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.[-]

Reid's recipe is an equivalent of the McKay correspondence in dimension three. It marks interior line segments and lattice points in the fan of the G-Hilbert scheme (a specific crepant resolution of $\mathbb{C}^{3} / G$ for $G \subset S L(3, \mathbb{C})$ ) with characters of irreducible representations of $G$. Our goal is to generalise this by marking the toric fan of a crepant resolution of any affine Gorenstein singularity, in a way that is compatible with both the G-Hilbert case and its categorical counterpart known as Derived Reid's Recipe. To achieve this, we foray into the combinatorial land of quiver moduli spaces and dimer models. This is joint work with Alastair Craw and Jesus Tapia Amador.[-]

Algebraic combinatorics studies combinatorial objects with an algebraic point of view, and conversely. As such, it is also a very fertile ground for experimental mathematics, involving both classical and new algorithms. I will discuss two topics: finite partially ordered sets and their invariants, and tree-indexed polynomials and power series. Finite partially ordered sets are discrete objects, that can be seen as directed graphs, but also possess an interesting representation theory. This leads to many difficult questions about a subtle equivalence relation, namely posets having equivalent derived categories. The theme of tree-indexed series, which can be traced back to Cayley, plays a role in the study of vector fields and ordinary differential equations. It is nowadays better understood in the framework of operads and can be considered as a nonassociative version of the study of alphabets, words and languages. Surprisingly maybe, rooted trees also appear in the study of iterated integrals, stemming out of the usual "integration-by-part" rule. I will describe the corresponding notions of algebras, without diving too much into the theory of operads. On the way, I will discuss some of the involved algorithms and their implementations.[-]

Algebraic combinatorics studies combinatorial objects with an algebraic point of view, and conversely. As such, it is also a very fertile ground for experimental mathematics, involving both classical and new algorithms. I will discuss two topics: finite partially ordered sets and their invariants, and tree-indexed polynomials and power series. Finite partially ordered sets are discrete objects, that can be seen as directed graphs, but also possess an interesting representation theory. This leads to many difficult questions about a subtle equivalence relation, namely posets having equivalent derived categories. The theme of tree-indexed series, which can be traced back to Cayley, plays a role in the study of vector fields and ordinary differential equations. It is nowadays better understood in the framework of operads and can be considered as a nonassociative version of the study of alphabets, words and languages. Surprisingly maybe, rooted trees also appear in the study of iterated integrals, stemming out of the usual "integration-by-part" rule. I will describe the corresponding notions of algebras, without diving too much into the theory of operads. On the way, I will discuss some of the involved algorithms and their implementations.[-]