There is a classification of smoothings with rational homology disk Milnor fibres of quasihomogeneous surface singularities. Their fundamental groups have been previously computed for all but three families of such singularities. In joint work with J. Wahl, we compute the fundamental groups in the remaining cases. According to Pinkham's method, the Milnor fibre may be obtained starting from a curve in the projective plane, performing a sequence of blowing-ups and then forgetting some exceptional components. We use this description to compute the fundamental groups in all three cases using the Zariski-van Kampen method. Exactly one of these families has non-abelian fundamental group. In that case, we can find an explicit smoothing of a complete intersection singularity in \(\mathbb{C}^4\) and a representation of that group in \(SL(4,\mathbb{C})\) so that the quotient smoothing has rational homology disk Milnor fibre.

Hirzebruch's L-class is well-known to occupy a central role in the topological classification of high-dimensional manifolds. The Goresky-MacPherson L-class of singular spaces can be assigned a similar role, but much less is known about its transformational properties. The talk will focus on the behavior of this class and related bordism orientation classes under Gysin restrictions associated to normally nonsingular embeddings. Our method involves the theory of mock bundles due to Buoncristiano-Rourke-Sanderson, as well as results jointly obtained with Laures and McClure. We will illustrate the result by calculations on Schubert variety examples. Similar Gysin restriction results can be obtained for Hirzebruch classes defined by Brasselet-Schürmann-Yokura through mixed Hodge modules.

In 1982, T. Yano proposed a conjecture about the generic \(b\)-exponents of an irreducible plane curve singularity. Given any holomorphic function \(f : (\mathbb{C}^2, \boldsymbol{0}) \longrightarrow (\mathbb{C}, 0)\) defining an irreducible plane curve, the conjecture gives an explicit formula for the generic \(b\)-exponents of the singularity in terms of the resolution data of \(f\). In this talk, we will present a proof of Yano's conjecture.

Let \(f: \mathbb{C}^2\to \mathbb{C}\) be a polynomial map. Let \(\mathbb{C}^2 \subset X\) be a compactification of \(\mathbb{C}^2\) where \(X\) is a smooth rational compact surface and such that there exists a map \(\Phi: X \to \mathbb{P}^1\) which extends \(f\). Put \(\mathcal{D}=X \setminus \mathbb{C}^2\); \(\mathcal{D}\) is a curve whose irreducible components are smooth rational compact curves and all its singularities are ordinary double points. The dual graph is a tree. We are interested in this tree, and we want to analyze its shape. At the end we will concentrate on the case where the generic fiber of \(f\) is rational. (Joint work with Daniel Daigle.)

I will discuss the Dolbeault moduli space of Higgs bundles on an algebraic curve over the complex numbers. Dolbeault moduli spaces are one of the ingredients in the Nonabelian Hodge Theory of the curve. Much is known and much is not known. I will focus on my particular point of view, from which I consider the cohomology ring of this space and some of the structures on it.
I will start by introducing the P=W conjecture, also as a motivation for recent joint works in progress with my current student Siqing Zhang, and with Davesh Maulik, Junliang Shen and Siqing Zhang.
The first work provides a cohomological shadow of a (quite non-existing) Nonabelian Hodge Theory for a curve over a finite field. The second applies the picture over a finite field to prove something about the picture over the complex numbers. Amusingly, we then use this result over the complex number to prove one over the finite field.

The classical Kirchhoff polynomial of a graph appears as a term in Feynman integrands, which motivates the study of the projective hypersurfaces they define. These, however, are highly singular and are known to have arbitrarily complicated motives.
I will describe some joint work with Delphine Pol, Mathias Schulze and Uli Walther that looks for order in this apparent chaos. In particular, we give conditions under which a graph hypersurface admits a torus action, which allows us to address a conjecture of Aluffi on the Euler characteristic of the graph hypersurface.

In this talk I will discuss various topological and algebro-geometric consequences of the existence of zeros of global holomorphic 1-forms on smooth projective varieties. Such consequences have been indicated by a plethora of results. I will present some old and new results in this direction. One highlight of the topic is an interesting connection between two sets of such 1-forms, one that arises out of the generic vanishing theory and the other that falls out of Hodge theory of algebraic maps. This is joint work with Feng Hao and Yongqiang Liu (arXiv:2104.07074).

Let X be a topological space homotopy equivalent to a finite CW-complex. The cohomology jump loci are certain subvarieties of the Betti moduli space of X that parametrize local systems satisfying a dimension condition on their cohomology. When X is a complex algebraic manifold, the study of these subvarieties has a rich history and, for local systems of rank one, their geometry is well understood. In this talk, I will present some results on cohomology jump loci (and more general "special sets") in the case when X is no longer assumed to be smooth. I will focus on the cases where X is normal or where X is projective and the mixed Hodge structure on \(H^1(X,\mathbb{Q})\) has no weight zero. If time permits, I will discuss the compatibility of the jump loci with the Hodge theory of completed local rings in the representation variety.

It is a well-known result that a hypersurface \(H\) in a smooth complex variety \(X\) has Du Bois singularities if and only if the pair \((X,H)\) is log canonical (a condition which can be reformulated as saying that the minimal exponent of \(H\) is at least \(1\)). After reviewing some basic facts about the Du Bois complex and the minimal exponent, I will describe some vanishing and non-vanishing results for the cohomology of higher graded pieces in the Du Bois complex when the minimal exponent of \(H\) is larger than \(1\). This is joint work with S. Olano, M. Popa, and J. Witaszek.

