We shall talk on conjugacy separability of standard arithmetic subgroups of a special orthogonal group. The same property for 3-orbifolds also will be discussed. This is a joint work with Angel Del Rio and Pavel Zalesskii.

I shall discuss prosoluble subgroups of free profinite products towards a complete characterization of them in terms of the intersections with the free factors.

The ring of differential operators on a cuspidal curve whose coordinate ring is a numerical semigroup algebra is shown to be a cocommutative and cocomplete left Hopf algebroid, which essentially means that the category of $D$-modules is closed monoidal. If the semigroup is symmetric so that the curve is Gorenstein, it is a full Hopf algebroid (admits an antipode), which means that the subcategory of those $D$-modules that are finite rank vector bundles over the curve is rigid. Based on joint work with Myriam Mahaman

Oscillator networks, including neuronal ensembles, can exhibit multiple cooperative rhythms such as chimera and cluster states. However, understanding which rhythm prevails remains challenging. In this talk, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of rotators with higher-order Fourier modes in the coupling. We show that three-cluster splay states with two distinct, coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units.

Moduli spaces and moduli stacks of bundles depend on several parameters for their construction. The moduli space and the moduli stack of vector bundles with fixed determinant both depend on the choice of an algebraic curve, a rank and a line bundle. Moduli spaces of parabolic vector bundles depend, in addition, on the choice of a set of parabolic weights, which act as stability parameters. Their geometry is known to depend non-trivially on the choice of these parameters.

I will discuss some recent forays into some counting problems for free objects. I will focus on free inverse semigroups and free regular $*$-semigroups. I will first discuss recent results joint with M. Kambites, N. Szakács, and R. Webb giving a precise rate of exponential growth of the free inverse monoid of arbitrary (finite) rank, which turns out to be given by a surprisingly complicated but algebraic number.

Calabi-Yau metrics can degenerate in a 1-parameter family by varying the complex structure, and a basic invariant is the dimension of the essential skeleton, which is an integer between 0 and n. The case of zero is the context of noncollapsed degeneration of Donaldson-Sun theory, while the case of n is the context of the SYZ conjecture.

We will discuss solutions to Problems 6.4, 6.10, 6.21, and 6.22 from the paper "Four Notions of Conjugacy for Abstract Semigroups" by João Araújo, Michael Kinyon, Janusz Konieczny, and António Malheiro.

The rank versus genus conjecture, proposed by Waldhausen in 1978, asks whether the rank of the fundamental group of a 3-manifold equals its Heegaard genus. While there are known counterexamples to this conjecture, it still holds for link exteriors in the 3-sphere. In this talk, we discuss this problem and we describe families of links for which this conjecture is true.

We deal with the question of the $\omega$-reducibility of pseudovarieties of ordered monoids. A pseudovariety of ordered monoids $\mathsf{V}$ is called $\omega$ reducible if, given a finite monoid $M$, for every inequality of pseudowords that is valid in $\mathsf{V}$, there exists an inequality of $\omega$-words that is also valid in $\mathsf{V}$ and has the same “imprint” in $M$. In other words, we investigate the $\omega$-reducibility of an “ordered version” of $\mathsf{V}$-pointlike pairs, where $\mathsf{V}$ is a pseudovariety of ordered monoids.