This is a joint work with Piotr Pragacz. We give Gysin formulas for partial flag bundles for the classical groups. We then give Gysin formulas for Schubert varieties in Grassmann bundles, including isotropic ones. All these formulas are proved in a rather uniform way by using the step-by-step construction of flag bundles and the Gysin formula for a projective bundle. In this way we obtain a comprehensive list of new universal formulas.

On the third exterior power of a vector space of dimension six we can define a natural (up to constant) non-degenerate skew-symmetric form associated to the wedge product. To the family of Lagrangian subspaces with respect to this form we associate two complete families of polarized IHS manifolds. These are the so-called double EPW sextics and double EPW cubes. The elements of both families are constructed as double covers of suitable Lagrangian degeneracy loci. It is known that the period domains of these two families are related by a natural quotient 2:1 map. In joint work with G. Kapustka and G. Mongardi we prove that this quotient map is compatible with the period maps of the two families. As a consequence, we deduce that EPW cubes are moduli spaces of objects on the Kuznetsov components of their associated Gushel-Mukai fourfolds.

Modules over Lie algebroids generalize Higgs bundles and vector bundles with integrable connections. I will show a few theorems concerning moduli spaces of such semistable modules. Some of them are completely new even in the case of usual semistable sheaves and allow, e.g., for a description of points of the moduli space of slope semistable sheaves.

Stability conditions on the Kuznetsov component of a Fano threefold of Picard rank 1, index 1 and 2 have been constructed by Bayer, Lahoz, Macrì and Stellari, making possible to study moduli spaces of stable objects and their geometric properties. In this talk we investigate the action of the Serre functor on these stability conditions. In the index 2 case and in the case of GM threefolds, we show that they are Serre-invariant. Then we prove a general criterion which ensures the existence of a unique Serre-invariant stability condition and applies to some of these Fano threefolds. Finally, we apply these results to the study of moduli spaces in the case of a cubic threefold X. In particular, we prove the smoothness of moduli spaces of stable objects in the Kuznetsov component of X and the irreducibility of the moduli space of stable Ulrich bundles on X. These results come from joint works with Song Yang and with Soheyla Feyzbakhsh and in preparation with Ethan Robinett.

We recall Voisin's specialization method for the study of the behavior of (stable) rationality of algebraic varieties under specialization and describe recent reformulations and extensions.

Unlike in the case of smooth projective surfaces, a minimal model of a log smooth surface with nonzero boundary may be singular. The construction of an almost minimal model, a modification of a usual MMP run, allows to delay the appearance of singularities. It allows to produce interesting curves lying outside the boundary and meeting it in a controlled manner. We will discuss the method and some of its applications, including a new strong structure theorem for normal log del Pezzo surfaces of rank 1 (from which a classification will follow).

Tuples of $n$ commuting matrices of size $d\times d$ form an algebraic variety${}^*$ that is both natural and useful in applications, for example to tensors. In the talk I will explain how surprisingly little is known and how one can investigate it using deformations of modules, via the classical ADHM construction and geometric/commutative algebra tools. This is a joint work with Klemen Šivic.

A vector bundle $\mathcal{E}$ on a smooth projective variety $(X,\mathcal{O}_X (1))$ is called ACM if $\mathrm{H}^i(X,\mathcal{E}(t))=0$ for all $t \in \mathbb{Z}$ and $0 < i < \mathrm{dim}(X)$. The structure of such bundles are known very explicitly for $\mathbb{P}^{n}$ and smooth guadrics $Q^n$. On the other side, there is not a structure theorem nor classification for non-ACM bundles on these varieties. In this talk, we will examine the structure of $\mathrm{H}_*^1 (X, \mathcal{E})$ for a rank 2 non-ACM bundle $\mathcal{E}$ where $X$ is $\mathbb{P}^2$ or $\mathbb{P}^1 \times \mathbb{P}^1$. Also, we will give a construction of weakly Ulrich bundles and supernatural bundles of any even rank on these varieties. This talk is based on a joint work with Filip Gawron.

Irregular Hodge theory was initiated by Deligne in 1984, with arithmetic motivations, to deal with Hodge properties in the context of connections with irregular singularities. It also proves useful for computing Hodge numbers of some classical motives. The talk will explain the origin and motivations, the main examples in exponential Hodge theory and examples of irregular Hodge-Tate structures related to Fano toric manifolds.

I will present a new criterion to test Whitney equisingularity and Thom's $(a_f)$ condition for certain families of (possibly non-isolated) hypersurface singularities that "behave well" with respect to their Newton diagrams. As a corollary, I will show that in such families all members have isomorphic Milnor fibrations. This is a joint work with Mutsuo Oka.

The family version of the Zariski multiplicity conjecture asserts that a family of isolated hypersurface singularities with constant Milnor number has constant multiplicity, too. I will explain a proof of this conjecture based on a new construction of the McLean spectral sequence. Given a holomorphic function $f$ on $X$ with isolated critical points, this sequence converges to the fixed-point Floer homology of the monodromy of $f$, and - in case $X$ is a Milnor ball, which was McLean's original setting - computes the multiplicity of $f$. To apply this result in families, we need to fix a Milnor ball $X$ for one member and keep the size for the nearby ones: this is where a new, general setting is used. To obtain this sequence, we make an A'Campo-style choice of a symplectic monodromy with good dynamics. This is a joint work with J. F. de Bobadilla.

