Program

Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots of the system in the complex n-torus is known to be the normalized volume of the convex hull of A (the BKK bound). Together with Frédéric Bihan and Jens Forsgård, we explore the following question: if the cardinality of A equals n+m+1, what is the maximum local intersection multiplicity at one point in the torus in terms of n and m? This study was initiated by Andrei Gabrielov in the multivariate case. We give an upper bound based on the computation of covolumes that is always sharp when m=1 and, under a generic technical hypothesis, it is considerably smaller for any dimension n and codimension m. We also present, for any value of n and m, a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions.

A germ at the origin of a closed subset set X of Rⁿ inherits two metrics from the ambient space: the inner metric, the distance between two points in X defined as the length of a shortest path in X connecting these points, and the outer metric, the distance in Rⁿ between the two points. A germ is Lipschitz Normally Embedded (LNE) if these two metrics are equivalent. Germs X and Y are inner (outer) Lipschitz equivalent if there is a homeomorphism X→Y bi-Lipschitz with respect to the inner (outer) metric. A surface germ is a closed germ of dimension 2 definable in a polynomially bounded o-minimal structure over R, e.g., semialgebraic or subanalytic. Although inner Lipschitz classification of surface germs was done by Lev Birbrair in 1999, outer Lipschitz classification is still an open problem. Even the simplest non-LNE surface germs may exhibit surprisingly complex outer Lipschitz geometry. I am going to review recent progress towards outer Lipschitz classification of definable surface germs, most of it joint work with Lev Birbrair. 

The World Singularity Theory Community lost two extremely important members: Maria Aparecisa Ruas and Le Dung Trang. I would like to dedicate the lecture to the memory of two of them and I am going to describe some results obtained by them together with the author. I will concentrate on the works on contact equivalence of functions and on the topological and Lipschitz Regularity Theorems. 

Hodge theory is a powerful tool for studying algebraic varieties, and it is particularly useful for proving vanishing results and for constructing moduli spaces. It is, however, highly sensitive to the presence of singularities: even very mild singularities can lead to substantial changes in the Hodge structure. This motivates the problem of quantifying (especially in a qualitative, structural sense) the effect of singularities on the Hodge-theoretic properties of individual varieties or of families.

The classical notions of mild singularities, rational and Du Bois singularities, reflect situations in which the “frontier” of the Hodge diamond behaves well. More recently, these notions have been refined to probe deeper Hodge levels, leading to the theory of higher Du Bois and higher rational singularities, developed in work of Mustata–Popa, M. Saito, Friedman–Laza, and collaborators.

In this talk, I will give an overview of these developments and discuss applications to local vanishing results and to the moduli theory of Calabi–Yau varieties.

Let (X, x) be a normal surface singularity and denote by L its link. The first complete classification of the finite dimensional representations of the fundamental group of L was done by McKay in the case of rational double point singularities. Later, Artin and Verdier, reformulate the McKay correspondence in a more geometrical setting. Their correspondence gives a complete classification of the indecomposable reflexive modules. In the case of quotient surface singularities, Esnault classified all the reflexive modules of rank one. Moreover, Esnault proved that quotient surface singularities are the only surface singularities with a finite number of indecomposable reflexive modules, such singularities are called Cohen–Macaulay finite representation type.

In this talk, we classify all the reflexive modules on quotient surface singularities. For this, we will use the Atiyah-Patodi-Singer theorem and the theory of secondary characteristic classes to construct our classification. As a consequence, the classification problem of reflexive modules over surface singularities of Cohen–Macaulay finite representation type is completely finished.

Joint work with José Antonio Arciniega-Nevárez and José Luis Cisneros-Molina.

Resolution of singularities in characteristic zero was established by Hironaka, whose approach relies on a sequence of blowups along carefully chosen centers. Although effective, this method involves a variety of non-canonical choices. In order to obtain a canonical procedure, Nash proposed the blowup that now bears his name, raising the question of whether its iteration suffices to resolve all singularities. In joint work with Federico Castillo, Daniel Duarte, and Maximiliano Leyton-Álvarez, we construct toric counterexamples showing that in dimensions at least four the iterated Nash blowup does not lead to a resolution. I will discuss these counterexamples and place them in the broader context of the resolution problem.

We study Calabi–Yau hypersurfaces X cut out by a general anticanonical divisor in a smooth projective toric Fano variety P_Δ associated with a smooth Fano polytope Δ. Building on the description of Cox rings of embedded varieties developed by Herrera–Laface–Ugaglia and on Batyrev’s classification of smooth toric Fano fourfolds in terms of Fano polytopes, we obtain criteria for finite generation of the Cox ring R(X) and for its birational geometry.

