In my talk I will recall the theory of entanglement criteria from Quantum Information Theory.
Then I will focus on the entanglement criteria based on tensor norms [Jivulescu, Lancien, Nechita 2022] which distinguishes among as it gives a unique description for well-known criteria such as realignment or SIC-POVM criterion. The tensor norm criterion solves in a positive way conjecture of Shang, Asadian, Zhu, and Guhne by deriving systematic relations between the performance of these two criteria. Connections to the problem of determining threshold points for various criteria are presented.[-]

I will explain how to combine tools of local tropical geometry and logarithmic geometry in order to study the structure of Milnor fibers of smoothings of isolated complex singularities, up to homeomorphisms. I will partly follow the paper “The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof”, written in collaboration with Marıa Angelica Cueto and Dmitry Stepanov.This course replaces a course on the same topic that should have been delivered by Angelica Cueto.[-]

I will explain how to combine tools of local tropical geometry and logarithmic geometry in order to study the structure of Milnor fibers of smoothings of isolated complex singularities, up to homeomorphisms. I will partly follow the paper “The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof”, written in collaboration with Marıa Angelica Cueto and Dmitry Stepanov.This course replaces a course on the same topic that should have been delivered by Angelica Cueto.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

The algebraic delta invariant, a number encoding the K-stability of a Fano variety, is a central theme of this Winter school. In the first lecture, T. Delcroix presents an analytic viewpoint on the delta invariant developped by Kewei Zhang, along with the rough ideas of the variational approach to existence of canonical Kähler metrics. In his second lecture, he extends this to the weighted Kähler setting (joint work with S. Jubert), allowing to deal with Kähler-Ricci solitons and more. [-]