Abstracts 

Paolo Aluffi

Title Newton–Okounkov bodies and Segre classes

Abstract We will present a novel approach to the computation of the Segre class of a subscheme of projective space, based on the construction of a suitable Newton-Okounkov body. The result may be viewed as a common generalization of results of Kaveh and Khovanskii and of an earlier result on Segre classes of monomial schemes. The construction of the Newton-Okounkov body is modeled on work of Lazarsfeld and Mustata.

Slides

Jean-Paul Brasselet

Title Hard Lefschetz and Hodge Riemann for polytopes

Abstract The lecture is a survey on previous results obtained in collaboration with Gottfried Barthel, Karl-Heinz Fieseler and Ludger Kaup (BBFK) and on results by Karl-Heinz Fieseler. Although these results may be considered "old", the techniques and methods are still current and likely to be used for further works. For non-simple polytopes, Stanley defined a generalized $h$-vector, using properties of intersection cohomology of general projective toric varieties. In order to prove the conjectured properties of this generalized $h$-vector various authors introduced a purely combinatorial virtual intersection cohomology for polytopes, 

inspired by equivariant intersection cohomology of projective toric varieties. Using Hodge-Riemann relations, this construction leads to the proof of a combinatorial Hard Lefschetz theorem.

Slides

László Fehér

Title Plücker formulas and motivic Chern classes of coincident root loci

Abstract Joint work with András P. Juhász.

Suppose that f is a degree d homogeneous polynomial in 3 variables. Then f defines a projective plane curve Z_f ⊂ P^2. Assuming that Z_f is non-singular, a famous formula of Plücker tells us that Z_f has 3d(d − 2) flexes. This can be reproved using equivariant cohomology classes of coincident root loci (which were calculated 20 years ago by Fehér–Némethi–Rimányi and Kőműves and also by Kazarian): if g is a degree d homogeneous polynomial in 3 variables, then g is a product of d linear factors, and Pol^d(C^2) is the disjoint union of the coincident root strata Y_λ for partitions of d, according to the multiplicities. For example the equivariant cohomology class of Y_{3,1,...,1} is

d(d − 2)( (d − 1)c_1^2  − (d − 4)c_2).

A simple geometric argument gives that the number of flexes can be obtained by substituting c_1^2 = c_2 = 1, obtaining the Plücker formula above. 

We can study similar problems in higher dimensions connected to tangent lines to degree d hypersurfaces, and there is a family of generalized Plücker formulas which contain the same information as the equivariant cohomology classes of coincident root loci.

The last Plücker formula calculates the Euler characteristics of the dual curve (which is the set of tangent lines) of Z_f. In modern terms this can be calculated from the equivariant Chern–Schwartz–MacPherson class c^{SM}( Y_{2,1,...,1}). The CSM class is a refinement of the cohomology class. The CSM classes of coincident root loci were calculated recently by Balázs Kőműves and the authors. It turns out that there is a wider class of Plücker formulas concerning the Euler characteristics of certain tangent line varieties which contain the same information as the CSM classes of coincident root loci.

The next natural step is to obtain information on the motivic χ_y-genus of these tangent line varieties. This can be done by calculating the motivic Chern classes of coincident root loci. We start to discover that there is an elaborate network of dependencies among these numbers governed by the geometry.

In this talk I present an algorithm calculating the motivic Chern classes of coincident root loci. We believe that these classes depend polynomially on d, just as the classical Plücker formulas. This conjecture is supported by computer evidence and some simple examples.

Nora Ganter 

Title Elliptic cohomology and Torus Actions

Abstract It has been known since the work of Quillen that cohomology theories with a theory of Chern classes give rise to (and are often determined by) a formal group law governing the behaviour of the first Chern class of line bundles under tensor product.The chromatic picture organises such cohomology theories according to the height of their formal group laws. The additive, multiplicative and elliptic formal group laws all relate to cohomology theories with Chern classes, namely cohomology, K-theory and various elliptic cohomology theories. When a torus action is added into the picture, it becomes more intuitive, and the geometry of the elliptic theory is finally emerging. These developments coincide with a range of applications in new settings. I will attempt to give an overview over these recent developments.

