# Abstracts

### The mechanics of a “twisted” brain

Valentina Balbi (UL)

Biological soft tissues are particularly common in nature. For instance, many organs in the human body such as the skin, the brain, the gastro-intestinal system are made of soft tissues. The brain, among all is particularly soft and delicate. Following an impact to the skull, brain matter can experience large stretches, possibly resulting in Diffuse Axonal Injury (DAI), which is the second leading cause of death from traumatic brain injury. Previous studies have focused on linear (uni-axial) stretches of brain to investigate DAI, but in reality brain matter undergoes a mix of deformation modes during an accident. This talk will focus on the mechanical behavior of the brain under torsion (twisting). In collaboration with University College Dublin, we collected data from torsion tests on (pigs) brain samples and modelled the experiments to finally quantify the elastic properties of the brain tissue. I will show that torsional impacts, such as a hook punch in boxing and a side impact in a car accident can also lead to dangerous levels of stretch compatible with DAI.

### Permutations, tensor products, and Cuntz algebra automorphisms

Francesco Brenti (Universita' di Roma “Tor Vergata”)

We introduce and study a new class of permutations which arises from the automorphisms of the Cuntz algebra. I will define this class, explain its relation to the Cuntz algebra, present results about symmetries, constructions, characterizations, and enumeration of these permutations, and discuss some open problems and conjectures. This is joint work with Roberto Conti and Gleb Nenashev.

### Chip-firing, toppling regimes, and combinatorial structures

Mark Dukes (UCD)

In this talk I will highlight some results from a series of papers that studied chip-firing on some well-known graphs. These studies reveal combinatorial structures that are in bijection with the “recurrent” or “critical” configurations, and help to illustrate the dynamics at the heart of chip-firing. We also present an analysis of these critical configurations on general graphs when the graph can be decomposed in two different ways.

### Riordan arrays and weighted lattice paths

Aoife Hennessy (Waterford IT)

This talk concerns bijections between Motzkin and Łukasiewicz paths arising from Riordan array decompositions. Bijections have been shown between Motzkin paths and Łukasiewicz paths with constant weights. We introduce a bijection between Motzkin paths and Łukasiewicz paths with non constant weighted steps and use these bijections to prove some simple combinatorial identities.

### Exact, free-surface equatorial flows

David Henry (UCC)

We construct a new exact solution to the governing equations for geophysical fluid dynamics, in the equatorial region of the ocean. The solution is presented in the terms of spherical coordinates, and represents a steady purely-azimuthal flow with a free-surface. Of particular note is that this solution accommodates a general fluid stratification: the density may vary both with depth and with latitude. Using a short-wavelength stability analysis we prove that flows defined by our exact solution are stable for a certain choice of the density distribution. This is joint work with Calin Martin, University of Vienna.

### Highly connected 7-manifolds with non-negative sectional curvature

Martin Kerin (WWU Münster/NUI Galway)

Although non-negative sectional curvature is one of the most fundamental notions in Riemannian geometry, there are few known constructions, and, hence, few known examples, of manifolds admitting such a Riemannian metric. In this talk, I'll briefly survey some of the known constructions, then describe a new construction which yields a rich family of non-negatively curved 7-manifolds, including many examples that exhibit interesting topological properties. For example, this family contains all exotic 7-spheres, as well as infinitely many spaces not even homotopy equivalent to previously known examples. This is joint work with Sebastian Goette (Freiburg) and Krishnan Shankar (Oklahoma/NSF).

### Convolutions on the complex torus

Marianne Leitner (TCD/DIAS)

"Quasi-elliptic" functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated and given by Eisenstein series. A related structure has appeared recently in the computation of Feynman integrals.

### Guessing passwords

Hazel Murray (Maynooth University)

A common method for attacking passwords is to guess them. If you know the distribution passwords are chosen with in a dataset then guessing will be optimal. But if we just have a sample of passwords from a given dataset we will show how these provide an insight into the distribution and allow us to optimise our guessing strategy.

### Alternating Signed Bipartite Graph Colourings

Cian O'Brien (NUI Galway)

This talk concerns a class of bipartite graphs that arise from alternating sign matrices. To an alternating sign matrix, we may associate an alternating signed bipartite graph, which has a vertex for each row and column of the matrix. Vertex $r_i$ is connected to vertex $c_j$ by a blue edge if there is a $1$ in the $(i, j)$ position of the matrix, and by a red edge if there is a $-1$. In this talk, we present results on when a given graph $G$ admits an edge colouring $c$ such that the coloured graph $G^c$ is an alternating signed bipartite graph.

### A result of Andrzej Granas (1929-2019)

Dónal O'Regan (NUI Galway)

I will present a very simple proof of Andrzej Granas's favourite result in a general setting. This proof hides a result of Jean Leray (1906-1998) and Juliusz Schauder (1899-1943) which I will also talk about.

### Ireland and its place in European Research in Mathematics Education

Maurice OReilly (DCU)

The 11th Congress of the European Society for Research in Mathematics Education (CERME11) took place in Utrecht in February 2019, two years after CERME10 was held in Croke Park, Dublin. This presentation will outline the context of this series of congresses and how they are organized. The overall contribution of Irish researchers in mathematics education to CERME10 and CERME11 will be compared with that from other participant countries.

### Groups, growth, and graphs

Tobias Rossmann (NUI Galway)

We consider the enumeration of conjugacy classes in families of groups constructed from graphs by means of classical constructions due to Baer and Tutte. Using techniques from a number of areas, we investigate associated zeta functions and use these to deduce strong uniformity results in the spirit of a famous polynomiality conjecture from 1960 due to Higman. This is joint work with Christopher Voll.

### Some Results on Completely Positive Matrices

Helena Šmigoc (UCD)

An $n\times n$ matrix $A$ is said to be completely positive if there exists a (not necessarily square) entry-wise nonnegative matrix $V$ such that $A=VV^T$. A matrix is called doubly nonnegative if it is both entry-wise nonnegative and positive semidefinite. Every completely positive matrix is clearly doubly nonnegative, however the converse is not true. In the first part of the talk, we will show that every integer $2 \times 2$ doubly nonnegative matrix $A$ is completely positive with an integer completely positive factorisation. In the second part, we will explore some completely positive factorisations connected to distance matrices.

### Genus $g$ Zhu Recursion for Vertex Operator Algebras

Mike Welby (NUI Galway)

This talk presents a Zhu reduction formula for $n$-point differentials for a vertex operator algebra (VOA) on a genus $g$ Riemann surface. Zhu reduction is a powerful technique for developing differential equations for the partition function $n$-point differentials for a VOA and differential forms on a Riemann surface. The theory so far has been developed at up to genus two, this work extends the results to all higher genera.