### Program

**2nd Week November 11-15, 2019**

The second week will be a conference on **"Topological and Analytical topics on Singularity Theory".**

**Talks**

Title: **On the quotient of Milnor and Tjurina numbers for surface singularities.****Patricio Almiron,** Universidad Complutense de Madrid**Abstract:** For isolated two-dimensional hypersurface singularities there are three important invariants: the Milnor number, \mu, the geometric genus,p_g, and the Tjurina number, \tau. In this talk, I will focus on the relationships between these three invariants due to classic J. Wahl results. From here I will present a positive answer to Dimca and Greuel's question: Is for any plane curve singularity \mu/\tau. In addition, I will give an intrinsic reason to these bound and I will link this problem with an old conjecture given by Durfee in 1978 about the geometric genus. I will finally discuss the possible extension of this question in the surface case and its relation with Durfee conjecture.

Title: **Bernstein-Sato functional equations, V-filtrations and multiplier ideals of direct summands****Josep Alvarez Montaner,** Universidad Politécnica de Cataluña**Abstract:** V-filtrations is an essential tool in the theory of D-modules over regular rings containing a field of characteristic zero. Its existence is essentially equivalent to the existence of Bernstein-Sato polynomials and, modulo a shift, the V-filtration of the ring along an ideal is the filtration given by the multiplier ideals associated to this ideal. In this talk we will present an extension of the theory of V-filtrations to the case of direct summands of a polynomial ring which are differentiably extensible.

Title: **Torsion divisors of plane curves and Zariski pairs****Enrique Artal Bartolo**, (Universidad de Zaragoza)**Abstract:** The influence of the arithmetic of Picard groups of smooth projective plane curves in the topology is presented. It is a joint work with S. Bannai, T. Shirane and H. Tokunaga.

Title: **Pencils in not Necessarily Normal Surfaces****Gonzalo Barranco Mendoza,** IMATE Cuernavaca**Abstract:** abstract

Title: **Yano's conjecture****Guillem Blanco***Abstract: abstract.pdf*

Title: **On p-adic string amplitudes in the limit p approaches to one****Miriam Bocardo** (Universidad de Guadalajara)**Abstract:** *Talk_Guanajuato_2019*

Title: **Topology of smoothings of non-isolated singularities of complex surfaces****Octave Curmi**, Aix-Marseille Université**Abstract:** *curmi abstract.pdf*

Title: **Embedding codimension of spaces of arcs****Roi Docampo **University of Oklahoma**Abstract:** A fundamental theorem of Drinfeld, Grinberg, and Kazhdan studies the singularities of the arc space of an algebraic variety. It states that the formal neighborhood at a non-degenerate arc decomposes as the product of an infinite-dimensional smooth piece and a finite-dimensional singular piece. In collaboration with Christopher Chiu and Tommaso de Fernex, we revisit this theorem by studying embedding codimensions in spaces of arcs. For a Noetherian local ring, the embedding codimension is the difference between the embedding dimension and the dimension, and can be thought as a measure of the singularities. In order to apply this notion to arc spaces, we extend it to arbitrary local rings and establish several general properties. Our main result for spaces of arcs is a converse of the Drinfeld-Grinberg-Kazhdan theorem: the two results, combined, provide a characterization of the non-degenerate arcs as those defining points on the arc space where the embedding codimension is finite. We also give bounds for these embedding codimensions, effectively controlling the size of the singular factor appearing in the decomposition of the formal neighborhood.

Title: **Mixed Hodge structures on Alexander modules.****Eva Elduque,** University of Michigan**Abstract:** Given an epimorphism from the fundamental group of a smooth quasi-projective variety U onto the integers Z, one naturally obtains an infinite cyclic cover of the variety. The homology groups of this infinite cyclic cover, which are endowed with Z-actions by deck transformations, determine the family of Alexander modules associated to the epimorphism. A folklore conjecture states that these Alexander modules, or rather their torsion parts with respect to the Z-actions, can be equipped with canonical mixed Hodge structures.

We give a positive answer to this conjecture in the case when the epimorphism is the induced map on fundamental groups of an algebraic map f from U into the punctured complex plane. Furthermore, we compare the resulting mixed Hodge structure to the limit mixed Hodge structure on the generic fiber of f. This is based on an ongoing project with C. Geske, L. Maxim, and B. Wang.

Title: **A simplicity criterion for normal isolated singularities****Tommaso de Fernex**, (University of Utah)**Abstract:** The link of an isolated singular point of a complex variety is an analytic invariant of the singularity. It is natural to ask how much information the link carries about the singularity; for instance, the link of a smooth point is a sphere, and one can ask whether the converse is true. Work of Mumford and Brieskorn has shown that this is the case for normal surface singularities but not in higher dimensions. Recently, McLean asked whether more structure on the link may provide a way to characterize smooth points, providing a positive answer in dimension three. In this talk, I will discuss how CR geometry can be used to define a link-theoretic invariant of singularities that distinguishes smooth points in all dimensions. The proof relies on a partial solution to the complex Plateau problem.

