Tuesday, 9 May 2023
14:30, Aula E (Ca’ Vignal 1).
Ingo Blechschmidt (Universität Augsburg)
Synthetic scheme theory: a simpler framework for algebraic geometry
Note: This talk has been postponed to a later date.
Abstract: The modern framework for algebraic geometry put forward by Grothendieck and his school has been enormously successful, providing the basis for many deep cornerstone results of the subject. Not least, we owe to this basis the proof of Fermat’s Last Theorem. Despite these successes, Grothendieck himself expressed discontent with his framework, and especially in recent years, concerns about the limitations and technical difficulties of the modern theory of algebraic geometry arose.1
With the benefit of hindsight, now that the mathematical content of the then-revolutionary new approach to algebraic geometry is well-understood, we propose an update to the foundations of algebraic geometry, called synthetic scheme theory, built on three postulates. These postulates capture essential geometric properties and allow us to reason constructively, avoiding the use of transfinite principles and other highly abstract concepts. Our hope is that this approach will allow for a clearer and more intuitive expression of the central notions and insights of algebraic geometry, requiring less technical machinery, will facilitate integrated developments, and promote computer-assisted proofs in the subject.
Crucially, our approach rests on the greater axiomatic freedom provided by constructive mathematics: The three postulates are inconsistent with classical logic.
16:30, Aula Gino Tessari (Ca’ Vignal 2).
Hans Leiß (Universität München)
_Algebraic Representation of the Fixed-Point Closure of *-continuous Kleene Algebras_
Abstract: The theorem of Chomsky and Schützenberger says that each context-free language L over a finite alphabet X is the image of the intersection of a regular language R over X+Y and the context-free Dyck-language D over X+Y of strings with balanced brackets from Y under the bracket-erasing homomorphism from the monoid of strings over X+Y to that of strings over X.
Within Hopkins’ algebraization of formal language theory we show that the algebra C(X) of context-free languages over X has an isomorphic copy in a suitable tensor product of the algebra R(X) of regular languages over X with a quotient of R(Y) by a congruence describing bracket matches and mismatches. It follows that all context-free languages over X can be defined by regular(!) expressions over X+Y where the letters of X commute with the brackets of Y, thereby providing a substitute for a fixed-point-operator.
This generalizes the theorem of Chomsky and Schützenberger and leads to an algebraic representation of the fixed-point closure of *-continuous Kleene algebras.
Perhaps the most pressing are that the currently framework heavily relies on the axiom of choice and other classical principles–even though high-level reasoning in algebraic geometry is often constructive–and that the framework references and requires gadgets of little geometric significance, such as injective or flabby resolutions. These foundational issues block integrated developments of algorithms in algebraic geometry, hindering us from extracting certified algorithms directly from proofs, and render computer formalization of algebraic geometry particularly challenging.↩︎