Némethi60

Conference program

	Monday	Tuesday	Wednesday	Thursday	Friday
09:00-09.30	Registration & opening
09:30-10.30	A. Dimca	P. Popescu-Pampu	J. Wahl	F. Loeser	K. Altmann
10:30-11.00	Coffee break
11:00-12.00	W. Veys	A. Melle	Z. Szabó	G-M. Greuel	R. Rimányi
12:00-12.10	Break
12:10-13:10	E. Artal	G. Farkas	M. Borodzik	K. Palka	M. Pe Pereira
13:10-15:10	Lunch break
15:10-16:10	T. Okuma	A. Parusinski	Free afternoon	D. van Straten
16:10-16:20	Break		Free afternoon	Break
16:20-17:20	L. Fehér	I. Luengo	Social programs	NA60
17:20-17:40	Welcome reception	Coffee break
17:40-18:40		D. Kerner
19:00				Conference dinner

A google calendar version of the program can be reached here: Némethi60 Calendar.

The book of abstracts can be downloaded from here.

Social events of the conference

Birkhäuser book table

All participants are kindly invited to come by and browse a selection of latest Birkhäuser/Springer books and journals.
Birkhäuser publishing editor attendance from Monday afternoon to Wednesday.

Welcome wine&cheese&sandwich reception

Date: Monday (May 27, 2019)
Location: MTA Rényi Institute of Mathematics

Conference dinner

Date: Thursday (May 30, 2019), 19.00
Location: Treffort Aula Restaurant

Titles & abstracts

K. Altmann (Freie Universität Berlin)

Infinitesimal qG-deformations of cyclic quotient singularities

The subject of the talk is two-dimensional cyclic quotients, i.e. two-dimensional toric singularities. We introduce the classical work of Kollár/Shephard-Barron relating the components of their deformations and the so-called P-resolutions, we give several combinatorial descriptions of both gadgets, and we will focus on two special components among them - the Artin component allowing a simultaneous resolution and the qG-deformations preserving the Q-Gorenstein property. That is, it becomes important that several (or all) reflexive powers of the dualizing sheaf fit into the deformation as well. We will study this property in dependence on the exponent r. While the answers are already known for deformations over reduced base spaces (char = 0), we will now focus on the infinitesimal theory. (Joint work with János Kollár.)

E. Artal Bartolo (Universidad de Zaragoza)

Cyclic covers of weighted projective planes; applications to weighted-Le-Yomdine singularities

It is well-known since Zariski that the Betti number of a cyclic cover of the projective plane curve ramified along a cuspidal-nodal curve depends on the position of their singularities. Following the work of Esnault-Viehweg work, this formula has been generalized for arbitrary curves by several authors. In this talk, we state a version of Esnault-Viehweg results which allows to state a formula for the Betti numbers of a cyclic cover of a weighted projective plane curve ramified along a curve, in terms of the behavior of curves of some fixed degrees with respect to some multiplier ideals on the singular point. This formula follows the lines of the classical one, with some corrections on the singular points of the weighted projective plane. This formula will give information about the monodromy of weighted-Le-Yomdine singularities of surface; rational cuspidal curves will provide surface singularities whose link is a rational homology sphere. This is a joint work in progress with J.I. Cogolludo and J. Martín-Morales.

M. Borodzik (University of Warsaw)

Real Seifert forms 20 years after

In 1995 András Némethi introduced so-called Hodge Variation Structures, that formed a bridge between Picard-Lefschetz theory and link theory. In this talk we show how the `Hodge-theoretical' point of view on knot theory originated by Némethi can help understand twisted Blanchfield pairings for links. This is a joint project with Anthony Conway and Wojciech Politarczyk.

A. Dimca (Université de Nice-Sophia Antipolis)

Generators of the Jacobian syzygy module and rational cuspidal curves

We give upper bounds for the number and degrees of generators of the module of Jacobian syzygies of a reduced plane curve. Then we relate these numbers to rational (nearly) cuspidal curves, and to curves realizing the maximum Tjurina number in a well known du Plessis-Wall inequality.

G. Farkas (Humboldt-Universität Berlin)

Syzygies, Koszul modules and topological invariants of groups

I will discuss the deep connection between the structure of the equations of certain algebraic varieties and Alexander invariants of groups. On the algebro-geometric side, this parallelism has recently led to a very simple proof of Green's Conjecture on syzygies of canonical curves, whereas on the topological side has produced a universal bound on the nilpoteny index of the fundamental group of non-fibred compact Kähler manifolds. Joint work with Aprodu, Papadima, Raicu and Weyman.

