We investigate sufficient conditions for the relative topological and Lipschitz triviality of analytic deformations of polynomial map germs. Motivated by the relative Kuiper-Kuo condition and Parusinski's inequality, we introduce new inequality criteria and establish structural theorems governing the triviality of such deformations. Furthermore, we demonstrate that invariance of the integral closure of Jacobian ideals also implies relative triviality.

In this talk, we discuss recent results on global invertibility problems for semialgebraic local diffeomorphisms. We focus on the non-properness set S_F, which plays an important role in questions related to the Jacobian Conjecture and Jelonek's Conjecture. First, we present criteria for bijectivity of nonsingular semialgebraic maps in terms of the surjectivity of certain linear differential operators associated with the map. Next, we discuss topological results on the non-properness set of a semialgebraic local diffeomorphism.

We consider a definable map-germ \( (f,g) :(\mathbb{R}^n,0) \rightarrow (\mathbb{R}^2,0) \). Under some conditions on \( f \) and \( g \), we establish Lê-Greuel type formulas, namely we relate the following quantities: $$\chi \left(\{f=\alpha\} \cap\{g=\delta\} \cap B_\varepsilon^n \right),$$ $$\chi \left(\{f=\alpha\} \cap\{g\ge \delta\} \cap B_\varepsilon^n \right)-\chi \left (\{f=\alpha\} \cap\{g\le \delta\} \cap B_\varepsilon^n \right),$$ where \( (\alpha,\delta)\) is a regular value of \( (f,g)\) and \( 0< | (\alpha,\delta) | \ll \varepsilon \ll 1 \), to Poincaré-Hopf indices of appropriate vector fields. Our method is based on Lagrange multipliers and some polar techniques. We provide a large class of weighted-homogeneous mappings for which our conditions are satisfied. Joint work with Juan Antonio Moya Pérez (University of Valencia).

We discuss the notion of properness of a polynomial map \( f: \mathbb{C}^n \to \mathbb{C}^n \) at a point of the target. We present a method to describe the set of non-proper points of \( f \) with respect to Newton polyhedra of \( f \) . We obtain an explicit precise description of such a set of \( f \) when \( f \) satisfies certain non-degeneracy condition.

The analytic lattice cohomology of a reduced curve germ categorifies its delta invariant and captures information about other invariants, including the Heegaard Floer link homology and the Cohen-Macaulay type. We show that these cohomology groups can be computed from filtrations on an appropriate collection of nested punctual Hilbert schemes; this allows us to define a theory of analytic lattice cohomology for finite maps between arbitrary-dimensional complex-analytic multigerms such that the lattice cohomology of a curve germ is retrieved as the cohomology of its normalization map. Based on joint work with András Némethi.

Optimizing distance from a data point (possibly coming from some measurements) towards an algebraic variety \( X \) (possibly representing some ideal model) is an important problem in applications. In the process of solving this problem using the tools of algebraic geometry, we have to make choices (either of the metric or the variety, etc.), whether we work over the reals or over the complex. These choices lead to several problems, including complexification, isotropicity, realification, and non-differentiability. A part of these problems manifests in discriminants, and they become computable, measurable, and detectable using the coefficients of the so-called "(Euclidean) Distance polynomial". Based on joint work with Luca Sodomaco, we will present recent results on these coefficients and their effectiveness in dealing with these problems.

Local zeta functions offer an arithmetic perspective on singularities, with their poles capturing refined geometric information. Igusa local zeta function provide a powerful bridge between arithmetic and the geometry of singularities. Its poles encode subtle invariants of the underlying hypersurfaces. In this talk, we focus on Igusa zeta function attached to a class of polynomials which arise naturally in the study of resolution of singularities in positive characteristic, highlighting the differences with the (zero) classical setting.

We show that the number of bifurcation points at infinity of a polynomial function \( f:\mathbb{C}^2\to \mathbb{C} \) is at most the number of branches at infinity of a generic fiber of \( f \) and that this upper bound can be diminished by one in certain cases.

We will present a series of results concerning the detection of the bifurcation locus of a polynomial map, results that we have obtained jointly with Mihai Tibăr over the past several years.