Title: **Surfaces with K^2=7 and p_g=4 whose canonical linear system has one base point****Juan Salvador Garza Ledesma**, IM-UNAM**Abstract:** The ultimate goal in the birational classification of minimal regular surfaces of general type with given K^2 and p_g, is to divide them into several strata of moduli and telling exactly how these strata are patched together. It is known, mainly by the work of Nick Shepherd Barron and Christopher Hacon, that the boundary points of the compactification of such strata correspond to surfaces with semi-log-canonical singularities. I will explain some possible applications of graded ring methods to explicitly construct such surfaces and their deformations.

Title: **Topology of the Saito foliation of a complex plane curve**.**Yohann Genzmer**, (Université Paul Sabatier, Toulouse)**Abstract:** The module of vector fields tangent to a given germ of complex plane curve S is free and of rank 2. This result of Kyoji Saito goes back to the 70's and identifies two vector fields naturally associated to S. We will study how the properties of these vector fields provide some informations about the curve and, in particular, the dimension of the generic stratum of its moduli space.

Title: **Differential Signature****Jack Jeffries **(CIMAT)**Abstract:** Many results have shown that properties of singularities measured by resolutions of singularities in characteristic zero have analogues in positive characteristic measured by the Frobenius morphism. One useful invariant in positive characteristic that does not yet have an exact characteristic zero analogue is the F-signature. In this talk, we discuss a new invariant for rings of characteristic zero or p>0 defined in terms of differential operators. This has many of the good properties of F-signature. We will discuss some applications of this invariant to singularities arising in the minimal model program as well as the topology of singularities. This is based on joint work with Holger Brenner and Luis Núñez-Betancourt and joint work with Ilya Smirnov.

Title: **Nash Blow-up in prime characteristic****L. Nuñez Betancourt,** CIMAT

Title: **Positive factorizations of some pseudo-periodic diffeomorphisms****Pablo Portilla,** CIMAT**Abstract: **A classical theorem by Lickorish says that any mapping class of a surface can be expressed as a composition of right-handed Dehn twists and their inverses. A natural question is, which automorphisms can be expressed only as a composition of right-handed Dehn twists? We motivate the importance of this question from different viewpoints: singularity theory, contact topology and mapping class groups. Then we will present some techniques within the theory of mapping class groups as well as an interplay between singularity theory and mapping class group that has implications in both fields.

Title: **The geometric genus and Seiberg--Witten invariant of Newton nondegenerate Weil divisors in a toric space****Baldur Sigurðsson,** UNAM, Cuernavaca**Abstract:** We define Newton nondegenerate Weil divisors in toric varieties and present a formula for the geometric genus in terms of the Newton diagram. In the case of normal Gorenstein surface singularity with rational homology sphere we show that the geometric genus can be obtained by a computation sequence, showing that the geometric genus is in this case a topological invariant.

We also present some unfinished work to prove a similar formula for the Seiberg--Witten invariant in this case, aimed at proving the Seiberg--Witten invariant conjecture of Némethi and Nicolaescu.

Title: **Hyperplane sections, polar curves and modifications in normal surfaces.****Jawad Snoussi,** IMATE Cuernavaca**Abstract: **Hyperplane sections and polar curves of a normal surface carry information on local geometry and topology of the singularity. They are related to the tangent cone and the limits of tangent spaces at the singular point, and naturally to the blow-up of the point and the Nash Modification. We will explain these concepts and relate them to some results comparing the two mentioned modifications

**Coloquium**

Title: **Zeta functions and the monodromy conjecture.****Wim Veys,** KU Leuven

**1st Week November 4-8, 2019**

**Courses**

Title: **Weighted projective planes and weighted Lê-Yomdine singularities.****Enrique Artal Bartolo** (Universidad de Zaragoza)**Abstract:** In this course, weighted projective planes are studied as surfaces with normal singularities. Some special curves will be studied using a family of Cremona transformations, leading to the concept of Zariski pairs (curves with the same "combinatorics" and distinct topology). With the concept of weighted blow-ups the notion of weighted Lê-Yomdine singularities will be introduced.