L. Fehér (Eötvös Lóránd University, Budapest)

Motivic Chern classes of singularities

Recent progress in calculation of motivic Chern classes and equivariant motivic Chern classes will hopefully make these invariants effective tools to study singular varieties and singularities.
In this lecture I talk about the simplest case: conical singularities.
Specialization of the motivic Chern class (y=0) gives a motivic K-theory fundamental class. This class is different from the sheaf-theoretic and the push-forward K-class for cones over smooth hypersurfaces.
An equivariant version of the study of cones leads to the motivic generalization of the "projective Thom polynomial" formula of Fehér-Némethi-Rimányi.

G-M. Greuel (TU Kaiserslautern)

Finite determinacy of matrices and ideals

We characterize ideals I in the power series ring R = K[[x1,...,xs]] that are finitely determined up to contact equivalence. Here two ideals I and J are contact equivalent if the local K-algebras R/I and R/J are isomorphic. If I is minimally generated by a1,..., am, we call I finitely contact determined if it is contact equivalent to any ideal J that can be generated by b1,...,bm with ai-bi in (x1,...xs)^k for some integer k. The main result says that I is finitely contact determined if and only if I is an isolated complete intersection singularity, provided dim(R/I) > 0 and K is an infinite field (of arbitrary characteristic). We give also computable and semicontinuous determinacy bounds. The above result is proved by considering left-right equivalence on the ring M_{m,n} of m x n matrices A with entries in R and we show that the Fitting ideals of a finitely determined matrix in M_{m,n} have maximal height, a result of independent interest. The case of ideals is treated by considering 1-column matrices. Fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in R. Some results are known in characteristic 0, but they need new (and more sophisticated) arguments in positive characteristic partly because the tangent space to the orbit of the left-right group cannot be described in the classical way. In addition we point out several other oddities, including the concept of specialization for power series, where the classical approach (due to Krull) does not work anymore. We report also on some open problems and a conjecture. (Joint with Thuy Huong Pham, to appear in J. of Algebra.)

D. Kerner (Ben-Gurion University of the Negev)

Tjurina modules for matrix singularities, finite determinacy, new singularity ideals

Let R be a local ring over a field of zero characteristic, e.g. power series in several variables. Consider the space of matrices with entries in R. Consider the action of contact group, the left-right multiplications and the coordinate changes. We study the corresponding Tjurina module, T^1, the tangent space to the miniversal deformation.
We obtain various bounds on localizations of T^1 and compute the set theoretic support of T^1, i.e. the radical of the annihilator of T^1. This brings the definition of an (apparently new) type of singular locus, the "essential singular locus" of a map/sub-scheme. It reflects the "unexpected" singularities of a subscheme, ignoring those imposed by the singularities of the ambient space. Unlike the classical singular locus (defined by a Fitting ideal of the module of differentials) the essential is defined by the annihilator ideal of the module of derivations.

F. Loeser (Sorbonne Université)

Motivic integration on the Hitchin fibration

Groechenig, Wyss and Ziegler have recently proved a conjecture of Hausel and Thaddeus concerning an equality between stringy Hodge numbers of moduli spaces of Higgs bundles for $\mathrm{SL}_n$ and $\mathrm{PGL}_n$. A crucial ingredient in their approach is the use of $p$-adic integration in the fibres of the Hitchin fibration. We will present a motivic version of their result which is obtained by using motivic integration. This is joint work with Dimitri Wyss.

I. Luengo Velasco (Univesidad Complutense de Madrid)

Post-quantum Cryptography with high degree polynomials

Post-quantum cryptography is the public-key cryptography resistant to future quantum computers. In this talk we will talk about a post-quantum cryptosystem called DME (Double Matrix Exponentiation) based on on high degree polynomials on a small number of variables that I have developed (using ideas of Algebraic Geometry), patented and present it to the NIST contest to choose the future post-quantum cryptography standard. I will also present some Conmutative Algebra open questions related with the algebraic cryptoanalysis of the scheme DME.

A. Melle Hernández (Univesidad Complutense de Madrid)

The minimal Tjurina number of irreducible germs of plane curves

In this talk a closed formula for the minimal Tjurina number of any equisingularity class in terms of the multiplicity sequence of the strict transform along a resolution is given. As a consequence a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and the Tjurina numbers for any irreducible germ of plane curve singularity. (Joint work with Maria Alberich-Carramiñana, Patricio Almiron and Guillem Blanco.)

K. Palka (IMPAN-Institute of Mathematics, Warsaw)

Singularities of planar curves

The number of singular points of a planar curve can be bounded in terms of homological invariants of the curve. But the existing bounds are far from optimal. We show how to improve them using the minimal model program for log surfaces.

A. Parusinski (Université de Nice-Sophia Antipolis)

Whitney’s fibering conjecture and arc-wise analytic equisingularity

We give a report on recent result on Zariski equisingularity including a construction of an arc-wise analytic stratification and the proof of Whitney's fibering conjecture. We apply this construction to give a stratified general position theorem in real and complex algebraic geometry. We also discuss the relation between Zariski equisingularity and Lipschitz stratification as well as several open problems.

