AnàlisiInner factors of Dirichlet space functions
Resum: Every function in the Dirichlet space on the unit disc has an inner/outer factorization. Which inner functions occur in this way? For Blaschke products, this is the old question of which subsets of the disc are zero sets for the Dirichlet space.
Structure of one-dimensional metric currents
Resum: The goal of the talk is to give an overview of the metric theory of currents by Ambrosio-Kirchheim, together with some recent progress. Metric currents are a generalization to the metric setting of classical currents. Classical currents are the natural generalization of oriented submanifolds, as distributions play the same role for functions. We present a structure result for metric 1-currents as superposition of 1-rectifiable sets in complete and separable metric spaces, which generalizes a previous result by Schioppa. This is based on an approximation result of metric 1-currents with normal 1-currents and a more refined analysis in the Banach space setting. This is joint work with D. Bate, J. Takáč, P. Valentine, and P. Wald (Warwick).
GeometriaSymplectic Tiling Billiards on Complete Affine Tori
Resum: Symplectic tiling billiards is a game played on a tiling of the plane that only relies on a notion of parallelism and lends itself to study under the more general non-Euclidean affine structures of the torus. In this talk we will address the classification of affine structures on the torus and how to use them to play symplectic tiling billiards. Demonstrations of the game will be provided in real time along with arguments to explain observed behavior of orbits. This is joint with with Fabian Lander.
Models Estocàstics i DeterministesConvergence in law for quasi-linear SPDEs
Resum: We consider the quasi-linear stochastic wave and heat equations in $\mathbb{R}^d$ with $d\in \{1,2,3\}$ and $d\geq 1$, respectively, and perturbed by an additive Gaussian noise which is white in time and has a homogeneous spatial correlation with spectral measure $\mu_n$. We allow the Fourier transform of $\mu_n$ to be a genuine distribution. Let $u^n$ be the mild solution to these equations. We provide sufficient conditions on the measures $\mu_n$ and the initial data to ensure that $u^n$ converges in law, in the space of continuous functions, to the solution of our equations driven by a noise with spectral measure $\mu$, where $\mu_n\to\mu$ in some sense. We apply our main result to various types of noises, such as the anisotropic fractional noise. We also show that we cover existing results in the literature, such as the case of Riesz kernels and the fractional noise with $d=1$.
Teoria d'AnellsKMS states on separated graph C*-algebras I
Resum: The study of KMS states on finite graph C*-algebras was initiated in 2013 by Kajiwara and Watatani, and simultaneously by an Huef, Laca, Raeburn and Sims. While the former focused on graphs with sinks and sources, the latter initially concentrated on strongly connected graphs. Subsequently, in 2014 and 2015, an Huef et al. fully characterized the simplex of KMS states for arbitrary finite graphs, including the reducible case.
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