AnàlisiQualitative rectifiability results from Usual Square Function Estimates
Resum: In 1993 David and Semmes proved that an Ahlfors-David regular $d$-dimensional set in $\mathbb{R}^{d+1}$ is uniformly rectifiable if and only if its associated usual square function is bounded on $L^2$. We make a step towards more qualitative statements in the plane by replacing the lower bound of the AD-regularity by positive upper density. Concretely, we prove that if the set satisfies a \textit{weak flatness} condition or is a subset of a quasicircle, then usual square function bounds in $L^2$ imply rectifiability. For the proof of the latter statement we investigate the corresponding \emph{extremal measures} called \emph{Cauchy-flat} measures. This is joint work with Ben Jaye.
Stability of the solvability of the Dirichlet problem under small bi-Lipschitz domain transformations
Resum: We show that small bi-Lipschitz deformations of a Lipschitz domain (with possibly large Lipschitz constant) preserve the solvability of the Dirichlet problem for the Laplacian with boundary data in $L^p$, for the same value of $p>1$. As a consequence, for all $p>1$, we obtain the solvability of the $L^p$ Dirichlet problem for $C^1$ perturbations of convex domains, thereby unifying two fundamentally different settings in which such results were previously known: convex and $C^1$domains. This is a joint work with Linhan Li and Jinping Zhuge.
Inner 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.
Geometria GeneralitzadaSymplectic Calabi-Yau manifolds, examples, questions and applications
Resum: In the first half of the talk I will explain what symplectic Calabi-Yau manifolds are, describe some open questions about them, and give a way to construct examples. In the second half of the talk I will describe some applications of these examples to the study of minimal surfaces in 4-manifolds and knots in 3-manifolds.
Geometria Tropical Real algebraic varieties close to non-singular tropical limits
Resum: In real algebraic geometry, determining the possible topologies of real varieties is an important and often challenging problem. A breaktrough was achieved in the 70's by Viro's combinatorial patchworking that allows to transform certain combinatorial data into a real hypersurface of prescribed topology. The non-archimedean and tropical reformulation of this technique is the starting point for our combinatorial description of real varieties in any codimension "close to non-singular tropical limits". In my talk, I will try to give an introduction to these topics (and the non-archimedean geometry in the background).
Real algebraic varieties close to non-singular tropical limits
Resum: In real algebraic geometry, determining the possible topologies of real varieties is an important and often challenging problem. A breaktrough was achieved in the 70's by Viro's combinatorial patchworking that allows to transform certain combinatorial data into a real hypersurface of prescribed topology. The non-archimedean and tropical reformulation of this technique is the starting point for our combinatorial description of real varieties in any codimension "close to non-singular tropical limits". In my talk, I will try to give an introduction to these topics (and the non-archimedean geometry in the background).
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 II
Resum: In this talk, we will continue with the presentation of an ongoing project, joint with J. Claramunt (UPC), E. Gillaspy (UM) and F. Lledó (UC3M) concerning the existence and structure of KMS states on finite separated graph C*-algebras. We will introduce a new method of building KMS-states on the graph C*-algebra C*(E,C), consisting in analizing the KMS-states on a certain quotient C*-algebra of C*(E,C), the so-called tame separated graph C*-algebra O(E,C). In this setting, we may apply a result of Neshveyev on groupoid C*-algebras in order to determine all the KMS-states on the tame C*-algebra O(E,C).
KMS 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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