In 1949 Fano published his last paper on 3-folds with canonical sectional curves (Rendiconti Accademia dei Lincei). There he constructed and described a 3-fold of the type \(X^{22}_3 \subset \mathbb{P}^{13}\) with canonical curve section, which we like to call Fano's last Fano. Together with Roberto Pignatelli we re-wrote Fano's ingenious construction, providing various (in our opinion missing) proofs, in modern language, trying to use results and techniques available at that time. Then we construct Fano's with modern tools, in particular via the Hilbert scheme of zero cycles on a rational surface; as a consequence we easily point out the corresponding example in the Mori-Mukai classification. It is easy to foreseen future generalizations of this construction.

We discuss several applications of (vanishing of) Koszul modules related to vector bundles or Gauss-Wahl maps to algebraic geometry. Based on joint works with G. Farkas, S. Papadima, C. Raicu, A. Suciu, J. Weyman.

Topological and algebraic properties of projective plane curves or line arrangements fixed by a finite group can be studied on the quotient of the projective plane by the action of the group, seen as an orbifold. More simple versions of the fundamental group can be computed, and also braid monodromy.

Using George Bergman's advice "what every ring theorist should know" and a 101 years old theorem, a complete presentation of the monoid \((\mathbb{C}[X], \circ)\) is given. Looking for nice projections between unordered configuration spaces, we will see why equilateral triangles should be removed.

We investigate the Hodge filtration on the localization of complex affine space along a hyperplane arrangement. We focus on its generation level: the first piece of the Hodge filtration that completely determines the rest. While natively a D-module theoretic problem, we instead employ birational techniques recently developed by Mustata and Popa. For generic arrangements we show that the generating level is zero and so the multiplier ideal completely determines the Hodge filtration. In the case of Q-divisors, where divisor = generic arrangement, we give a (easy) combinatorial formula for the generating level. This is a first batch of results from an ongoing project.

Contact loci are sets of arcs on a smooth variety with prescribed contact order along a fixed hypersurface. They appear in motivic integration, where motivic zeta functions are generating series for classes of contact loci in appropriate Grothendieck groups. We give an overview of recent results on the topology of contact loci, relating their cohomology with the Floer cohomology of monodromy iterates, and their irreducible components with dlt models.

Topological complexity and recent generalizations are invariants motivated by the motion planning problem from robotics. I will discuss one of these generalizations, parameterized topological complexity, in the context of bundles involving fiber-type hyperplane arrangements.

Cohomology jump loci and resonance varieties provide refined invariants of groups and spaces as well as insight into finiteness properties. These have been long-running themes in Alex Suciu's work, and I will survey some of his central contributions to the area.

Let \( f : U \to \mathbb{C}^*\) be an algebraic map from a smooth complex connected algebraic variety \(U\) to the punctured complex line \( \mathbb{C}^*\). Using \(f\) to pull back the exponential map \( \mathbb{C} \to \mathbb{C}^*\), one obtains an infinite cyclic cover \(U^f\) of the variety \(U\). The homology groups of this infinite cyclic cover, which are endowed with \(\mathbb{Z}\)-actions by deck transformations, determine the family of Alexander modules associated to the map \(f\). In previous work jointly with Geske, Herradon Cueto, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of Alexander modules. In this talk, we will talk about work in progress aimed at generalizing this theory to abelian covering spaces of algebraic varieties which arise in an algebraic way, i.e. from maps \(f:U\to G\), where \(G\) is a semiabelian variety. Joint work with Moises Herradon Cueto.

Inspired by developments in the theory of resonance varieties we consider the following question: Given a space X and vector bundles E and F together with a map \(f:Sym^2(E)\to F\), what is the virtual class of the locus of those points in X for which the kernel of f contains a symmetric tensor of prescribed rank? The answer, given in terms of the Chern classes of E and F turns out to have a remarkable number of applications to moduli theory. Based on joint work with Rimányi respectively Jensen and Payne.

We survey known results on the question whether two conjugacy classes of homomorphisms of a surface group onto some group G are equivalent by the action of the mapping class group. We show that there exist infinitely many finite simple nonabelian groups G for which there are at least two equivalence classes.

