World Logic Day Workshop 2022
January 17, 2022
Professor
Tohoku University, Sendai, Japan.
Title of the talk: Classifying theorems: reverse mathematics and its multiple viewpoints
Abstract: The program of reverse mathematics was initiated by H. Friedman in the 1970s and developed by S. Simpson and others. Its goal is to classify mathematical theorems by comparing them with set-existence and other types of axioms in the setting of second-order arithmetic. The strength of axioms can be measured by several different viewpoints, such as consistency strength or computability strength. For example, consider the Peano existence theorem for ordinary differential equations and the Hilbert basis theorem for polynomial rings. Then the former is stronger in the sense of consistency strength while the latter is stronger in the sense of computability. Recently, the field of reverse mathematics has been widely expanding with newer perspectives. In this talk, we overview the classification studies of reverse mathematics and related topics.
Time: 11.00-12.00 UTC
Professor
University of Udine, Italy.
Title of the talk: Classifying theorems: the approach of Weihrauch reducibility
Abstract: Many mathematical theorems assert that given an object x with certain properties there exists an object y related to an x in a prescribed way. Very often y is not unique. A typical example is the intermediate value theorem: given a continuous function x:[0,1]->R with x(0)·x(1)<0 there exists y in [0,1] such that x(y)=0. It is then natural to ask how hard it is, given x (an instance of the problem) to find a correct y (a solution for x). We can view the theorems as partial multi-valued functions that can be compared in the framework of Weihrauch reducibility: the main idea is to reduce one multi-valued function to another one by a computable and uniform procedure- This approach leads to a classification of theorems which is much more fine-grained than the one provided by reverse mathematics. This area of research has seen a tremendous increase of activity in the last 15 years and the talk will present an overview of some of its achievements.
Time: 12.00-13.00 UTC
Professor
Kazan Federal University, Russia.
Title of the talk: Computable Structures and Isomorphisms
Abstract: The talk will be devoted to a survey of recent research on the algorithmic complexity of isomorphisms between computable presentations of countable algebraic structures. The first direction of studies is related to the notions of computable categoricity, relative categoricity, degrees of categoricity, algorithmic and spectral dimensions. The second direction is related to the notion of a punctual presentation of a structure and the punctual analogs of the categoricity notions from above.
Time: 13.00-14.00 UTC