Fern\'andez de Bobadilla, N. Budur,Q.T. Le).
====================================================================================
Goryunov, Victor
Vanishing cycles of matrix singularities
The talk is about holomorphic map germs M : (Cs,0) → Matn, where the target is the space of either square, or symmetric, or skewsymmetric n × n matrices. The target contains the set ∆ of all degenerate matrices, and our main object will be the vanishing topology of M−1(∆). Our attention is on the singular Milnor fibre of M, that is, the local inverse image V of ∆ under a generic small perturbation of M. The variety V is highly singular, but, according to Lˆe Du ̃ng Tr ́ang’s theorem, it is homotopic to a wedge of (s − 1)dimensional spheres.
The talk will start with introduction of local models for the spheres vanishing in the matrix context.
We will then prove the μ = τ conjecture formulated by Damon for corank 1 mapgerms M with a generic linear part, and a generalisation of this conjecture to the matrix version of boundary function singularities.
Bifurcation diagrams of matrix singularities will also be discussed, and a rather unexpected appearance of the discriminants of certain ShephardTodd groups as such diagrams will be highlighted.
If time permits, possible approaches to the study of the monodromy will be mentioned.
====================================================================================
Victor Kulikov
On rigid germs of finite morphisms of smooth surfaces
Abstract: In the talk, questions related to deformations of germs of finite
morphisms of smooth surfaces and a correspondence between the set of
rigid germs of finite morphisms of smooth surfaces and the set of Belyi
rational functions will be discussed.
====================================================================================
======================= Wednesday =========================================
Kyoji Saito
Primitive forms without metric structure and integrable hierarchy
Abstract: We introduce primitive forms without higher residue structure.
Just as a classical primitive form induces a flat structure (Frobenius
manifold structure), they induces flat structure without metric. They,
further, induce cerain integrable hierarchies which generalize Gelfand
Dikii Hierarchy.
====================================================================================
GertMartin Greuel
On Semicontinuity of Singularity Invariants in Families of Formal Power Series
Abstract: The problem we are considering came up in connection with the classification of singularities in positive characteristic.
Then it is important that certain invariants like the determinacy can be bounded simultaneously in families of formal power series parametrized by some affine algebraic variety. In contrast to the case of analytic
or algebraic families, where such a bound is well known, the problem is rather subtle, since the modules defining the invariants are quasifinite but not finite over the base space. In fact, in general the fibre dimension is not semicontinuous and the quasifinite locus is not open. However, if we pass to the completed fibers in a family of modules we can prove that their fiber dimension is semicontinuous under some mild conditions. We prove this in a rather general framework by introducing and using the completed and the Henselian tensor product. Finally we apply this to the Milnor number and the Tjurina number in families of hypersurfaces and complete intersections and to the determinacy in a family of ideals.
====================================================================================
======================== Thursday ==========================================
Shihoko Ishii
A new bridge between positive characteristic and characteristic 0
Abstract:
One bridge between singularities in positive characteristic and the singularities in
characteristic 0 is already established in the following form:
``Fsingularities in characteristic p for p>>0” <==> ``birational singularities in characteristic 0”
In my talk I will show a challenge to establish a bridge in a different form:
``birational singularities in characteristic p (fixed) "==>
``construct a birational singularities in characteristic 0 with the same properties”.
I will show the present status of this direction.
I will also show applications when the challenge is successful.
====================================================================================
Klaus Altmann
Universal extensions of semigroups
Homogeneous deformations of toric singularities can be
understood as extensions of the associated semigroups. They
will form their own category, and the surprising result is
that this category contains an initial object.
That is, unlike to the geometric singularities obtained from
taking the Spec of their semigroup algebras, the semigroups
themseves are not obstructed. We will demonstrate this for
Pinkham's example of the cone over the rational normal curve
of degree 4.
====================================================================================
Stanislaw Janeczko
Geometric and algebraic restrictions of differential forms
Abstract: By algebraic and geometric conormals and tangents we study germs of differen tial forms over singular varieties. The geometric restriction of differential forms to singular varieties is introduced and algebraic restriction of differential forms with vanishing geometric restrictions, called residual algebraic restrictions, are investi gated. Residues of plane curvegerms, hypersurfaces, Lagrangian varieties as well as the geometric and algebraic restriction via a mapping were calculated. The natural exact sequence 0 → R•(Z) → A•(Z) → G•(Z) → 0 defines the residues R•(Z) = G•(Z, M)/A•(Z, M) with A•(Z) = Λ•(M)/A•(Z, M) and G•(Z) = Λ•(M)/G•(Z, M), where G•(Z,M) are geometric restrictions and A•(Z,M) are algebraic restrictions to Z.