Slides

Daniel Holmes

Title A new tool for GKM spaces and their Gromov–Witten invariants

Abstract Using equivariant localization on the moduli space of stable maps, it is known how to calculate equivariant Gromov—Witten invariants of GKM spaces by summing over certain decorated trees. I will explain how to do this solely in terms of the GKM graph and introduce a new tool implementing these calculations. Example applications include GW invariants of local models of non-isolated P1’s in 3-space, some equivariant quantum products of Tolman/Eschenburg/Woodward’s twisted flag manifold, as well as equivariant Seidel elements / shift operators. Joint work with Giosuè Muratore.

Slides

András Juhasz

Title Chern-Schwartz-MacPherson classes of coincident root strata and related invariants of varieties of tangent lines

Abstract The coincidence of roots defines a GL(2)-invariant stratification of the vector space of homogeneous degree d polynomials in two complex variables. We give a new, recursive formula for the equivariant Chern-Schwartz-MacPherson (CSM) class of these coincident root strata (CRS). Using this method, we can show that the d-dependence of these classes is polynomial.

For a generic degree d hypersurface the set of its tangent lines with a prescribed set of intersection multiplicities, e.g. bitangents or inflection lines, can be identified as a coincident root locus of a certain section of some vector bundle. This enables us to deduce information about these varieties of tangent lines from GL(2)-equivariant CSM classes of CRS. For example, we get that the Euler characteristics of (generic sections of) these varieties of tangent lines is also a polynomial in d.

Slides

Allen Knutson

Title Some pipe dream formulae with closed curves

Abstract We recall a generalization of matrix Schubert varieties to the rectangular case, and study their conormal varieties. Under a 321-avoiding assumption (that we'll motivate), we give a formula for the classes of these varieties as a sum over "Temperley-Lieb pipe dreams". Each of these uses only bump tiles and upside-down bump tiles, and has a nonlocal contribution based on 2 to the number of closed loops. This is joint with Paul Zinn-Justin.

Matrix Schubert varieties have another generalization: consider the finitely many B_{2n}-orbit closures on {M in Mat_2n, M^2 = 0}. I describe work in progress on a pipe dream formula for these, which again involves closed loops unless the orbit lies inside upper triangular matrices. This is joint work in progress with a big crowd at McMaster University.

Anatoly Libgober

Title Rationally Elliptic Algebraic Varieties and Singularities

Abstract A topological space is rationally elliptic if the sum of the ranks over Q of the homotopy groups and the cohomology groups is finite. I  will describe the background, results and conjectures on classification of rationally elliptic smooth and singular algebraic varieties following my recent preprint: "Families of singular algebraic varieties that are rationally elliptic spaces".

Slides

Yuze Luan

Title Components of nested Hilbert scheme of 3 and n points on C^2

Abstract Using torus action on the Hilbert scheme of points and the corresponding Białynicki-Birula cell decomposition, we compute the dimension and components of the nested Hilbert scheme of 3 and n points on the affine plane C^2. All components have dimension n-1. This generalizes the work of Bulois and Evain that all components of Hilbert scheme of 2 and n points are of dimension n-1.

Laurentiu Maxim

Title Higher singularities of local complete intersections via characteristic classes 

Abstract Higher analogues of rational and Du Bois singularities were introduced recently through Hodge theoretic methods, and applied in the context of deformation theory, birational geometry, etc. In this talk, I will give a brief overview of these singularities, and explain how they can be studied through the lens of characteristic classes in the case of local complete intersections. (Joint work with Bradley Dirks and Sebastian Olano).

Slides

Leonardo Mihalcea

Title A quantum localization theorem for Grassmannians

Abstract The Yang-Baxter algebra arising in the study of a five-vertex lattice model is given by relations of the form

RTT=TTR, where R is the R-matrix, and T is the transfer matrix. These may be regarded as operators acting on the direct sum of torus-equivariant (quantum) K theory rings of the Grassmannians of linear subspaces in C^n. In joint work with V. Gorbounov and C. Korff, we give geometric interpretations for R and T in terms of operators in the equivariant quantum K theory ring of Grassmannians, and related `quantum=classical' convolutions. For example, the R-matrix gives the left Weyl group action, and the quantum trace of T gives the quantum K multiplication by the lambda_y class of the dual of the quotient bundle. As an application, we obtain a quantum version of the localization theorem, where the fixed point classes are quantized to the eigenvectors of the quantum trace of T.

