Mathematical Research Seminar - Archive

Bi-coset graphs, which were originally dened by Du and Xu in 2000, completely characterize all semi-transitive graphs and can be viewed as generalizations of Haar (di-)graphs (sometimes called bi-Cayley (di-)graphs). In this talk, we will nd strong constraints on the automorphism groups of bi-coset graphs and digraphs.

Come and explore the infinite possibilities of mathematics with us!

For any scientific endeavour, the discussion of ethics tends to fall into two categories. One is ethics in the practice of the science; the other is ethics for academics in their own conduct and as regards publication and dissemination of their work. Many universities have well thought through policies and training for their students and staff.

But, what about ethics for mathematicians? Gradually, the world is realising that mathematicians rule the world – mathematics is essential in almost every walk of modern life. Yet, often, mathematics is omitted from discussions on ethical issues. Frameworks for providing training in ethics to our mathematics students are often lacking, particularly regarding the practice of mathematics outside academia. On the other hand, there has been some development in ethical guidance for academic publishing of mathematics.

I have been a member of the Ethics Committee of the European Mathematical Society since 2018. This has opened my eyes to some issues, which I would like to share, together with some thoughts for the future of the ethical agenda.

We kindly request your presence at this extraordinary event, as your attendance will undoubtedly contribute to the vibrancy and richness of the ensuing discussions.

A signed graph can be described as an undirected graph whose edges are assigned a + or − sign; the edges are said to be positive or negative according to whether they have + or − sign, respectively.

Signed graphs have been introduced by Harary to develop and formalize a psychological theory proposed by Heider for analysing the network of relations in a group of people. In particular, balanced signed
graphs are privileged worlds where stability reigns, thanks to two disjoint subgraphs each one internally positive, while externally negative. Balance is actually hard to find in ordinary life; several relaxations are studied in
the literature in order to better suit reality. In this spirit, we introduce two novel discrete structures: fruitful coverings and antagonism colorings.

Based on a joint work with Andrea Vietri and Manuela Montangero.

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I will introduce the notion of slice regular function on a (symmetric) domain of the quaternions and will give results on the structure of the space of slice regular functions, focusing in particular on the *-product and the structure of the zeroes of the elements of this space.
Then I will present the analogous of the exponential operator, together with sine and cosine, and give some (unexpected) statements on the behaviour of these operators. (JW with A. Altavilla)

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Generalised pp-waves with purely axial and purely tensor torsion of parallel Ricci curvature have their particular physical interpretation. The spinor field which completely determines the complexified curvatures of those spacetimes, satisfies the massless Dirac equation.

We consider the massless Dirac operator on a $3$-manifold which describes a single massless neutrino living in a 3-dimensional compact universe. The eigenvalues of the massless Dirac operator are interpreted to be the energy levels of that massless particle.

We break the spectral symmetry  of the massless Dirac operator on a $3$-torus using the perturbation of the Euclidean metric.

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Syzygies are objects invented and utilized by David Hilbert in 1890 to study relations among polynomial equations, and have played a big role in the development of modern algebraic geometry. The quest to understand patterns of syzygies is both challenging and interesting, and sometimes reveals unexpected connections to other branches of mathematics. In this talk, I will describe recent joint work with David Eisenbud on monomials with linear syzygies.  It turns out that certain fractal structures that appear in many contexts, from game theory to number theory, play a significant role in building optimal examples.

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Let $S_n(\mathbb{F}_2)$ be the set of all $n\times n$ symmetric matrices with coefficients from the binary field $\mathbb{F}_2=\{0,1\}$, and let $SGL_n(\mathbb{F}_2)$ be the subset of all invertible matrices.
Let $\tilde{\Gamma}_n$ be the graph with the vertex set $S_n(\mathbb{F}_2)$, where two matrices $A, B \in S_n (\mathbb{F}_2)$ form an edge if and only if $\text{rank}(A-B)=1$.
Let $\Gamma_n$ be the subgraph in $\tilde{\Gamma}_n$, which is induced by the set $SGL_n(\mathbb{F}_2)$. If $n=3$, $\Gamma_n$ is the Coxeter graph.
It is well-known that is a distance function  on  $\tilde{\Gamma}_n$ is given by
$$d(A,B) =
  \begin{cases}
    \text{rank}(A-B),       & \quad \text{if } A-B  \text{ is nonalternate or zero,}\\
    \text{rank}(A-B)+1,   & \quad \text{if } A-B \text{ is alternate and nonzero.}
  \end{cases}
$$
Even the Coxeter graph shows that the distance in $\Gamma _n$ must be different.
The main goal is to describe the distance function on $\Gamma_n$.
Joint work with Marko Orel.

We look forward to sharing the passion for math with you!