AnàlisiSolving inverse spectral problems with Schur's algorithm for bounded analytic functions
Resum: The half-line Dirac operators with $L^2$-potentials can be characterized by their spectral data. It is known that the spectral correspondence is a homeomorphism: close potentials give rise to close spectral data and vice versa. We prove the first explicit two-sided uniform estimate related to this continuity in the general $L^2$-case. The proof is based on an exact solution of the inverse spectral problem for Dirac operators with $\delta$-interactions on a half-lattice in terms of the Schur's algorithm for analytic functions. Joint work with Pavel Gubkin (St. Petersburg).
Col·loquiLa grandesa dels punts petits: una excursió a la teoria aritmètica de l'equidistribució
Resum: L'alçada d'un punt algebraic mesura la seva complexitat aritmètica és a dir, la complexitat de les seves coordenades. Les altures són una eina fonamental en la teoria dels nombres, i tenen un paper clau en resultats cèlebres com el teorema de Faltings, que va establir la conjectura de Mordell de 1922 que les corbes algebraiques de gènere almenys dos només tenen un nombre finit de punts racionals. Entre les seves moltes propietats, destaca l'equidistribució de punts petits. Aquest fenomen no és només bonic per si mateix, sinó que també té aplicacions a problemes sobre interseccions improbables com la conjectura de Bogomolov. En aquesta xerrada explicaré què significa l'equidistribució de punts petits (inclòs l'exemple clàssic d'arrels de la unitat sobre la circumferència unitat), esbossaré per què es compleix en diversos contextos i revisaré alguns resultats recents (i no tan recents).
Models Estocàstics i DeterministesWell-posedness of some dissipative semilinear SPDEs
Resum: I shall report on some recent results on existence and uniqueness of mild solutions to a class of semilinear stochastic evolution equations with additive noise. The linear part of the drift term is the generator of a compact semigroup of contractions, while the nonlinear part is only assumed to be the superposition operator associated to a decreasing function. Generalizations to the case where the semigroup of contractions is not compact will also be discussed.
Teoria d'AnellsLengths, ranks, and Gabriel dimension
Resum: In the first part of the talk we will recall a few classical results about length functions in Grothendieck categories, proved by Peter Vámos in the sixties. More concretely, we will show that there is a bijective correspondence between irreducible discrete length functions and the points of the so-called atom spectrum, recently introduced by Kanda, and we will use this to classify all length functions in categories with Gabriel dimension.
The Cuntz semigroup of rings with stable rank one
Resum: The Cuntz semigroup of a C*-algebra A with stable rank one enjoys a key structural property due to Coward, Elliott and Ivanescu: the order relation in Cu(A) among countably generated Hilbert A-modules is simply the order of isometric embeddings: $[X] \leq [Y]$ if and only if $X$ is isomorphic to a submodule of $Y$, and $[X] = [Y]$ iff $X$ is isomorphic to $Y$.
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