An algebraic or complex analytic subset in \(\mathbb{C}^n\) has 2 natural metrics: the outer metric (restriction of the euclidean metric) and the inner metric (natural extension of the riemannian metric on the smooth part). These metrics considered up to bilipschitz mappings are analytic invariants, that is, they do not depend on the complex analytic embedding.
Recently there is an intense activity in bilipschitz geometry of germs and degenerations, enriching and providing finer information on problems that were studied previously from the topological viewpoint (for multiplicity invariance of the germ or certain equisingularity notions).
In the works [1] and [2] we develop a new metric algebraic topology, called the Moderately Discontinuous Homology and Homotopy, in the context of subanalytic germs in \(\mathbb{R}^n\) (with a supplementary metric structure) and more generally of (degenerating) subanalytic families. This theory captures bilipschitz information, or in other words, quasi isometric invariants, and aims to codify, in an algebraic way, part of the bilipschitz geometry.
A subanalytic germ is topologically a cone over its link and the moderately discontinuous theory captures the different speeds, with respect to the distance to the origin, in which the topology of the link collapses towards the origin. Similarly, in a degenerating subanalytic family, it captures the different speeds of collapsing with respect to the family parameter.
The MD algebraic topology satisfies all the analogues of the usual theorems in Algebraic Topology: long exact sequences for the relative case, Mayer Vietoris and Seifert van Kampen for special coverings...
In this talk, I will present the most important concepts in the theory and some results or applications that we got until the present.
[1] (with J. Fernández de Bobadilla, S. Heinze, E. Sampaio) Moderately discontinuous homology. To appear in Comm. Pure App. Math. Available in arXiv: 1910.12552
[2] (with J. Fernández de Bobadilla, S. Heinze) Moderately discontinuous homotopy. Submitted. Available in ArXiv:2007.01538

I will explain the proof of a positivity conjecture of Mihalcea-Singh, concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassmannian \(LG(n,2n)\). Building on the work of Boe and Fu we give a positive recursion for the local Euler obstructions, and use it to show that they provide a positive count of admissible labelings of certain trees, analogous to the ones describing Kazhdan-Lusztig polynomials. Unlike in the case of the Grassmannians in types A and D, where the local Euler obstructions are strictly positive (when they can be), for \(LG(n,2n)\) the obstructions are often zero, and we give a classification for when vanishing occurs. Restricting to the big opposite cell in \(LG(n,2n)\), which is naturally identified with the space of \(n \times n\) symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification. Joint work with Paul LeVan.

We give an introduction and overview about the different functorial theories of characteristic classes of singular spaces in the complex algebraic context. Then we explain the proof of a non-characteristic pullback formula for motivic Chern and Hirzebruch classes of mixed Hodge modules based on Verdier specialization and some micro-local sheaf theory of Kashiwara-Schapira. Applications include an intersection formula for transversal intersections, as well as a proof of a weak form of a conjecture about the agreement of the intersection homology Hirzebruch class (for the parameter \(y = 1\)) with the homology L-class for a complex projective algebraic variety.

I will speak about joint work with Marcelo Aguilar and Aurelio Menegon. We consider a holomorphic map-germ \((C^{n+1},0) \to (C,0)\) with a non-isolated critical point at \(0\), and study how the topology of the fiber \(f^{-1}(t) \cap B_r\) changes as we go from the non-critical levels to the special fiber \(f^{-1}(0) \cap B_r\).

This talk is a survey of some ideas and problems about singularities of Schubert varieties. I will discuss conjectures contained in old joint work with Li Li (Oakland U.), and with Alexander Woo (U. Idaho).

Let \(F=(f_1,\dots,f_p)\) be a tuple of polynomials on \(\mathbb{C}^n\). The multivariate (strong) monodromy conjecture predicts that polar components of the topological zeta of \(F\) are components of the Bernstein-Sato ideal of \(F\). We explain how to construct an auxiliary polynomial \(g\) such that the monodromy conjecture holds for the tuple \((f_1,\dots,f_p,g)\) as well as for the polynomial \(f_1\dots f_p g\). Based on joint work with Nero Budur.

The Denef-Loeser topological and motivic zeta functions are analytic invariants of holomorphic map germs \(f:\mathbb{C}^n\to \mathbb{C}\), which are usually computed from embedded resolutions of \(f\). They codify some information about the topology of the Milnor fiber of the zero locus. More concretely, the Monodromy Conjecture predicts that any pole of these zeta functions is related with an eigenvalue of the monodromy at some point of \(f^{-1}(0)\).
In this talk, we introduce some recent techniques that we have developed for the study of these zeta functions for \(\mathbb{Q}\)-divisors over orbifold varieties: a change of variables formula from relative canonical divisors, as well as a closed formula using compositions of weighted blowing-ups. Finally, we discuss some potential applications of the previous techniques on the study of the Monodromy Conjecture for some surface singularities.
This is a joint work Edwing LEON-CARDENAL (UNAM), Jorge MARTIN-MORALES (UNIZAR-CUD) and Wim VEYS (KU Leuven).