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.[-]

Matrices whose coefficients are univariate polynomials over a field are a basic mathematical object which arises at the core of fundamental algorithms in computer algebra: sparse or structured linear system solving, rational approximation or interpolation, division with remainder for bivariate polynomials, etc. After presenting this context, we will give an overview of recent progress on efficient computations with such matrices. Next, we will show how these results have been exploited to improve complexity bounds for a selection of problems which, interestingly, do not necessarily involve polynomial matrices a priori: computing the characteristic polynomial of a scalar matrix, performing modular composition of univariate polynomials, changing the monomial order for multivariate Gröbner bases.[-]

D-finite functions play a prominent role in computer algebra because they are well suited for representation in a symbolic software system, and because they include many functions of interest, such as special functions, orthogonal polynomials, generating functions from combinatorics, etc. Whenever one wishes to study the integral or the sum of a D-finite function, the method of creative telescoping may be applied. This method has been systematically introduced by Zeilberger in the 1990s, and since then has found applications in various different domains. In this lecture, we explain the underlying theory, review some of the history and talk about some recent developments in this area.[-]

Matrices whose coefficients are univariate polynomials over a field are a basic mathematical object which arises at the core of fundamental algorithms in computer algebra: sparse or structured linear system solving, rational approximation or interpolation, division with remainder for bivariate polynomials, etc. After presenting this context, we will give an overview of recent progress on efficient computations with such matrices. Next, we will show how these results have been exploited to improve complexity bounds for a selection of problems which, interestingly, do not necessarily involve polynomial matrices a priori: computing the characteristic polynomial of a scalar matrix, performing modular composition of univariate polynomials, changing the monomial order for multivariate Gröbner bases.[-]