A. du Plessis and C.T.C. Wall have obtained lower and upper bounds for the Tjurina number of projective plane curves. In some cases, the curves having maximal Tjurina numbers are exactly the free curves. In the talk we will survey various new results in this area, as well as a number of open problems.
The case of line arrangements is very interesting and will be given special attention. One of the open problems is a stronger form of K. Saito's Conjecture, saying that the freeness of a line arrangement is determined by the combinatorics.

Let \(X\) be a closed semialgebraic set of dimension \(k.\) If \(n\ge 2k+1\), then there is a bi-Lipschitz and semialgebraic embedding of \(X\) into \(\mathbb{R}^n.\) Moreover, if \(n \ge 2k+2\), then this embedding is unique (up to a bi-Lipschitz and semialgebraic homeomorphism of \(\mathbb{R}^n\)). We also give local and complex algebraic counterparts of these results.

We describe a refinement of the bounds on the number of reducible fibers in a pencil of curves on a smooth projective surface in the case when the classes of components of reducible members belong to a fixed subset of Neron-Severi group. A very special case of these results is a theorem by the speaker and S. Yuzvinski on pencils of plane curves containing members which are unions of lines. Joint work with J.I. Cogolludo.

We construct the analytic lattice cohomology associated with the analytic type of any complex normal surface singularity. It is the categorification of the geometric genus of the germ, whenever the link is a rational homology sphere. It is the analytic analogue of the topological lattice cohomology, associated with the link of the germ whenever it is a rational homology sphere. This topological lattice cohomology is the categorification of the Seiberg-Witten invariant, and conjecturally it is isomorphic with the Heegaard Floer cohomology.
We compare the two lattice cohomologies: in some simple cases they coincide, but in general, the analytic cohomology is sensitive to the analytic structure. We expect a deep connection with deformation theory. We provide several basic properties and key examples, and we formulate several conjectures and problems.

Let \((X, 0) \subset (\mathbb{C}^n,0)\) be a germ of a reduced hypersurface. Assume that there is a sequence of regular elements (i.e., elements that are in the maximal ideal but not in the square of the maximal ideal, called HYPERPLANE), \(h_i\), such that their intersections with \(X\) is isolated (hypersurface) singularity and their Milnor numbers tend to infinity. Then they proved that there is a hyperplane h such that its intersection with X has non isolated singularity (equivalently, Milnor algebra is infinite dimensional). Here we provide different classes replacing hyperplanes, where a similar limiting theorem hold. We will discuss many examples and counter examples in this talk. This work is an on going project with Mihai Tibar and Mohit Upmanyu.

In this talk we present several invariants of two variables holomorphic functions with emphasis on bi-Lipschitz invariants. (Joint work with Mihai Tibăr and Piotr Migus.)

We first recall the construction of the dual MacPherson Chern class transformation via characteristic cycles. Then we explain our related work with Mihai Tibar on representing MacPherson cycles in case of an ambient manifold with trivial tangent bundle, as well as a corresponding global index theorem in the affine context. Finally we discuss recent work with Paolo Aluffi, Leonardo Mihalcea and Changjian Su on the positivity of dual Segre-MacPherson classes. Applications are given to semi-abelian varieties, the Behrend constructible function, Schubert cells in flag manifolds and the complement of a projective hyperplane arrangement.

We consider polygons in the affine plane.
We'll study the (signed) Area function on this space with several types of constraints.
- all edge lengths fixed (Linkages),
- the total edge length (perimeter) fixed (rope),
- the vertices lie on a circle.
We will give geometric criteria for critical points of Area and their Morse types. We will encounter various attributes from singularity theory: Cerf-diagram, the Eisenbud-Levine-Khimshiashvili signature formala, non-smooth analysys and will end up with some represenation theory.
This is joint work with Gaiane Panina, George Khimshiashvili and Wilberd van der Kallen.

We study the lower central series, the Alexander invariants, and the cohomology jump loci of groups arising as split extensions with trivial monodromy in first homology with appropriate coefficients. We use these techniques to gain further understanding of the Milnor fibration of the complement of a hyperplane arrangement and the fundamental group of its Milnor fiber.