Since the seminal work of Nambooripad (among others), people have been interested in representing semigroups by categories. This led to the well-known equivalence between inverse semigroups and inductive groupoids (the ESN theorem), or the less-known equivalences between the category of regular semigroups, the category of regular inductive groupoids, and the category of cross-connections.

The variational principle states that the topological entropy of a compact dynamical system is a supremum of measure-theoretic entropies of invariant measures supported on this system. Therefore, one may ask whether we can get a similar formula for the topological entropy of a dynamical system restricted to the level sets, which are usually not compact.

Mirror symmetry is a somewhat mysterious phenomenon that relates the geometry of two distinct Calabi-Yau manifolds. In the realm of trying to understand this relationship an equation for a connection on a line bundle in a Kahler manifold appeared. This is commonly called the deformed Hermitian-Yang-Mills equation and I will explain what it is and the current joint work with Benoit Charbonneau and Rosa Sena-Dias which explicitly solves this equation on a specific setting. This helps in understanding the problem of the existence of solutions and ruling out possible stability conditions.

I will present some results that were obtained in collaboration with Csaba Schneider (Universidade Federal de Minas Gerais) concerning groups that have a solvable subgroup of prime-power index. Under weak conditions such groups are solvable and, when they are not, the index of their solvable radical is asymptotically small. 
 

Keywords: Solvability and Fermat primes

In this talk, I will discuss recent joint work with Alexeev-Engel-Schaffler that provides explicit, combinatorial descriptions of stable degenerations of numerically polarized Enriques surfaces of degree 2. We show that this can be identified with a semitoroidal compactification of the Hodge-theoretic period domain, leveraging Alexeev-Engel’s seminal work on compactifications of moduli spaces of K3 surfaces via integral-affine geometry.

I will explain how to express the space of almost complex structures J(M) on a six-manifold M as the quotient of a space of sections of a sphere bundle over M by an S^1 action. This leads to a computation of the rational homotopy type of J(M) in most cases, as well as a simple description of the homotopy type in others, such as when M is the six-sphere. In dimension 6, two orthogonal, almost complex structures generically intersect at a finite number of points inside twistor space. I will give a formula for the intersection number and discuss some applications.

Given a group $G$ and a partial factor set $\sigma $ of $G,$ we introduce the twisted partial group algebra $\kappa_{par}^{\sigma}G,$ which governs the partial projective $\sigma $-representations of $G$ into algebras over a field $\kappa .$ Using the relation between partial projective representations and twisted partial actions we endow $\kappa_{par}^{\sigma}G$   with the structure of a crossed product by a twisted partial action of $G$ on a commutative subalgebra of $\kappa_{par}^{\sigma}G.$   Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral se

The simple positive energy representations for an affine vertex algebras associated to higher rank Lie algebras of type $A$ and admissible levels whose top spaces are induced modules of cuspidal representation of type $A$ are extensively studied. Note that all simple weight modules with finite-dimensional weight spaces are classified. However, the general classification of simple admissible weight modules of type $A$ is not know. In this talk, we explain the recent developments in the general open problem.

Finite automata are classical machines that recognise regular languages. However, many other devices characterising this class are known. These alternative models may represent regular languages in a significantly more concise way than classical recognisers. In this talk I will present an overview of old and recent results on formal systems for regular languages from a descriptional complexity point of view, that is by considering the relationships between the sizes of such devices.
 

The recurrence coefficients of a generalised symmetric Freud weight are positive solutions of a discrete equation in the discrete Painlevé I hierarchy. They also satisfy a coupled system of nonlinear differential equations. Such orthogonality weights also arise in the context of Hermitian matrix models and random symmetric matrix ensembles. In this talk I will report on properties of such special solutions of this integrable system of equations in the dP-I hierarchy, explaining the connections to other areas of mathematics.