I will discuss the first order theory of homeomorphism groups of manifolds. I will show how the internal logic of the homeomorphism group of a manifold determines the various homotopy functors of a manifold, including the fundamental group and homology, as well as the various properties of these functors, such as residual finiteness of the fundamental group.

Let \(X\) be a real analytic manifold. A function \(f \colon X \to \mathbb{R}\) is said to be curve-analytic if it is real analytic when restricted to any locally irreducible real analytic curve in \(X\). We prove that every curve-analytic function with subanalytic graph is actually real analytic. To accomplish this task, we give a criterion for an arc-analytic function to be real analytic. A function is called arc-analytic if it is real analytic along any parametrized real analytic arc. We also obtain analogous results for Nash manifolds and Nash functions, in which case the assumption of subanalyticity is superfluous. Joint work with W. Kucharz.

We give a characterization of plus-one generated (POG) arrangements (of lines in the complex projective plane) in terms of Ziegler restrictions and we show that the possible splitting types onto a projective line for the bundle of logarithmic vector fields associated to a POG arrangement are determined by the exponents of the arrangement. Based on joint work with Takuro Abe and Denis Ibadula.

We fix an isolated curve singularity. The construction of the lattice homology associated with such a curve starts with the identification of certain cubical spaces \(\{S_n\}_n:\) their homology determines the lattice homology. We will introduce a natural filtration of these spaces, they will induce spectral sequences for every \(S_n\). Each page of the spectral sequence provides a set of new invariants of the curve. The spectral sequence converges to the lattice homology, whose Euler characteristic is the delta invariant. The multivariable Poincaré series of the first page can be identified with the multivariable motivic Poincaré series. In the case of plane curve singularities (when the embedded link determines its link Floer homology as well) from the first page one can read this link Floer homology as well.

In this talk I will present some of the major contributions of Laurentiu Păunescu to Singularity Theory.

Koszul modules were originally introduced by Papadima and Suciu as topological invariants of groups, but more recently a more algebraic theory has emerged, which has resulted in several interesting consequences to the study of algebraic varieties and that of finitely generated groups. The goal of my talk is to introduce the basic theory of Koszul modules and their resonance varieties, and to discuss a number of fundamental invariants and some open problems related to them. The talk is based on projects with M. Aprodu, G. Farkas, S. Papadima, A. Suciu, K. VandeBogert, J. Weyman.

AA well-known topological problem linking the Theory of Singularities to that of Hyperplane Arrangements is the so-called \(K(\pi,1)\)-conjecture for Artin groups, proved in the 70s by Deligne in the spherical case. The introduction of the so-called "dual structure" was an important tool for [Paolini-Salvetti, Proof of the \(K(\pi,1)\)-conjecture for affine Artin groups, Inv. Math., 2021]. Exposing this approach and possible generalizations, we show that it works for all rank three groups, deducing some standard properties on their structure.

In this talk, I will prove Fukui-Kurdyka-Paunescu's Conjecture, which says that subanalytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. This is joint work with Alexandre Fernandes and Zbigniew Jelonek.

Using equivariant Todd (resp. Hirzebruch) classes of toric varieties, we translate the equivariant Hirzebruch-Riemann-Roch theorem for simplicial projective toric varieties into various (weighted) Euler-Maclaurin type formulae for simple lattice polytopes. This is joint work with S. Cappell, L. Maxim and J. Shaneson.

Given an immersed hypersurface \(M\) and a point \(x \in \mathbb{R}^n\), what can be said about the number of normals trough \(x\) ? We show that under mild conditions this number exceeds the sum of the Betti numbers with at least 2 or 4. This follows from a study of the geometry and singularities of the focal surface of \(M\) and the Morse theory of the quadratic distance function. This is related to the classical conjecture that for any convex body \(K \subset \mathbb{R}^n\) there exists a point in the interior of \(K\) which belongs to at least 2n normals. The conjecture is known to be true for \(n = 2, 3, 4\). This was recently reviewed by work of Martinez-Maure and Grebennikov \(\&\) Panina. (A reference for the talk is: Gaiane Panina, Dirk Siersma: "Concurrent normals of immersed manifolds", Communications in Mathematics 31 (2023), no. 3, 1-8.)