It is widely recognized that the equilibrium of dynamic real-world systems is essential for the maintenance of life on our planet, and that the equilibrium of the economy and financial systems is essential for ensuring better living conditions for all human beings. The study of equilibrium of real-world dynamical systems is a difficult task. One difficulty is related to the presence of randomness or uncertainty. From a mathematical point of view, relevant dynamical systems incorporating randomness or uncertainty can be described by solutions of stochastic differential equations. The main goal of this talk is to introduce the notion of an equilibrium state, or a stationary statistical solution, to these random dynamical systems. Starting from the concept of Gaussian measure, we define the Gaussian Wiener process, which allows the introduction of stochastic dynamical systems, described by solutions of stochastic differential equations. Special emphasis will be given to the Ornstein-Uhenbeck stochastic process, its associated semigroup, and the corresponding invariant measure.

According to the Vagner- Preston Theorem, every inverse semigroup is isomorphic to a subsemigroup of an appropriate symmetric inverse semigroup _{I}^{X}. Hence the study of subsemigroups of the symmetric inverse semigroup _{I}^{X}, for some set X, is of high importance. We focus essentially in two inverse semigroups: the inverse semigroup of partial isometries of the star graph _{S}^{n} and the inverse semigroup of partial isometries of the cycle graph _{C}^{n}.

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Fernanda Cipriano obtained her PhD in Mathematics from the Faculty of Science of the University of Lisbon in 2001. She is currently an Associate Professor with Habilitation in the Department of Mathematics of FCT NOVA. She is also an Integrated Researcher and a member of Executive Committee of the research Center for Mathematics and Applications, NOVA MATH.

Her main research topics are stochastic and deterministic differential equations, stochastic analysis, and optimal control with applications in fluid mechanics and financial mathematics. According to Mathscinet, she has 33 articles, the most of them published in top ranking journals.

At present, she is teaching the course Stochastic Differential Equations to PhD and Master students, as well as Stochastic Calculus applied to Mathematical Finance to Master students. At this moment she is supervising four PhD thesis (two of them are joint supervisions).

I obtained my Bachelor Degree in Mathematics at NOVA School of Science and Technology (FCT NOVA). In 2019 I started my Master Degree in Mathematics and Applications at FCT NOVA, with a specialization in Pure Mathematics, where I worked under the supervision of Professor Vítor Hugo Fernandes. I am currently a PhD student at FCT NOVA under the supervision of Professors António Malheiro and Alan Cain.