Enumeration of rooted constellations and hypermaps through quantum matrix integrals

We present a new method for enumeration of rooted constellations and other objects these can represent, specifically rooted hypermaps and maps. We derive a closed-form generating function enumerating rooted hypermaps with one face and a fixed number of darts, partitioned by number of edges, and vertices. We derive an algorithmic procedure for calculating generating functions enumerating all rooted hypermaps for fixed number of darts, partitioning by number of edges, vertices and faces, as well as an analogous procedure for enumerating rooted maps for fixed edge count. We also look at the enumeration problem for general rooted constellations, but do not calculate generating functions. Using these results we find recursion relations for calculating the total number of rooted hypermaps, maps and constellations of any given degree. This method is based on matrix integration tools originally developed in the study of bipartite quantum systems, specifically in calculating mean properties of their subsystems, where the averaging is over all possible pure states of the overall system. We present this work first, studying the mean von Neumann entropy of entanglement between the quantum system's two subsystems. We look at an unproven entropy approximation proposed by Lubkin (1978), derived from an infinite series expansion of the entropy which was not known to be convergent. We prove that this series is convergent if and only if the subsystem being studied is of dimension two, by deriving closed-form expressions for the series terms and finding their limiting behaviour. In light of this we examine the validity of Lubkin's approximation rigorously, confirming the limit in which it is valid, but deriving a more accurate approximation in the process

Using character varieties: Presentations, invariants, divisibility and determinants

If G is a finitely generated group, then the set of all characters from G into a linear algebraic group is a useful (but not complete) invariant of G . In this thesis, we present some new methods for computing with the variety of SL2C -characters of a finitely presented group. We review the theory of Fricke characters, and introduce a notion of presentation simplicity which uses these results. With this definition, we give a set of GAP routines which facilitate the simplification of group presentations. We provide an explicit canonical basis for an invariant ring associated with a symmetrically presented group\u27s character variety. Then, turning to the divisibility properties of trace polynomials, we examine a sequence of polynomials rn(a) governing the weak divisibility of a family of shifted linear recurrence sequences. We prove a discriminant/determinant identity about certain factors of rn( a) in an intriguing manner. Finally, we indicate how ordinary generating functions may be used to discover linear factors of sequences of discriminants. Other novelties include an unusual binomial identity, which we use to prove a well-known formula for traces; the use of a generating function to find the inverse of a map xn ∣→ fn(x); and a brief exploration of the relationship between finding the determinants of a parametrized family of matrices and the Smith Normal Forms of the sequence

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library