AnàlisiOn the behavior of harmonic measure density ratios at points of vanishing codimension-one density
Resum: In this talk, we study the behavior of the density ratios of harmonic measure at points of vanishing codimension-one density. In particular, we show that at $\omega$-almost every point with vanishing density, the density ratios exhibit arbitrarily large oscillations as the radius tends to $0$. As a consequence, the density ratio function fails to be almost monotone. This result holds for general open sets $\Omega\subset \mathbb{R}^{n+1}$.
GeometriaA commutativity result for Gromov-Hausdorff limits of Hilbert geometries
Resum: Due to the Klein model of hyperbolic geometry, one may identify the hyperbolic plane with the interior of a conic (let's say, a disk $D$) in the projective plane $\mathbb{RP}^2$, and the group of hyperbolic isometries with the group of projective automorphisms of $\mathbb{RP}^2$ preserving $D$. By deforming $D$ in the class of convex and bounded open subsets of $\mathbb{RP}^2$, one can still realize the pair of the deformed domain $\Omega$ and its group of projective automorphisms $\mathrm{Aut}(\Omega)$ as a metric space. Namely, there is a projectively invariant metric $d_{\Omega}$ on $\Omega$, called the Hilbert metric, so that $\mathrm{Aut}(\Omega)=\mathrm{Isom}(\Omega,d_\Omega)$. In particular, for the disk $D$, $d_D$ coincides with the hyperbolic metric. In the cases where $\mathrm{Aut}(\Omega)$ admits a discrete subgroup $\Gamma$ so that $\Omega/\Gamma$ is compact, we say $\Omega$ is divisible. This is of course the case of $D$, indeed any compact hyperbolic surface can be realized as such a quotient $D/\Gamma$.
Invariantes de tipo finito y filtración de Johnson.
Resum: En esta charla presentaré resultados recientes obtenidos en colaboración con Ricard Riba (UdG) sobre la interacción entre la teoría de invariantes finitos, la filtración de Johnson del mapping class group de una variedad y la filtración inducida sobre el conjunto de las esferas de homología entera. En particular demostraré que la esfera de Poincaré no puede ser construida, mediante escición de Heegaard, por un elemento arbitrariamente lejos en la filtración de Johnson.
Geometria TropicalScheme theory from principal localizations
Resum: I will explain how one can reconstruct all scheme theory for (semi)rings without topological spaces from general principles using just localization at elements.
Teoria d'AnellsRevisiting pure *-homomorphisms
Resum: In this talk, I will revisit joint work with Joan Bosa where we introduced the notion of pure *-homomorphisms. The focus will not be so much on the results that we obtained, but rather on an interesting consequence of our main theorem: The composition of any two pure *-homomorphisms factors, up to an appropiate equivalence relation, through a pure object.
TopologiaA proof of the Burnside $p^{\alpha}q^{\beta}$ Theorem without character theory
Resum: This Burnside Theorem (whose first proof dates back to 1904) states that if the order of a finite group G is divisible by only two different primes, then G must be solvable and its original proof involves a crucial use of character theory. In the 70’s, starting from a work of J. G. Thompson on N-groups, it was given a “completely group-theoretical” proof by Goldschmidt and Bender – then simplified by Matsuyama. In this talk I will highlight the key ingredients of this proof and I will retrace its main steps.
Derived defect basis for some infinite discrete groups
Resum: In this project we establish conditions on a discrete group G and a commutative ring spectrum R that ensure that the family of (co)induction functors between module categories over the algebras of R-valued cochains on classifying spaces, taken along a specified family of subgroups of G, is jointly conservative. This is a joint work with Juan Omar Gómez.
Higher Semiadditivity (after Hopkins, Lurie, and Harpaz)
Resum: A category is \emph{pointed} when it has a zero object, that is, when the limit and the colimit of the empty diagram exist and coincide. A pointed category is \emph{semiadditive} when it has finite biproducts, that is, when limits and colimits of diagrams indexed by finite sets exist and coincide. In the land of $\infty$-categories, these two conditions are but the first two levels of a sequence of increasingly strong ``higher semiadditivity'' conditions, which require limits and colimits indexed by $n$-finite spaces to exist and coincide, thus enabling the ``integration'' of families of maps indexed by $n$-finite spaces. In this introductory talk, I will explain these notions and their relation to categories of spans, following Harpaz.
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