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.)

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.)

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.

Resum: Kummer lines extend the classical notion of lines of curvature to the geometry of line congruences. Their defining directions are governed by binary differential equations, providing a natural connection between differential geometry and dynamical systems. In this talk, we introduce the geometric meaning of Kummer lines, their associated fundamental forms, and their relationship with classical curvature lines. We also explain how the Lie–Cartan lifting translates the study of these equations into the analysis of a vector field. Our main focus is the emergence of limit cycles from families of periodic leaves on the sphere and the Clifford torus. Using first- and second-order averaging, we relate the existence and hyperbolicity of these cycles to simple zeros of averaged displacement functions. We consider perturbations of both the underlying surface and the director field, obtaining a hyperbolic equatorial cycle on the sphere and arbitrarily many hyperbolic cycles on suitable deformations of the Clifford torus.

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.

Resum: Rigidity results for group operator algebras have unsurprisingly followed diverging paths in the settings of C$^*$-algebras and von Neumann algebras. Famously, group von Neumann algebras of icc amenable groups have no memory of the group, while C$^*$-superrigidity results have managed to tackle different classes of amenable groups. Although it is natural to expect there should exist a broader variety of non-amenable C$^*$-superrigid groups than W$^*$-superrigid, it was previously unknown whether there exist non-amenable groups that are C$^*$-superrigid but not W$^*$-superrigid. In joint work with Juan Felipe Ariza Mejía and Ionut Chifan, we bridge this gap by constructing a continuum of non-amenable groups exhibiting this phenomenon.

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.

Resum: Decomposition spaces, also called 2-Segal spaces, are simplicial spaces that generalise (nerves of) posets, monoids, categories, and operads, and they are supposed to be the most general setting supporting the construction of incidence (co)algebras and the Möbius inversion principle. The aim of the talk is to motivate the theory and explain the basic ideas. This is meant as background material for topics that may come up later in the seminar, in particular a talk at some point on quasi\-symmetric functions, and maybe talks on higher Segal spaces.

(This is old work with Imma Gálvez and Andy Tonks; the recommended entry point is the survey "Decomposition spaces in Combinatorics".)

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.

Resum: Hyperbolic surfaces have historically been played as drums. In this talk, we are going to play a hyperbolic surface as a marimba: we assign notes to a finite collection of closed geodesics and, following a random geodesic on the surface, we play a note when the corresponding closed curve is hit. We investigate to what extent the melodies produced determine (geometric properties of) the hyperbolic surface. Joint work with Juan Souto.

Resum: The aim of this talk is twofold: first, to provide an overview of various Moser-type arguments that have appeared in the literature, particularly in Lie theory and Poisson geometry, and to highlight their significance and impact; and second, to discuss the Moser trick more specifically in the setting of regular foliations, with a view to a possible extension beyond this setting.

Link to connect: https://teams.microsoft.com/meet/3377224720319840?p=bCPVW42jPuZQMAIG64
Meeting ID: 337 722 472 031 984 0
Passcode: TX7sw93T