Two natural questions that arise in tensor area are the computation of the rank and the identifiability of a specific tensor. In this talk I will provide some answers to these problems in the case of symmetric tensors, by using tools from classical Algebraic Geometry.

In my talk I will describe recent developments in classifying algebraic varieties in arithmetic settings. I will start by explaining the background for complex varieties and providing motivation for the study of their arithmetic analogues. The work is partially based on recent breakthroughs in arithmetic geometry and commutative algebra.

The strength of a homogeneous polynomial is the smallest length of an additive decomposition as sum of reducible forms. It is called slice rank if we additionally require that the reducible forms have a linear factor. Geometrically, the slice rank corresponds to the smallest codimension of a linear space contained in the hypersurface defined by the form. Due to this relation, it is well-known and easy to compute the value of the general slice rank and also to show that the set of forms with bounded slice rank is Zariski-closed. In this talk, I will present the following results from recent joint works with A. Bik, E. Ballico and E. Ventura: (1) the set of forms with bounded strength is not always Zariski-closed: this is an asymptotic result in the number of variables proved by using the theory of polynomial functors; (2) for general forms, strength and slice rank are equal: this is proved by showing that the largest component of the secant variety of the variety of reducible forms is the secant variety of the variety of forms with a linear factor.

I will discuss properties of $p$-adic étale cohomology of $p$-adic analytic spaces (comparison with de Rham type cohomologies, duality). This is based on a joint work with Pierre Colmez and Sally Gilles.

The Betti numbers of Hilbert schemes of points in the plane were first computed by Ellingsrud and Strømme. In this talk, we'll let a finite cyclic group act on the plane, and consider the analogous problem of computing the Betti numbers of the fixed loci under the induced action on the Hilbert scheme of points. The main result is a new product formula for the Betti numbers in the case where no element of the group fixes the symplectic form on the plane. This is joint work with Paul Johnson.

It is a classical result that two general quadratic forms can be simultaneously diagonalized in a unique way. For forms of higher degrees, it is a long-standing problem to classify when we get uniqueness. In this talk I'll present the solution for forms in three variables. Our strategy is to translate the problem into the study of a certain linear system on a projective bundle on the plane.

I will discuss recent progress on the existence of minimal models and Mori fiber spaces for generalized pairs. I will also explain the close relationship between the existence of minimal models and the existence of weak Zariski decompositions for generalized pairs. This is joint work with Vladimir Lazić.

Following Simpson, non-abelian Hodge theory studies representations of the fundamental group of a smooth projective complex variety by way of a non-abelian generalisation of the Hodge decomposition. In this talk, I will discuss an analogue of this theory in the p-adic world: I will explain why Scholze's pro-étale site naturally appears in this context, and how it can be used to study p-adic representations of the étale fundamental group of p-adic varieties in terms of "pro-étale vector bundles". This leads to a kind of Hitchin fibration that gives a non-abelian generalisation of the Hodge--Tate sequence of p-adic Hodge theory.

The variety of uniform matrix product states arises both in algebraic geometry as a noncommutative analogue of the Veronese variety, and in quantum many-body physics as a model for a translation-invariant system of sites placed on a ring. Using methods from linear algebra, representation theory, and invariant theory of matrices, we study the linear span of this variety. This talk is based on joint work with Claudia De Lazzari and Harshit Motwani.

Let $k$ be a field and $F$ be its algebraic closure. A $k$-algebra $B$ is said to be an $\mathbf{A}^n$-form over $k$ if $B \otimes_k F$ is isomorphic to the polynomial ring $F[Y_1, \dots, Y_n]$.

It is well-known that separable $\mathbf{A}^1$-forms over $k$ are isomorphic to the polynomial ring $k[Y]$ and that there exist non-trivial purely inseparable $\mathbf{A}^1$-forms over fields of positive characteristic. A nontrivial result of T. Kambayashi establishes that separable $\mathbf{A}^2$-forms over $k$ are also isomorphic to the polynomial ring $k[Y_1, Y_2]$. However, for $n >2$, it is not known whether every separable $\mathbf{A}^n$-form is necessarily isomorphic to the polynomial ring $k[Y_1, \dots, Y_n]$.

In this talk, we shall discuss a partial solution to this problem for the case $n=3$. We shall also discuss $\mathbf{A}^2$-forms over commutative rings.

The study of surfaces of general type and their moduli goes back to the beginning of the 20th century. If the invariants are chosen carefully, then we have a good understanding of the moduli space in a classical sense. We will concentrate on the particular case of regular surfaces with geometric genus 2, and canonical degree 1. One can try to compactify the classical moduli space by admitting surfaces with certain mild singularities. I will describe some of the behaviours and complications which arise when we do this. This is ongoing joint work with Marco Franciosi, Rita Pardini, Julie Rana, and Sönke Rollenske.

Schubert calculus roughly speaking is a theory organizing cohomological invariants of Schubert varieties into an algebraic structure. The behaviour of such invariants in families is governed by the torus actions on the homogenous spaces. Therefore the torus equivariant theories are proper objects to study. They have a rich structure: they are modules over the algebras spanned by characters and Hecke-type operations. Starting from the classical divided differences, Demazure-Lusztig operations, I will report on the recent development in $K$-theory and elliptic theory.

This talk is based on joint work with John Christian Ottem. I will explain a generalization of results by Shinder on variation of stable birational types in degenerating families, and how this can be used to extend Schreieder's non-stable rationality bounds from hypersurfaces to complete intersections in characteristic zero. This technique yields new results already for complete intersections of quadrics.