Title: **Geometric monodromy via tête-à-tête graphs****Pablo Portilla** (CIMAT)**Abstract:** In 2010, N. A'Campo introduced tête-à-tête graphs with the motivation of modeling the geometric monodromy of isolated plane curve singularities. These graphs are spines of surfaces with non-empty boundary

equipped with a metric that satisfies a particular property. In this minicourse, we will give the basic definitions of tête-à-tête graphs and we will see how they model mapping classes of topological surfaces. We will answer natural

questions such as, which mapping classes can be realized by a tête-à-tête graphs? are they enough to fulfill A'Campo's original motivation? We will see that it is not the case which motivates the introduction of mixed

tête-à-tête graphs that allow to model a bigger class of surface automorphisms. In particular, we will see that this class coincides with that of monodromies associated with reduced germs defined on isolated complex surface

singularities.

The results mentioned in this course are a joint work with Javier Fernández de Bobadilla, María Pe Pereira and Baldur Sigurdsson.

Title: **Hodge theory of matroids****Botong Wang** (University of Wisconsin, Madison)**Abstract:** The goal of these lectures is to give an introduction to matroid theory, and to present some recent progress using ideas from algebraic geometry, more specifically, Hodge theory. The first result is the log-concavity conjecture by Heron-Rota-Welsh. This is proved by Adiprasito, Huh and Katz by relating the log-concavity to the Hodge-Riemann relations. The second result is the “top-heavy” conjecture by Dowling-Wilson. For a realizable matroid, the conjecture is proved using hard Lefschetz theorem. For an arbitrary matroid, this is an on-going joint work with Braden, Huh, Matherne and Proudfoot.

Title: **Strings, Local Zeta Functions, Etcetera****Wilson Zuñiga Galindo** (Cinvestav, Querétaro)**Abstract:** *AbstractTalkWZG.pdf*

**Coloquium:**

Title: **Ultrametricidad en Física****Wilson Zuñiga Galindo** (Cinvestav, Querétaro)**Abstract:** *AbstractColWZG.pdf*

**Talks and Pretalks**

(Pretalk) **The Tjurina number for plane curve singularities****Patricio Almiron,** Universidad Complutense de Madrid**Abstract:** On this talk I will review some basic definitions and facts about deformations of plane curve singularities. I will mainly focus on different interpretations of the dimension of the miniversal deformation of a plane curve singularity. From this, we are able to provide a closed formula for the minimal Tjurina number in an equisingularity class for any germ of irreducible plane curve singularity by using the results of Wall and Genzmer.

-(1st pretalk) **Periods of integrals in the Milnor fiber****Guillem Blanco**, (Universidad Politécnica de Cataluña) *Abstract: abstract1.pdf*

-(2nd pretalk) M**ultivalued forms on P1 and cohomology with local coefficients****Guillem Blanco**, (Universidad Politécnica de Cataluña) *Abstract: abstract2.pdf*

-(3rd pretalk) **The monomial curve and its deformations****Guillem Blanco**, (Universidad Politécnica de Cataluña) *Abstract: abstract3.pdf*

-(1st & 2nd pretalks) **Topology of the Saito foliation of a complex plane curve****Yohann Genzmer**, (Université Paul Sabatier, Toulouse)

Title: **F-volumes****Sandra Lisett Rodríguez Villalobos** UNAM)**Abstract:** In this talk, we will talk about the definition and properties of the F-volume of a sequence of ideals. This is a generalization of the F-threshold of an ideal. In particular, we will discuss the relation of this invariant with singularities and the Hilbert-Kunz multiplicity. This is joint work with Wágner Badilla-Céspedes and Luis Núñez-Betancourt.

Title: **Asymptotic expansion for higher bilinear forms on the Milnor algebra induced by Grothendieck duality****Miguel Angel de la Rosa**, (Universidad de Tabasco)

Title:**Configurations of points and new Zariski pairs of line arrangements****Juan Viu-Sos,** IMPA**Abstract:** A couple of combinatorially equivalent line arrangements with different topology is called a Zariski pair. In the last decades, only three Zariski pairs were known, all of them defined over non-trivial number fields, and each of them distinguished by using computer calculations at some step.

In this talk, we present a method to distinguish Zariski pairs admitting real equations, consisting on the weight counting of points of the dual configuration over the real projective space. From this, we construct "by hand" a new Zariski pair of 13 lines, exhibiting interesting both topological and arithmetical properties, as well as other 9 new examples.

Title: **On the log canonical threshold and numerical data for plane curve singularities.****Wim Veys,** KU Leuven**Abstract: **We show some properties of numerical data of an embedded resolution of singularities for plane curves, which are inspired by a conjecture of Nguyen and motivated by a conjecture of Igusa on exponential sums. This talk is also propaganda for the language of Eisenbud-Neumann diagrams

Title: **Strings and Local Zeta Functions****Wilson Zúñiga Galindo**, (Cinvestav, Querétaro)