GeometriaGeometría de los espacios de foliaciones holomorfas en el plano proyectivo complejo
Resum: Estudiamos el espacio de foliaciones holomorfas de un grado dado del plano proyectivo complejo bajo la acción de su grupo de automorfismos: órbitas, degeneraciones y cuencas de atracción. El cerrado de Zariski de foliaciones con simetrías infinitesimales contiene las foliaciones homogéneas y las dos únicas órbitas de dimensión mínima, seis, para las que damos criterios de degeneración. Mediante la aplicación de Gauss estudiamos después las foliaciones convexas, las de web dual plano y las de Galois. Para las homogéneas, estas propiedades se traducen en condiciones sobre una aplicación racional de la recta proyectiva: ser críticamente fija, satisfacer condiciones locales en sus puntos críticos no fijos y ser de Galois (clasificación de Klein). Como aplicación, clasificamos las foliaciones convexas reducidas en grado bajo y proponemos conjeturas sobre índices de Camacho–Sad.
Geometria TropicalThe semiring of global sections is global
Resum: During last year we studied the construction of the Sp(ectrum) of a semiring A, as a locally semiringed space $Sp(A) = (X,\mathcal{O}_X)$. One may take the global sections $\Gamma A=\mathcal{O}_X(X)$ to recover a semiring, with a canonical map $\iota \colon A \to \Gamma A$. We say that $A$ is global if $\iota$ is an isomorphism (for example, all rings are global). In this talk I will do a nonempty subset of: (1) give examples of global and non-global semirings, (2) show that hard idempotent semirings are global, and (3) prove that \Gamma A is global for any semiring A. (Disclaimer on use of AI: (3) and some examples in (1) are digestions of arguments found by Claude.)
Sistemes DinàmicsHomological data on the periodic structure of self-maps on wedge sums
Resum: The Lefschetz fixed point theory and its generalizations are important tools in the study of the periodic structure of self-maps on spaces with ice structure, e.g. compact polyhedra. In this talk we use this approach to study the set of periods for continuous self-maps on a wedge sum of topological manifolds, which exhibits a particular combinatorial structure. We explicitly compute the Lefschetz numbers, the Dold coefficients and consider the corresponding set of algebraic periods. Moreover, we study the special case of maps on a wedge sum of tori, and highlight the homological obstructions that arise in defining these maps. We present a characterization of all the possible continuous maps (up to homology) on the wedge sum of tori.
Teoria d'AnellsOperadors diferencials sobre anells d'invariants de grups finits
Resum: En aquesta xerrada donarem una introducció a la teoria d'anells d'operadors diferencials sobre l'anell de polinomis i alguna de les seves aplicacions a l'àlgebra commutativa i la teoria de singularitats. A continuació veurem quines parts de la teoria es poden estendre al cas d'anells d'invariants de grups finits.
TopologiaHigher Semiadditivity, part 2
Resum: Abstract (for part 1 and part 2): A category is 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 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 ∞-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 these two introductory talks, I will explain these notions and their relation to categories of spans, following Harpaz.
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