10:00 - 11:00

Type theoretic semantics for first order logic, Oskar Berndal

Whereas the semantics of first order logic are well-understood, many questions remain regarding the semantics of type theory. There is not even an established and unified notion of what precisely is a type theory.

In a recent work by Uemura, a general notion of type theories is proposed together with semantics for these type theories. The aim of this work is to present a type theory within this framework such that its semantics recovers the semantics for first order logic.

The main obstacle is the mismatch between what one takes as a morphism in the semantics: In first order logic one takes the functional relations whereas in type theory one essentially takes its terms. In order to bridge this gap we introduce terms for definite descriptions to the type theory.

11.00-12.00

The spectral theorem for normal operators and applications, Marko Kocic

This paper aims to present the spectral theorem for normal operators and then describe the bilateral shift operator on a sequence space using the spectral theorem. To do so, the notion of Hilbert spaces needs to be defined first, and then we study bounded operators on a Hilbert space. The structures of bounded operators which are presented are the adjoint, the inverse, and the spectrum of bounded operators.

Further the spectral theorem for self-adjoint operators is presented, which then gets extended to the spectral theorem for normal operators. Finally we apply the spectral theorem for normal operators to the bilateral shift operator.