The features of a group being finitely generated or finitely presented are, respectively, the n=1 and n=2 cases of the finiteness property Fn. Towards the end of the last century, the question of when these finiteness conditions descend to subgroups of G led to the discovery of the sets Σn(G) (and their homological counterparts Σn(G;A), for A a ℤ[G]-module). Each set Σn(G) is a collection of homomorphisms G → R, refining property Fn, in the sense that G has type Fn precisely if Σn(G) contains the zero map.

During the last half century, a large proportion of number theorists have been occupied with the Langlands program, which proposes the existence of certain correspondences between automorphic and Galois representations, going vastly beyond quadratic reciprocity or class field theory. We will discuss the origins of the Langlands program and try to explain how methods from higher category theory, representation theory, and also $p$-adic geometry have played a role in transforming the field during the last two decades.

Title: Totally positive skew-symmetric matrices

Title: 2-Representations of affine type A Soergel bimodules: some observations and examples.  

Let C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, C is defined in k^2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least 2. We use this data to construct 3 pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp.