Fall 2020 Seminars
Time: Monday, December 7, 2–3 PM (CST, UTC -6)
Speaker: Jason Parker (Brandon University)
Title: Isotropy Groups of Quasi-Equational Theories
Abstract: In , my PhD supervisors (Pieter Hofstra and Philip Scott) and I studied the new topos-theoretic phenomenon of isotropy (as introduced in ) in the context of single-sorted algebraic theories, and we gave a logical/syntactic characterization of the isotropy group of any such theory, thereby showing that it encodes a notion of inner automorphism or conjugation for the theory. In the present talk, I will summarize the results of my recent PhD thesis, in which I build on this earlier work by studying the isotropy groups of (multi-sorted) quasi-equational theories (also known as essentially algebraic, cartesian, or finite limit theories). In particular, I will show how to give a logical/syntactic characterization of the isotropy group of any such theory, and that it encodes a notion of inner automorphism or conjugation for the theory. I will also describe how I have used this characterization to exactly characterize the ‘inner automorphisms’ for several different examples of quasi-equational theories, most notably the theory of strict monoidal categories and the theory of presheaves valued in a category of models. In particular, the latter example provides a characterization of the (covariant) isotropy group of a category of set-valued presheaves, which had been an open question in the theory of categorical isotropy.
 J. Funk, P. Hofstra, B. Steinberg. Isotropy and crossed toposes. Theory and Applications of Categories 26, 660–709, 2012.
 P. Hofstra, J. Parker, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Computer Science 341, 201–217, 2018.
Time: Monday, September 28, 2–3 PM (CDT, UTC -5)
Speaker: Bogdan Chornomaz
Title: Introduction to Möbius functions
Abstract: We will give a brief introduction to Möbius functions on lattices and discuss their basic properties. We will also prove that lattices with nonvanishing Möbius function are SSP and that geometric lattices have a nonvanishing (moreover, alternating) