Geometria TropicalCàlculs explícits en semianells
Resum: El desenvolupament de la teoria d'esquemes sobre semianells ensopega amb alguns obstacles que la fan diferent de la teoria sobre els anells. La construcció de contraexemples (i en alguns casos les demostracions de teoremes) requereixen poder manipular elements i ideals en semianells amb algunes propietats "universals".
Models Estocàstics i DeterministesLong-time behavior and numerical analysis of SDEs with singular and dissipative coefficients
Resum: In this talk we will discuss a particular class of stochastic differential equations (SDEs) driven by a fractional Brownian motion (fBm) with small Hurst parameter. To be more specific, we study the long-time behavior of equations with both a singular and a dissipative drift, motivated by the last developments in regularization by noise theory. This field has shown a huge progress since the introduction of the Stochastic Sewing Lemma in 2020 by Khoa Lê. The main idea of regularization by noise is to use the highly oscillatory behavior of the fBm to prevent the solution from spending too much time in the regions where the drift may be ill-defined, thus considerably extending the theory of well-posedness to a broader class of SDEs. Nevertheless, the long-time behavior for these kind of equations is much less understood, mainly due to the non-Markovian nature of the noise. In his 2003 article, Martin Hairer used the theory of stochastic dynamical systems (SDS) to overcome this problematic when the drift is smooth and dissipative. His idea is to couple the solution with the past history of the noise, thus obtaining a lifted dynamic that is Markovian over an infinite dimensional space. In recent works, these two frameworks (regularization by noise theory and the theory of SDS) have been merged successfully, hence showing the existence of a unique invariant measure for SDEs with singular and dissipative coefficients driven by the fBm. However, other important features related to the invariant measure are less understood, such as the existence of a density and/or numerical approximations. In this talk we will discuss existence and Gaussian bounds verified for such density. Furthermore, we will introduce a tamed-Euler scheme to approximate the solutions in the long-time, as well as the invariant measure itself.
Teoria d'AnellsGlobal and local triviality for extensions of ample groupoids
Resum: Using the duality between the categories of (boolean) inverse semigroups and ample groupoids, we state the (local or global) triviality of the central term of any extension of ample groupoids in terms of (continuous) twisted crossed products.
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