N3.12

Abstract: Triangulations of surfaces serve as important examples for cluster theory, with the natural operation of “diagonal flips” encoding mutation in cluster algebras and categories. In this talk we will focus on the combinatorics of mutation on marked surfaces with infinitely many marked points, which have gained importance recently with the rising interest in cluster algebras and categories of infinite rank. In this setting, it is no longer possible to reach any triangulation from any other triangulation in finitely many steps. We introduce the notion of mutation along infinite admissible sequences and show that this induces a preorder on the set of triangulations of a fixed infinitely marked surface. Finally, in the example of the completed infinity-gon we define transfinite mutations and show that any triangulation of the completed infinity-gon can be reached from any other of its triangulations via a transfinite mutation. The content of this talk is joint work with Karin Baur.

Abstract: We define the Hochschild complex and cohomology of a monoid in an Ab-enriched monoidal category. Then we interpret some of the lower dimensional cohomology groups and discuss when the cohomology ring happens to be graded-commutative.

Firstly I will outline Dirac cohomology for graded Hecke algebras and the branching rules for the projective representations of $S_n$. Combining these notions with the Jucys-Murphy elements for $\tilde{S}_n$, that is the double cover of the symmetric group, I will go through a method to completely describe the spectrum data for the Jucys-Murphy elements for $\tilde{S}_n$. If time allows I will also explain how this spectrum data gives rise to a a concrete description for the matrices of the action of $\tilde{S}_n$.

I will try to give a brief description of the use of group theory and character theory in chemistry, specifically vibrational spectroscopy. Defining the group associated to a molecule, how one would construct a representation corresponding to such a molecule and the character table associated to this. Then, time permitting, I will go in to the deconstruction of the data from spectroscopy; finding such a group and hence molecule structure. 

Two polyhedra are said to be scissors congruent if they can be subdivided into the same finite number of polyhedra such that each piece in the first polyhedron is congruent to one in the second. In 1900, Hilbert asked if there exist tetrahedra of the same volume which are not scissors congruent. I will give a history of this problem and its proofs, including an incorrect 'proof' by Bricard from 1896 which was only rectified in 2007.

The problem of computing the Galois group of an irreducible, rational polynomial has been studied for many years. I will discuss the methods developed over the years to approach this problem, and give some examples of them in practice. These methods mainly involve constructing and factorising resolvent polynomials, and thereby determining better upper bounds for the conjugacy class of the Galois group within the symmetric group, i.e. describe its action on the roots of the polynomial explicitly. I will describe how using approximations to the zeros of the polynomial allows us to construct resolvents, and in particular, how using p-adic approximations can be advantageous over numerical approximations, and how this can yield a direct and systematic method of determining the Galois group.