P. Popescu-Pampu (Université de Lille)

A tropical and logarithmic study of Milnor fibers

I will present joint work with Maria Angelica Cueto and Dmitry Stepanov, explaining how to combine tools from tropical and logarithmic geometries in order to understand the structure of Milnor fibers of complex singularities.

R. Rimányi (UNC Chapel Hill)

From counting partitions to the structure of motivic characteristic classes

The 19th century idea of a "Durfee squares" produced an effective combinatorial trick to count partitions. (In fact, one of its modern applications is the h-index measuring mathematicians’ productivity.) It has been reinterpreted in Donaldson-Thomas theory as the comparison of two ways of calculating DT invariants of the A_2 quiver. In this talk, we will explore the Donaldson-Thomas quantum dilogarithm identities and their infinite-variable generalizations via motivic characteristic classes.

M. Pe Pereira (Univesidad Complutense de Madrid)

Moderately discontinuous homology

I will introduce a new notion of metric homology for subanalytic germs which is a bilipschitz subanalytic invariant. It applies in particular to analytic germs with both the inner and the outer metric and it gives analytic invariants of the germ. I will give the basic definitions and give some examples. In particulat it keeps the puisseux pairs of a plane branch and some of the known obstructions for being metrically conical. This is a joint work with J. Fernández de Bobadilla, S. Heinze and J.E. Sampaio.

D. van Straten (Johannes Guttenberg Universität Mainz)

The Hamiltonian Normal Form

We describe an iteration leading to a normal form for Hamiltonian systems near a Morse critical point that is suitable for the analysis invariant tori. Under a Bruno condition on the frequency vector the iteration is convergent. We point out some consequences. This is joint work with M. Garay.

Z. Szabó (Princeton University)

Link Floer homology, Thurston norm and bordered algebras

In a recent joint work with Peter Ozsvath, we extended the bordered algebraic approach to Link Floer homology. The lecture will explain some of the new ingredients, and discuss a relationship between these invariants and the Thurston norm.

T. Okuma (Yamagata University)

Normal reduction numbers of normal surface singularities

We discuss the normal reduction number of the local ring of normal surface singularities. This invariant is also defined in terms of cohomology of line bundles on a resolution space. It is known that a normal surface singularity is rational if and only if the normal reduction number is one. In this talk, we show fundamental properties of normal reduction number, and then give an upper-bound in terms of the geometric genus and formulas for some special cases. This is a joint work with Kei-ichi Watanabe and Ken-ichi Yoshida.

W. Veys (KU Leuven)

Which exceptional divisors contribute to jumping numbers?

The multiplier ideals of a hypersurface encode subtle information about its singularities. They induce important discrete invariants of such a singularity, its so-called jumping numbers, including the log canonical threshold. Given a hypersurface singularity (D,0), one can establish a complete list of 'candidate jumping numbers' in terms of an embedded resolution of (D,0). More precisely, every exceptional (prime) divisor of the resolution induces some candidate jumping numbers. However, this list is in general much too large, and it is a challenge to determine the actual jumping numbers in the list. For instance, typically many exceptional divisors induce only false candidates. When D is a curve, Smith-Thompson and Tucker found geometric characterisations of the exceptional divisors that contribute to actual jumping numbers. In particular, the ones that contribute are precisely those that already occur in a certain partial log resolution of (D,0), its so-called log canonical model. We investigate if and how this result generalizes to higher dimensions. This is joint work with Hans Baumers.

J. Wahl (UNC Chapel Hill and Duke University)

Complex surface singularities with rational homology disk smoothings

Consider a complex normal surface singularity (V,0) with a smoothing whose Milnor number is 0, i.e., the Milnor fibre has no rational homology. Such a (V,0) must be a rational singularity, and all cyclic quotient singularities of type p2/(pq-1) (0 < q < p, (p,q) = 1) have a unique such smoothing. In the 1980's, we discovered three triply-infinite and six singly-infinite families of such singularities, all weighted homogeneous. Later work of Stipsicz, Szabó, Bhupal, and myself proved that these were the only weighted homogeneous examples. In his unpublished PhD thesis, our student Jacob Fowler made substantial progress on remaining questions about these examples, such as counting the number of distinct smoothings in each case; calculating the fundamental group of the Milnor fibre (it is finite but can be non-abelian); determining the analytic type when there is a modulus in the resolution graph. We will describe these results as well as some recent progress on a few unsettled issues. We have conjectured that the above are the only surface singularities with rational homology disk smoothings. These questions have analogues in symplectic/contact geometry issues (such as the relation to the existence of symplectic fillings of the links), with related results by Stipsicz and others.