Toru Ohmoto

Title The Stiefel–Whitney homology class for a category

Abstract From homotopy theory viewpoint, Tom Leinster (2008) introduced the notion of Euler characteristic for a finite category (that has been generalized to the theory of magnitude,  nowadays). It would be natural to ask what is `integration with respect to that Euler characteristic measure' ? Also in a similar purely combinatorial context with finite categories, we discuss about a counterpart to the (Sullivan's) Stiefel-Whitney homology class. 

Slides

Irma Pallares

Title On L-classes for singular varieties

Abstract Several characteristic classes of manifolds have their counterparts in the singular case. In fact, there are often different characteristic classes for singular varieties that recover the same class in the smooth case. In this talk, we will discuss various L-classes for singular varieties, how they are related, and their connection to rational homology manifolds. The talk is based on joint works with J. Fernández de Bobadilla and M. Saito.

Slides

Constantin Podelski

Title The Chern-Mather class of Theta divisors and the Tannakian Schottky Problem

Abstract Using the Tannakian category arising from convolutions of perverse sheaves on an abelian variety, one can attach to a principally polarized abelian variety (ppav) a reductive group along with a representation. The Tannakian Schottky Problem asks whether this group and representation characterize Jacobians in the moduli space of ppav's. We show that this holds in dimension up to 5. The main tool in the proof is a new criterion detecting Jacobians, relying on the Tannakian formalism and the Chern-Mather class of the Theta divisor. 

Tim Seynnaeve

Title Complete quadrics: Schubert calculus for Gaussian models

Abstract Let $\mathcal{L}$ be a generic linear space of symmetric matrices over the complex numbers. By inverting all invertible matrices in this space, we obtain an algebraic variety. Computing the degree of this variety is a natural geometric question in its own right, but is also interesting from the point of view of algebraic statistics: the number we obtain is the so-called maximum likelihood degree (ML-degree) of the generic linear concentration model. In 2010, Sturmfels and Uhler conjectured that if we fix the dimension of $\mathcal{L}$, this ML-degree is a polynomial in the size of the matrices. Using Schubert calculus and intersection theory on the space of complete quadrics, we were able to prove this polynomiality conjecture, and to write an algorithm that can compute these polynomials efficiently. This talk is based on joint works with Rodica Dinu, Laurent Manivel, Mateusz Michałek, Leonid Monin, and Martin Vodička. 

Slides

Julianna Tymoczko

Title Springer fibers and web graphs

Abstract The Springer fiber of a linear operator X is the subvariety of the flag variety consisting of flags that are fixed by the action of X.  Springer fibers do not generally admit global torus actions but they have geometric structure connected to important combinatorial objects associated both with flags (especially permutations) and with matrices (especially the partitions at come from Jordan blocks).  We describe a collection of labeled combinatorial graphs called webs that can be used to parametrize the cells of certain families of Springer fibers.  Time permitting, we also discuss open questions, including questions related to the representation theory of webs.

Slides

Vikraman Uma

Title Equivariant K-theory of cellular toric varieties and related spaces

Abstract In this talk we describe the T_{comp}-equivariant topological K-ring of a complete T-cellular toric variety. We further show that K_{T_{comp}}^0(X) is isomorphic as an R(T_{comp})-algebra to the ring of piecewise Laurent polynomial functions on the associated fan denoted PLP(\Delta). Furthermore, we compute a basis for K_{T_{comp}}^0(X) as a R(T_{comp})-module and multiplicative structure constants with respect to this basis. This talk is based on my recent paper of the similar title to be published in the IMPAN journal Fundamenta Mathematicae (see math arXiv: 2404.14201.v1). If time permits I shall also briefly describe the recent extensions of the results to the relative setting with applications to the description of the equivariant K-ring of toroidal horospherical embeddings in my recent preprint math arXiv: 2409.05719.

Slides

Paul Zinn-Justin

Title Pipe dreams, lower-upper varieties and Schwartz-MacPherson classes

Abstract We introduce several generalisations of certain combinatorial models called pipe dreams, namely generic pipe dreams and hybrid pipe dreams, and show how they compute equivariant cohomology classes of the lower-upper varieties of Knutson, and equivariant CSM classes of Schubert cells and related spaces. If time permits, we'll discuss the extension to K-theory. This is joint work with A. Knutson.