Schwartz-MacPherson classes were first defined in 1965 by Marie-Helene Schwartz, in the framework of obstruction theory, using "radial" vector fields and Whitney stratifications. Although their definition is easy to understand, using Whitney stratifications requires delicate and very technical constructions. The use of Lipschitz stratifications makes it possible to simplify the construction of the radial vector fields and therefore the construction of the classes, remaining within the framework of obstruction theory, as Marie-Helene Schwartz did. This is a joint work with Tadeusz Mostowski and Nguyen Thi Bich Thuy.

In a series of papers joint with C. Geske, M. Herradón Cueto, L. Maxim and B. Wang, we developed a Hodge theory for certain (co)homological objects associated to abelian covering spaces of smooth complex algebraic varieties. These covering spaces are smooth complex analytic varieties, but are not algebraic varieties in general. In this talk, we will focus on describing the main properties of these mixed Hodge structures, the kind of information that the filtrations contain, and some applications.

Given a normal crossings degeneration \(f:(X,\omega_X)\to \Delta\) of compact Kahler manifolds, in recent work with T. Pelka we have shown how to associate a smooth locally trivial fibration \(f_A:X_A\to \Delta_{log}\) over the real oriented blow up of the disc \(\Delta\). It is moreover endowed with a closed 2-form \(\omega_A\) giving it the structure of a symplectic fibration. The restriction of \(\omega_A\) to every fibre of \(f_A\) "at positive radius" (that is over a point of \(\Delta\setminus \{0\}\)) is the modification by a potential of the restriction of \(\omega_X\) to the same fibre. The construction can be regarded as a symplectic realization of A'Campo model for the monodromy and has found the following applications:
(1) We can produce symplectic representatives of the monodromy with very special dynamics, and based on this and on a spectral sequence due to McLean prove the family version of Zariski's multiplicity conjecture.
(2) If f is a maximal Calabi-Yau degeneration we can produce Lagrangian torus fibrations over a the complement of a codimension 2 set over the (expanded) essential skeleton of the degeneration, satisfying many of the properties conjectured by Kontsevich and Soibelman.
In the talk I will highlight the main aspects of the construction, and at the end of it, depending on time we will present some of the applications (1) or (2).

We will recall some basics of deformation theory, and of Deligne's theory of mixed Hodge complexes, that he invented to put mixed Hodge structures on the cohomology of algebraic varieties. We will show how the deformations of a mixed Hodge complex can themselves be given the structure of a mixed Hodge complex, and how this can be used to construct mixed Hodge structures on Alexander modules. This is joint work with C. Geske, E. Elduque, L. Maxim and B. Wang.

The Hilbert scheme of points in the complex plane is a classical object of study in algebraic geometry. McKay correspondence provides an isomorphism between its K-theory (or cohomology) and the space of symmetric functions, creating a bridge between geometry and combinatorics. Multiplication by a class in the K-theory induces an endomorphism of the space of symmetric functions. In the cohomological case, compact formulas for such maps were found by Lehn and Sorger. The K-theoretical case was studied by Boissiere using torus equivariant techniques. He proved a formula for multiplication by the class of tautological bundle and stated a conjecture for the remaining generators of K-theory. In the talk, I will show how torus action simplifies this problem and prove the conjectured formula using restriction to a one-dimensional subtorus. I will also discuss the multiplicative structure on the space of symmetric functions induced by the product in K-theory. This is a joint work with M. Zielenkiewicz.

Restricting a constructible function to a hypersurface complement, its characteristic cycle is classically described by the sharp operation introduced by Ginzburg. Recently Laurentiu et al. find that the sharp operation is equivalent to the logarithmic compactification of the characteristic cycle when the divisor is SNC. We explore the possiblity of extending their result when the divisor is a free divisor in the sense of Saito. In particular, we show that the CSM class of the hypersurface complement is equal to the Chern class of the logarithmic tangenet bundle when the divisor is free and strongly Euler homogeneous. This is a joint work with Xiping Zhang.

Given a bounded constructible complex of sheaves \(F\) on a complex abelian variety, we prove an equality relating the cohomology jump loci of \(F\) and its singular support. As an application, we identify two subsets of the set of holomorphic 1-forms with zeros on a complex smooth projective irregular variety \(X\); one from Green-Lazarsfeld's cohomology jump loci and one from the Kashiwara's estimates for singular supports. This is a joint work with Yajnaseni Dutta and Feng Hao.

I will give a brief overview of Jörg Schürmann's contributions to mathematics, with an emphasis on characteristic classes for singular varieties.

I will speak about a proof of the existence of universal polynomials associated to multi-singularity loci of maps and also demonstrate some application.

In this talk, we will introduce a characterization of rational homology manifolds in terms of cubical hyperresolutions and discuss its applications in the theory of characteristic classes.

