Joachim Jelisiejew

University of Warsaw

Algebraic geometry of tensors
Complexity theory has a pool of open questions related to tensors. Recently, tools of algebraic geometry, such as deformation theory, proved to be very effective for some of those problems. However, many open questions of varying difficulty remain. In the talk I will review some of the recent methods and mention a number on less known open problems. The talk is partially based on the joint work with Landsberg, Pal and with Mandziuk.

Martí Lahoz

Universitat de Barcelona

Stability conditions in families
Stability conditions on triangulated categories, introduced by Tom Bridgeland between 15 and 20 years ago, have found many applications to questions unrelated to derived categories.
In this talk, will give the definition of stability condition and explain why when they exist they come with a deformation space that is a complex manifold. Fixing a numerical class, this complex manifold is eqquipped with a wall and chamber structure and I will survey some applications of the so-called wall-crossing techniques. I will finish by explaining why a stability condition corresponds to a polarization on a non-commutative variety and what we need to talk about stability conditions in a family of varieties over a base.
The talk is partially based on joint work with Bayer, Macrì, Nuer, Perry and Stellari.

Eduardo de Lorenzo Poza

KU Leuven - BCAM

The Arc-Floer conjecture and plane curves
Given an isolated hypersurface singularity, we may associate to it two very different invariants. On the one hand we have contact loci, essentially the sets of arcs that have a specific order of contact with the singularity. On the other hand there is the monodromy of the Milnor fibration, which may be endowed with a symplectic structure.
The Arc-Floer conjecture predicts that the cohomology of the contact loci coincides up to a shift with the Floer homology of the monodromy iterates. In this talk we will explain the origin of this conjecture and what is known about it, and we will explore the key ingredients of the proof of the conjecture in the case of plane curve singularities, the only case in which the conjecture is known to hold. This is joint work with Javier de la Bodega, https://arxiv.org/abs/2308.00051.

Irma Pallarés Torres

KU Leuven

Characteristic classes and singular varieties
Characteristic classes of manifolds are usually cohomology classes associated to the tangent bundle of the manifold. Hirzebruch's theory unifies three important theories of characteristic classes of smooth manifolds: Chern classes, Todd classes and Thom-Hirzebruch L-classes. These three classes, Chern, Todd and L-classes, were individually extended to singular varieties. In 2010, Brasselet, Schürmann and Yokura extended Hirzebruch classes to singular varieties using natural transformations, and unified in a functorial sense three theories of characteristic classes of singular varieties: the Chern transformation of MacPherson-Schwartz, the Todd transformation of Baum-Fulton-MacPherson and the L-transformation of Cappell-Shaneson. In this talk, we will discuss Hirzebruch classes and delve into the relationships between these transformations.

Elvira Pérez Callejo

Universitat Jaume I

Algebraic integrability of planar polynomial vector fields and bounded negativity on rational surfaces
Throughout the thesis we obtain several necessary conditions for algebraic integrability of a planar polynomial vector field. Moreover, we describe several algorithms to decide on algebraic integrability (under certain conditions) and to compute a rational first integral in the positive case. We use, as a main tool, the extension of the vector field to a foliation on a Hirzebruch surface. Finally, we prove some results related to bounded negativity on rational surfaces.

Javier Sánchez González

Universidad de Salamanca

Étale Fundamental Group of Schematic Spaces
Schematic spaces are ringed posets that present an algebro-geometric behaviour similar to that of quasi-separated schemes. There are enough of them to model all such schemes and, furthermore, a whole class of more general locally ringed spaces. My thesis' main results were the construction of an étale fundamental group for schematic spaces and the proof of its corresponding Seifert-Van Kampen Theorem and homotopy exact sequence, which recover and generalize their classical homonyms. In this talk, I will introduce some key aspects of the theory of schematic spaces that allow us to prove the aforementioned results. This will take us on a tour through Category Theory, Descent Theory and the very foundations of the world of schemes.

Luis José Santana Sánchez

Universidad de Valladolid

The geometry of \(\mathbb P^n\) blown-up at points on a rational normal curve of degree \(n\)
Mori dream spaces are varieties with a finite Mori chamber decomposition, for which the Mori program can be carried out for any divisor. An instance of these varieties are blow-ups of the complex projective space at points on a rational normal curve as shown by Castravet and Tevelev. In this talk, we will study the geometry of these spaces and see how it can be completely understood by its stable base locus.

Irene Spelta

Centre de Recerca Matemàtica - CRM

On the geometry of the Torelli locus
The construction that associates to an algebraic curve \(C\) its Jacobian \(JC\) (as a principally polarized abelian variety) defines a morphism \(j : M_g → A_g\), which, by a famous theorem of Torelli, is injective on geometric points. In this talk, we will discuss the interplay between the geometry of \(M_g\) and of \(A_g\). In particular, we will focus on the (local) symmetric geometry of \(A_g\), and we will consider certain special (totally geodesic) subvarieties. We will explain how this is linked to a conjecture by Coleman and Oort, and we will analyse explicit examples.
RGAS Universitat de Barcelona IMUB