In the case of definable (in an o-minimal structure) maps with values in \( \mathbb{R}^k, k>1 \) an analogue of K-L inequality was proven recently. Contrary to the case \( k=1 \), in general, a desingularizing function \( \psi \) does not extend continously to the boundary of the domain of \( \psi \). In the subanalytic case we expect that this phenomenon can not happen for maps satisfying Thom's \( a_f \) condition. Ongoing project with M. Bilski and L. Păunescu.

We describe a classification of reducible curves with components in very ample linear systems with small degree of discriminant for which the fundamental groups of the complement admit a free quotient of a rank larger than a certain constant depending on the surface.

I will explain how various facets of singularity theory can be used to understand the algebraic optimization degrees appearing in nearest point problems. (Based on joint work with J. Rodriguez, M. Tibăr and B. Wang.)

A polynomial map \( f: \mathbb{C}^{n+1} \to \mathbb{C} \) is known to be a topologically locally trivial fibration outside the preimage of a finite set of bifurcation values. The topology of its fibers is influenced both by affine singularities and by the singularities of \( f \) "at infinity". We will survey classical results and present new ones concerning the topology of general and atypical fibers, with particular emphasis on the role of singularities at infinity. Joint work with Laurentiu Maxim.

This talk presents a Newton polygon interpretation of the higher bi-Lipschitz invariants for plane holomorphic function germs. The main result shows that a natural augmented relative Newton polygon attached to each gradient canyon is a bi-Lipschitz right-equivalence invariant: its intermediate edges encode the second-level Henry-Parusinski-type invariants, while its top edge recovers the polar multiplicity of the canyon. (Joint with Laurențiu Păunescu and Mihai Tibăr.)

We compare the topological Milnor fibration and the motivic Milnor fibre of a regular complex function. We present two equivalent constructions: the first extends the classical Kato-Nakayama log-space, while the second is a version of the real oriented deformation to the normal cone. We recover A'Campo's model of the topological Milnor fibration by taking the quotient of the motivic Milnor fibration by suitable powers of \( R > 0 \) and showing that this determines the classical motivic Milnor fibre.
We also provide precise formulas that express how the motivic and logarithmic Milnor fibrations change when blowing up. As an application, we demonstrate that the motivic Milnor fibre is defined as an element of a suitable Grothendieck ring, without the requirement that the Lefschetz motive be invertible. This is joint work with J.-B. Campesato and G. Fichou.

We find new bi-Lipschitz invariants for holomorphic functions of two complex variables. Our new invariants can distinguish function germs that are otherwise indistinguishable by previous methods, as illustrated by the examples given in Section 4 of our paper. (Joint with Piotr Migus and Mihai Tibăr.)

There is long history in the relation between the critical points of the distance function and concurrent normals to a submanifold in Euclidean space. The study of caustics and counting the number of normals play a important role. In this talk we will give a general approach to the study of critical points of the distance function to a PL submanifold X. Examples are: polygons in the plane and in space and polygonal surfaces in 3-space (not necessarily convex), etc. What is the relation between normals and critical points ? Are generic singularities Morse and if so what is the index ? We will discuss the bifurcation set and show that for a knotted closed PL-curve there are at least 10 concurrent normals. Also for a convex simple polytope there is a point at least 10 concurrent normals. What can be said about the ED-degree?

Homogeneous hypersurfaces with non-isolated singularities provide a natural setting in which global Milnor fibrations encode subtle topological information. A particularly rich and tractable class arises from complex hyperplane arrangements, where the interplay between singularity theory, combinatorics, and algebraic topology can be studied in a concrete way.
In this talk, I will focus on the Milnor fiber \( F(\mathcal{A}) \) of a complex hyperplane arrangement \( \mathcal{A} \), and on the question of its 1-formality in the sense of rational homotopy theory. I will describe a combinatorial dichotomy governing this property. On one hand, in the absence of multinets, broad classes of arrangements satisfying suitable combinatorial constraints give rise to 1-formal Milnor fibers. On the other hand, the presence of reduced 3-multinets produces explicit obstructions to 1-formality.
The key mechanism on the non-formal side is a failure of the Tangent Cone Theorem for cohomology jump loci, detected via the interaction between monodromy eigenspaces and the mixed Hodge structure on \( H^1(F(\mathcal{A});\mathbb{C}) \). This leads to a combinatorial criterion for non-1-formality and an explicit infinite family of examples, given by the monomial arrangements \( \mathcal{A}(3k,3k,3) \).