Let \((X,0)\) be the germ of an equidimensional analytic set in \((\mathbb{C}^n,0)\) and \(f=(f_1,f_2)\) a map-germin to the plane defined on \(X\). We investigate topological invariants associated to the pair \((f,X)\), among them, the Chern number of families of differential forms associated to \(f\). The topological information provided by these invariants is useful, although difficult to calculate. For a 2-dimensional ICIS \((X,0) \subset (\mathbb{C}^n,0)\), we apply our results to give an alternative description for the number of cusps \(c(f|X)\) of an stabilization of an A-finite map germ \(f=(f_1,f_2):(X,0) \to (\mathbb{C}^2,0)\).

We know that classical Chern classes for complex manifolds play a key role for manifolds theory. In this talk we shall discuss the case of singular varieties.

The K-theoretic motivic Chern class defined by Brasselet, Schurmann, and Yokura is a generalization of the Chern-Schwartz-MacPherson class in homology. Motivated by some computations for the stable bases of the cotangent bundle of the flag variety and the Iwahori-Whittaker functions for the dual group, we calculated the Chevalley formula for the motivic Chern classes of the Schubert cells. The proof uses the alcove walk algebra defined by Ram. This is a joint work with Mihalcea and Naruse.

If \(X\) is a closed aspherical manifold of dimension \(2d\), the Singer-Hopf conjecture predicts that \((-1)^d \cdot \chi(X)\geq 0\). We prove this conjecture for projective manifolds whose fundamental groups admit an almost faithful linear representation. In fact, we prove a stronger statement that all perverse sheaves on \(X\) have nonnegative Euler characteristics. The main ingredients of the proof are non-abelian Hodge theories both in the archimedean and non-archimedean setting and a vanishing cycle functor for multi-valued one forms. This is joint work with Ya Deng.

Very few characteristic classes of smooth complex algebraic varieties admit extension for singular varieties. The most general is the Hirzebruch class, but if one restricts the class of admissible singularities one can go further. I will recall the elliptic classes of Borisov and Libgober which work well for varieties with Kawamata log-terminal singularities. Interesting explicit formulas can be obtained for the quotient singularities. With further modifications it can be applied successfully for Schubert varieties.

A classical result of Kashiwara and Malgrange gives an explicit Riemann-Hilbert correspondence for Deligne nearby cycles along a hypersurface in a complex manifold, which then enables us to calculate the eigenvalues of monodromy actions of Milnor fiber cohomologies algebraically. In this talk, I will first discuss the Riemann-hilbert correspondence for Alexander complexes of Sabbath along a finite set of hypersurfaces, generalizing the result of Kashiwara and Malgrange. Then I will explain how to use it to get topological cohomology jumping loci by using the correspondence, as well as the multivariable monodromy zeta function if time allowed.

We prove a conjecture of Q. Shi, Y. Wang, and H. Zuo on the maximal spectral number of isolated hypersurface singularities using the self-duality of the Jacobian ring. We study Hodge-theoretic invariants such as Hodge ideals and Tjurina spectrum, which provide insights into the topology and geometry of the singularities. We also discuss the variance of the Tjurina spectrum and give counterexamples to a generalized Hertling conjecture. Our results suggest potential connections between these Hodge-theoretic invariants and the characteristic classes of singular spaces, which could be an interesting direction for future research.

In the study of singular spaces, the Chern class is an important ingredient. In this talk we will discuss the Chern classes of certain group orbits whose closures are singular. We will talk about some computations, and show some interesting patterns observed. We will also explain how such patterns are related to the group actions.

In my talk, I will describe the action of the Weyl group of a semi simple linear group $G$ on cohomological and K-theoretic invariants of the generalized flag variety $G/B$. I will study the automorphism $s_i$, induced by the reflection in the simple root and calculate the formula for this automorphism. Moreover, I will expand this formula in the basis consisting of structure sheaves classes of Schubert varieties. I will provide effective formula for the approximation of this expansion, which in the case of $G$ being a special linear group is more exact.

The tropical geometry of BNS invariants and its application Abstract: We study the tropicalization of the Bieri-Neumann-Strebel invariant on the maximal abelian covering of a CW-complex and compare this with the jumping ideal and the Alexander ideal. As an application, we give a restriction on Kahler groups.

Quiver Grassmannians parametrize subrepresentations of fixed dimension inside a fixed quiver representation. In my talk I will focus on quiver Grassmannians of a nilpotent representation of the equioriented cyclic quiver defined by the same matrix over each arrow in the quiver. These spaces have a known GKM-variety structure and a stratification into affine cells via the Bialynicki-Birula decomposition. I will talk about my result regarding smoothness of closed Bialynicki-Birula cells in a special case.

Matroid Schubert varieties are analogies of the ordinary Schubert varieties in the sense that they played a similar role in the Kazhdan-Lusztig theory of matroids. The local Euler obstruction is a topological invariant attached to any complex algebraic varieties, and Macpherson defined his Chern class by using the local Euler obstruction. In this talk, we will calculate the local Euler obstruction numbers of matroid Schubert varieties, and examine when they are positive.