Many conceptual advances in algebraic geometry in mixed characteristics have led to considerable developments of this area and to discovering new domains of inquiry. Some aspects and perspectives on those developments will be discussed in this talk. This is a preparatory lecture for the ICM talk of B. Bhatt.

In this preparatory lecture I will introduce and state the fundamental properties of some objects and invariants relevant in the study of a module using free resolutions. We will start from the first appearance of these ideas in the work of Hilbert (and even earlier of Cayley), until the recently introduced Boij-Soderberg theory. Time permitting, we will also discuss briefly some bounds and conjectures on the aformentioned invariants.This is a preparatory lecture for the ICM talk of Irena Peeva.

We discuss the introductory part of a lecture by Avi Wigderson on complexity results for operator scaling problems via alternating minimization and the relation to singular matrix spaces. This is a preparatory lecture for the ICM talk of Avi Wigderson.

This lecture is based on Federico Ardila's preprint "The geometry of geometries: matroid theory, old and new" (https://arxiv.org/pdf/2111.08726.pdf). After reviewing basic definitions related to matroids, in this talk we introduce some key invariants of a matroid and we discuss recent achievements of matroid's theory. This is a preparatory lecture for the ICM talk of Federico Ardila.

We will focus on 3 examples in Richard Schwartz's Survey Lecture on Billiards. Given a polygon in the plane, consider a point mass moving at unit speed which has perfectly elastic collisions on the boundary. One simple first question is "Is there a periodic trajectory?" On a triangle, this is still not a fully solved problem. Schwartz showed via a computer aided proof that all triangles up to an angle of 100 degrees have a periodic trajectory. More recently we now know that periodic trajectories exist on all triangles up to 112.3 degrees. When the polygon has rational angles (for example the regular pentagon) there is always a periodic trajectory. In fact the periodic trajectories on the regular pentagon are completely classified and give some pretty pictures too. We'll finish by discussing what happens when we glue multiple polygons, for example gluing 12 regular pentagons to make a dodecahedron.This is a preparatory lecture for the ICM talk of Richard Schwartz.

Let Q be a smooth projective quadric of dimension at least 1 over a number field k. Then Q has a k-point if and only if it has points over the completions of k at every place. For example, the conic Q: $x^2+y^2 = -1$ has no real solutions, and therefore no rational points. However, the genus 1 curve $2y^2 = x^4 - 17z^4$ has solutions over the reals and the p-adics for every prime p, but does not have any rational points. Here, there is an obstruction to the local-global principle called the Brauer-Manin obstruction. In this talk, we will give an introduction to the Brauer-Manin obstruction, weak approximation, and local-global principles.This is a preparatory lecture for the ICM talk of Olivier Wittenberg.

In this talk, we will survey some results concerning the distribution of groups that arise in number theory. In particular, we shall discuss limit distributions, universality laws and survey recent results and developments in arithmetic statistics.This is a preparatory lecture for the ICM talk of Melanie Wood.

The Netflix problem is the problem that asks one to predict the ratings a user would give a movie or TV show given previous ratings the user has given, without any other information about the user. This problem is an example of matrix completion. Matrix completion is the problem of filling in missing entries to a data matrix under the assumption that the data lies on a low dimensional subspace. The first part of this talk will highlight some results and solution methods for this problem. In the second part of this talk I will turn to the problem of nonlinear matrix completion and highlight recent results in a paper of Florentin Doyens, Coralia Cartis and Armin Eftekhari. For this part, instead of assuming linear relations among the columns of our data matrix, we assume they obey nonlinear relations (i.e. they lie on the union of subspaces or some other algebraic variety). This part will give an overview on how one can formulate this problem as well as algorithms to solve them.This is a preparatory lecture for the ICM talk of Coralia Cartis.