I will aim at giving an idea of some aspects of Semigroup Theory, that have caught my interest along the years. Semigroups of transformations have inspired the study of different types of abstract semigroups, the most well-known being the symmetric group on a set X, i.e., the set of bijections on X with the composition operation, followed by the symmetric inverse semigroup of all injective partial maps on X. The finite case provides an important bridge between algebra and Theoretical Computer Science. I will talk about congruences and classes of semigroups of various kinds, mentioning questions raised by their study. The regular and the non-regular cases will be considered. 

The Bethe Ansatz Equations (BAE) are the cornerstone of Quantum Integrability. They are a system of coupled polynomial equations, whose order is 2 elevated to the number of degrees of freedom of the model. For an Integrable Quantum Field Theory, the BAE are thus a formidable infinite system of nonlinear transcendental equations. It was however discovered that some solutions can be expressed as spectral determinants of certain linear differential operators. Whether all solutions admit such a representation is called the ODE/IM correspondence conjecture. For one important Quantum Field Theory known as the Quantum KdV model, the ODE/IM conjecture has been almost completely proven and I will present the main ideas beyond the proof. I will also explain that the ODE/IM correspondence is a manifestation of a general very deep and still unknown duality, which is still up for grabs. 

Thanks to the development of information technologies, the last decades have seen a considerable growth of interest in the statistical theory of shape and its application to many and diverse scientific areas. Often the diagnosis of a pathology, or the description of a biological process mainly depend on the shapes present in images of cells, organs, biological systems, etc., and mathematical models which relate the main features of these shapes with the correct outcome of the diagnosis, or with the main kinetic parameters of a biological system are difficult to be retrieved.  In materials science optimization for quality control require methods of statistical shape analysis.  From the mathematical point of view, shape analysis uses a variety of mathematical tools from differential geometry, geometric measure theory, stochastic geometry, etc. Quite recently, instruments from algebraic topology have been introduced for shape description, giving rise to a new field of research called Topological Data Analysis, which is also proving useful to develop techniques of so called “explainable machine learning”, which are more transparent and interpretable of the traditional machine learning techniques.  In this talk some case studies will be presented, in which different statistical methods for random shapes are applied to solve problems coming from life sciences or from industrial applications.  Particular focus will be given to projects developed in collaboration with the European Consortium for Mathematics in Industry (ECMI). 

Mathematical modeling and simulations of the human cardiovascular system is a challenging and complex wide-range multidisciplinary research topic that has seen a remarkable growth in the last few years. This field, with a strong socio-economic impact, is rapidly progressing motivated by the fact that cardiovascular diseases are a major cause of morbidity and death worldwide. In this talk we will consider some mathematical models and simulations of the cardiovascular system and comment on their significance to yield realistic and accurate numerical results, using stable, reliable and efficient computational methods. Results on the simulation of some image-based patient-specific clinical cases will also be presented.  Working in Cardiovascular Mathematics is a very pleasant experience as one has the opportunity to join theory and practice with the final aim of improving healthcare. 

 In recent years we saw a steady progress in almost every classic NLP task yet as  the models become more flexible, versatile and expressive new frontiers arise. More minute and intricate tasks such as textual style transfer, narrative generation and better common sense reasoning are getting into focus of researchers. In the talk we overview several research topics paying special attention to computational creativity, theory of semantic information and effectiveness of long context representations. We will also discuss how modern NLP models are applied in modern financial markets.