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 $\Omega\subset \mathbb{R}^{n+1}$.
GeometriaInvariantes 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.
TopologiaDerived 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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