This is a joint work with Goo Ishikawa.
====================================================================================
Guangfeng Jiang
Free subarrangements of SHI and ISH arrangements
Abstract: The cones over SHI arrangements are proved to be free by Athanasiadis.
Recently Abe, Suyama and Tsujie proved the freeness of the ISH arrangement.
They also showed conditions on the freeness of the deleted ISH arrangements
which contain the braid arrangment as subarrangement.
In this talk, we are interested in the subarrangements SHI(G) and ISH(G)
of the SHI and ISH arrangements associated with a graph G,which may not contain all hyperplanes of the braid arrangement.We prove necessary and sufficient conditions on the graph G tomake the cones of SHI(G) and ISh(G) free.
====================================================================================
Ngo Viet Trung
Depth functions of homogeneous ideals
Abstract: A classical result of Brodmann says that the numerical nonnegative function depth R/I^t, where I is a homogeneous ideal in a polynomial ring R, is always convergent, i.e. it is constant for t large enough. In 2005, Herzog and Hibi posed the conjecture that the function depth R/I^t can be any numerical nonnegative convergent function. This talk will will give an affirmative answer to this conjecture and shows a similar result for the symbolic powers I^(t) of I, namely that for any numerical positive function f(t) which is periodic for t large enough, there is a homogeneous ideal I in a polynomial ring R such that depth R/I^(t) = f(t) for t > 0. It is still an open question whether the function depth R/I^(t) is always periodic for t large enough.
====================================================================================
Antonio Campillo
Poincare series of matroids
Abstract: Multiindex filtrations on the local ring of singularities have an associated Poincaré series which in many cases have direct information on their topology of geometry. These is the case of the natural filtration for plane curve singularities for which the Poincaré series coincides with the Alexander polynomial of the link. We show how for a natural filtration on polynomial rings provides a Poincaré series of matroids which has complete direct information of their discrete structure. This is a joint work with Ricardo Podestá (UNC, Argentina).
====================================================================================
======================== Friday ===========================================
Xiaotao Sun
A MiyaokaYau type inequality of surfaces in characteristic $p>0$
Abstract: For minimal smooth projective surfaces $S$ of general type, we prove $K^2_S\le 32\chi(\sO_S)$ and give examples of $S$ with $$K^2_S=32\chi(\sO_S).$$
This proves that $\chi(\sO_S)>0$ holds for all smooth projective minimal surfaces $S$ of general type, which answers completely a question of ShepherdBarron.
Our key observation is that such MiyaokaYau type inequality follows slope inequalities of a fiberation $f:S\to C$.However, we will gives examples of
$f:S\to C$ with nonsmooth generic fibers of arithmetic genus $g\ge 2$ such that
$$K^2_{S/C}<\frac{4g4}{g}{\rm deg}f_*\omega_{S/C},$$
which are counterexamples of Xiao's slope inequality in case of positive characteristic. This is a joint work with Gu Yi and Zhou Mingshuo.
====================================================================================
Sabir GuseinZade
Poincaré polynomials of filtrations and algebraic links in the Poincaré sphere and in the links of simple surface singularities.
The Alexander polynomial in several variables is defined for links in threedimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the $E_8$surface singularity $(S,0)$ in $(C^3,0)$ with the 5dimensional sphere $\S_{\epsilon}^5$ of radius $\epsilon$ in $C^3·. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in $(S,0)$ with the sphere $\S_{\epsilon}^5$ of radius $\epsilon% small enough. We discuss to which extend the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It turns out that, if the strict transform of a curve in $(S,0)$ does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding $E_8$diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere is determined by the Poincaré series of the filtration defined by the corresponding curve valuations. The Poincaré series of a filtration on the ring of germs of functions on a surface can be computed as a certain integral with respect to the Euler characteristic over the space of effective Cartier divisors. One can consider the corresponding integral over the space of effective Weil divisors. It is a power series with rational exponents: "the Weil Poincaré series". We describe to which extend the Weil Poincaré series of a collection of curve filtrations on a simple surface singularity determines the resolution of the curve and thus the topology of the corresponding algebraic link. The corresponding questions are considered for divisorial valuations on the simple surface singularities.
The talk is based on a joint work with A.Campillo and F.Delgado.
====================================================================================
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