For dimensions n = 3 or 4 we prove a dichotomy: if X is a smooth general anticanonical hypersurface in P_Δ, then either X is a Mori dream space, or Δ contains four vertices v1, v2, v3, v4 satisfying

(1) v1 + v2 = v3 + v4 = 0, or

(2) v1 + v2 = 0 and v3 + v4 = v1,

in which case the birational automorphism group Bir(X) is infinite. In many cases covered by the first alternative we compute R(X) explicitly: under mild combinatorial hypotheses on the vertices of Δ we give explicit presentations of R(X) by generators and relations. We also introduce a surface-based criterion to detect when a rational polyhedral cone coincides with Eff(X); combined with McKernan’s result that a Calabi–Yau threefold is a Mori dream space if and only if its effective cone is rational polyhedral, this allows us to handle the remaining cases not covered by the explicit Cox ring computations.

This is joint work with Michela Artebani and Luca Ugaglia.

A reflection mapping is a singular holomorphic mapping obtained by restricting the quotient mapping of a complex reflection group. We study the analytic structure of double point spaces of reflection mappings. In the case where the image is a hypersurface, we obtain explicit equations for the double point space and for the image as well. In the case of surfaces in $\C^3$, this gives a very efficient method to compute the Milnor number and delta invariant of the double point curve.

Joint work with J. R. Borges-Zampiva, G. Peñafort-Sanchis and J. N. Tomazella.

In this talk, we investigate strongly bi-Lipschitz trivial families and 2-dimensional sections germs in quasi-homogeneous complex sets with isolated singularity and with certain regular properties. As an application, we determine completely which Pham-Brieskorn hypersurface germs are inner metrically conical and which ones are Lipschitz Normally Embedded. This is a joint work with Maria Ruas and Edson Sampaio.

This is a joint work with P. González Pérez, M. González Villa, and E. León Cardenal, and we use the concepts introduced in the talk of M. González Villa.

There are two outstanding conjectures for the Igusa, topological and motivic zeta functions (the monodromy and holomorphy conjectures) which are still open. Using  the contents of the González-Villa's talk we present the proofs of these conjectures for Lê-Yomdin singularities of surfaces. For suspensions of curves the monodromy conjecture  is quite trivial and most cases of the holomorphy conjecture have been settled by Denef and Veys. We have completed the proof of the remaining casesthanks to the formulas relating the different twisted topological zeta functions of the curves and the suspensions.

A normal 2-dimensional singularity that admits a smoothing whose Milnor fiber is a rational homology disc is called QHD singularity. Examples of them are Wahl singularities 1/n^2(1,na-1) where gdc(n,a)=1. It is conjectured by Jonathan Wahl that QHD singularities must be weighted homogeneous. Bhupal and Stipsicz classified all of them. Most of them are not even log canonical. Degenerations of nonsingular projective surfaces into a surface with only QHD singularities are the mildest possible normal degenerations. Kawamata classified these degenerations for Wahl singularities, which he called moderate degenerations. I will describe a joint ongoing work with Marcos Canedo (UC Chile) which classifies all QHD degenerations of elliptic surfaces. As a byproduct we recover all the non-normal degenerations of Dolgachev surfaces recently found by Donggun Lee and Yongnam Lee in https://www.arxiv.org/abs/2509.07467.

I will introduce the Nash problem (and its generalized version), which concerns a correspondence between resolutions of singularities of a variety X and the arc space of X. After giving some historical context, I will present results on these problems for certain d-dimensional varieties equipped with a torus action of dimension d−1. I will then discuss how these results contribute to the broader understanding of the problem. (This is joint work with David Bourqui and Kevin Langlois.)

I will present a formalism to pullback Cartier modules and their singularity theory along regular morphisms. As an application, I will present a generalization of a result of Felipe Pérez regarding the discreteness of F-jumping numbers of mixed test modules and that their constancy regions are p-fractals.

An important problem in enumerative geometry is counting rational curves that interpolate a configuration of points in P^2, leading to Gromov-Witten invariants (over algebraically closed fields) and Welschinger invariants (over the real numbers). Recently, Kass, Levine, Solomon, and Wickelgren constructed "quadratic" invariants that work over an (almost) arbitrary base field. The “inconvenience” is that these new invariants are no longer numbers, but quadratic forms whose rank and signature recover the previously mentioned invariants. In a current work with Erwan Brugallé and Kirsten Wickelgren, we study these invariants in the framework of so-called Witt-invariants and show that, conversely, the quadratic invariants can be recovered from Gromov-Witten and Welschinger invariants. In my talk, I want to give an introduction to this topic.

We present a theory of Bernstein-Sato polynomials for meromorphic functions and we relate them to poles of Archimedean local zeta functions and jumping numbers of multiplier ideals. Joint work with Manuel González-Villa, Edwin León-Cardenal and Luis Núñez-Betancourt.

This is a joint work with E. Artal, P. González Pérez, and E. León Cardenal.

