Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the number field case and on a way to strengthen it assuming a height conjecture. During the second part we will focus on function fields of positive characteristic and describe a new result obtained in a joined work with Pacheco.[-]

This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories [2, 5, 6]. I will then proceed to explain the concept of wall-crossing, both in theory, and in examples [1, 2, 4, 6].

- Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction [7].

- Applications : Moduli spaces of sheaves on $K3$ surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on $K3$ surfaces [1, 8].[-]

In the first part, we describe the canonical model structure on the category of strict $\omega$-categories and how it transfers to related subcategories. We then characterize the cofibrant objects as $\omega$-categories freely generated by polygraphs and introduce the key notion of polygraphic resolution. Finally, by considering a monoid as a particular $\omega$-category, this polygraphic point of view will lead us to an alternative definition of monoid homology, which happens to coincide with the usual one.[-]

Duality in projective geometry is a well-known phenomenon in any dimension. On the other hand, geometric triality deals with points and spaces of two different kinds in a sevendimensional projective space. It goes back to Study (1913) and Cartan (1925), and was soon realizedthat this phenomenon is tightly related to the algebra of octonions, and the order 3 outer automorphisms of the spin group in dimension 8.
Tits observed, in 1959, the existence of two different types of geometric triality. One of them is related to the octonions, but the other one is better explained in terms of a class of nonunital composition algebras discovered by the physicist Okubo (1978) inside 3x3-matrices, and which has led to the definition of the so called symmetric composition algebras.
This talk will review the history, classification, and their connections with the phenomenon of triality, of the symmetric composition algebras.[-]

Exact categories were introduced by Quillen in 1970s as part of his seminal work on algebraic K-theory. Exact categories provide a suitable enlargement of the class of abelian categories (for example, an extension-closed subcategory of an abelian category inherits the structure of an exact category) in which one "can do homological algebra". Recently, motivated also by questions in algebraic K-theory, Barwick introduced the class of exact infinity-categories, relying on the newly-developed theory of infinity-categories developed by Joyal, Lurie and others. This new class of mathematical objects includes not only the exact categories in the sense of Quillen but also the stable inftinty-categories in the sense of Lurie (the latter are to be regarded as refinements of triangulated categories in the sense of Verdier). The purpose of this lecture series is to motivate the theory of exact infinity-categories and sketch some of its applications. Familiarity with the theory of infinity-categories is not expected.[-]

Khovanov and Seidel introduced in the early 2000's an action of the braid group by autoequi-valences on the homotopy category of projective modules over the zig-zag algebra. This categorical action descends to the Burau representation, one of the most famous braid representations, but unlike the classical story, the lifting is faithful. It is interesting to notice that simultaneously, the Burau representation was also extended into a faithful finite-dimensional linear representation by Lawrence, Krammer and Bigelow, proving the linearity of the braid group.
I will review the basic constructions, both at the level of vector representations and at the ca-tegorical level. We will discuss possible extensions of these from classical braids (type A) to larger Artin-Tits groups, spherical or not, and try to relate Khovanov-Seidel's construction to Soergel bimodules and categorified quantum groups. I will also try to emphasize several metric aspects that appear in an elegant way from the categorical setting, with an emphasis on Bridgeland's stability conditions. Along the way, I would like to list several open questions and problems that I care about, hoping that someone in the audience will come up with a good idea.[-]

I will give an introduction to the amplituhedron, a geometric object generalizing the positive Grassmannian, which was introduced by Arkani-Hamed and Trnka in the context of scattering amplitudes in N=4 super Yang Mills theory. I will focus in particular on its connections to cluster algebras, including the cluster adjacency conjecture. (Based on joint works with multiple coauthors, especially Evan-Zohar, Lakrec, Parisi, Sherman-Bennett, and Tessler.)[-]