The workshop is organized in honor of Marie-Françoise Bidaut-Véron and Laurent Véron for their important contributions in partial differential equations and in celebration of their 70th birthday. The workshop will provide an environment to exchange and discuss recent developments on singular problems associated to quasilinear equations and related topics.

More info here.

We will continue online on Thursday, May 14, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Soichiro Fujii

A unified framework for notions of algebraic theory

Abstract:
Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads. These variants of universal algebra are called notions of algebraic theory. In this talk, we present a unified framework for them. The key observation is that each notion of algebraic theory can be identified with a monoidal category, in such a way that algebraic theories correspond to monoid objects therein. To incorporate semantics, we introduce a categorical structure called metamodel, which formalises a definition of models of algebraic theories. We also define morphisms between notions of algebraic theory, which are a monoidal version of profunctors. Every strong monoidal functor gives rise to an adjoint pair of such morphisms, and provides a uniform method to establish isomorphisms between categories of models in different notions of algebraic theory. A general structuresemantics adjointness result and a double categorical universal property of categories of models are also shown.

We will continue online on Thursday, May 7, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Christian Espindola

Topos-theoretic completeness theorems

Abstract:
In this talk we will delve into the background details of the previous talk by introducing syntactic proof systems and their categorical semantics, including the construction of syntactic categories and $\kappa$-classifying toposes, as well as the role of certain properties of Grothendieck topologies and Kripke-Joyal semantics. We will then study some topos-theoretic completeness theorems for certain infinitary logics that generalize results of Deligne and Joyal.

We will continue online on Thursday, April 30, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Christian Espindola

A topos theoretic proof of Shelah’s eventual categoricity conjecture

Abstract:
Assuming the Generalized Continuum Hypothesis (GCH), we give a topos-theoretic proof of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC) with amalgamation. The proof is based on infinitary generalizations of Deligne and Joyal's completeness theorems for certain infinitary intuitionistic logics, by means of which it is possible to attack the question of categoricity in a topos-theoretic way. Using recent results on the Scott adjunction developed by Henry and Di Liberti, we prove under GCH that an AEC with amalgamation which is categorical in a pair of cardinals is also categorical in all cardinals in between. Under some extra natural assumptions on the AEC, we also deduce a new downward categoricity transfer. We also explain how these methods can be adapted to shed some light on the categoricity spectrum of more general accessible categories.