We present far reaching generalizations of formulas for the topological zeta functions of suspensions of $n$-dimensional hypersurfaces by 2 points due Artal, Cassou-Noguès, Luengo and Melle. These results are generalized to the motivic level and for arbitrary suspensions, by using a stratification principle and classical techniques of generating functions. The same strategy is also used to obtain formulas for the motivic zeta functions of some families of non-isolated singularities related to the resolution of superisolated, and Lê-Yomdin singularities.  The Euler characteristic specialization of the new formulas relate the topological zeta functions of the suspension with the (twisted) topological zeta functions of the original $n$-dimensional hypersurface via the Jordan's totient function. These formulas are a key tool for the results which will be presented in Artal's talk.

We work with a 1-parameter family σ : X → ∆ of isolated hypersurface singularities of fibre dimension 2. We show that if the Milnor number is constant, then any semistable model, obtained from σ after a sufficiently large base change must satisfy nontrivial restrictions. Those restrictions are in terms of the dual complex, Hodge structure and numerical invariants of the central fibre. To achieve this, we make use of the Steenbrink spectral sequence degenerating to the cohomology of a generic fibre of the resolution, endowed with the limit mixed Hodge structure.

The study of singular complex analytic hypersurfaces in the local context, i.e., those defined by a germ of a holomorphic function, dates back to the fundamental work of Milnor. He proved a fibration theorem which established the existence of the associated Milnor fiber (local). For the case where the hypersurface possesses an isolated singularity, Milnor introduced an important invariant: the Milnor number. This number plays a foundational role in the theory of Singularities.

In an extension of this theory, Massey introduced the Lê numbers in the setting of non-isolated singularities. These numbers have support on the singular set and, in the case of an isolated singularity, they coincide with the Milnor number.

Moving in a different direction, Bruce and Roberts extended the definition of the Milnor number of a function germ to its restriction to a variety germ, called the Bruce-Roberts number of germ with respect to variety.

Concerning germs of mappings, specifically for the case of an Isolated Complete Intersection Singularity (ICIS), Hamm proved that an analogous invariant is also well-defined. Hamm's Milnor number is defined as the degree of the middle homology of the corresponding fiber of an associated fibration to the germ.

Along this line of research, other numbers will be presented , such as the Segre numbers, for which explicit formulas exist for specific families of mappings. Furthermore, an extension of the Bruce-Roberts number to the context of germs of mappings restricted to a variety will be introduced, demonstrating, in certain cases, a relationship with Hamm's Milnor number. To conclude, potential avenues for future investigation will be highlighted.

This talk concerns recent progress toward understanding singularities of complex hypersurfaces through the valuative behaviour of its defining polynomial. Fix a polynomial f in C[x1,...,xn] and consider the variety Vf contained in A^n_C with a possible singularity at the origin. Working over the algebraic closure K of C((t)), we study the function x ↦ v(f(x)) on the infinitesimal neighbourhood m_K^n, with the aim of extracting canonical invariants of the singularity that do not depend on coordinate choices.

The central idea is to measure how v(f(x)) varies as x approaches the various strata of the singular locus of Vf(K). For this, we introduce a family of functions

lambda_0, ..., lambda_{n-1} : K^n \ Vf(K) → Q,

where lambda_d(x) encodes the minimal radius of a ball around x on which the function v ∘ f becomes "(≤ d)-riso-trivial", a notion of local triviality defined using risometries, a non-Archimedean enhancement of isometries. These lambda_d behave like valuative distances to successively worse singular strata, even though such strata cannot, in general, be canonically chosen.

Our main result is a piecewise linearity theorem: the set

Delta_0 = { (lambda_0(x), ..., lambda_{n-1}(x), v(f(x))) | x in m_K^n \ Vf(K) }

is a finite union of graphs of linear functions whose domains are semi-linear subsets of Q^n. Thus each hypersurface gives rise to a canonical, combinatorial, coordinate-independent object in Q^{n+1}. Although reminiscent of a tropicalization, this object is different in nature. It provides a new invariant for singularities, defined in terms of non-archimedean geometry plus a bit of model theory of valued fields.

This is joint work with Immanuel Halupczok and David Bradley-Williams.

Calabi-Yau varieties are objects of great interest in algebraic geometry and also play a significant role in string theory. Their importance is due, among other aspects, to the possibility of constructing pairs of families of varieties that are symmetric under mirror symmetry. When these varieties arise as hypersurfaces in toric Fano varieties, classical results by Batyrev, Berglund-Hübsch-Krawitz, among others, provide constructions of families of Calabi–Yau varieties that are mirror to each other, using the combinatorial features of the underlying varieties. In this talk, we will explain how this theory can be extended to complete intersections, yielding a more general construction that produces new examples of dual families of Calabi-Yau varieties. This is joint work with Michela Artebani (Universidad de Concepción) and Robin Guilbot (Université de Toulouse).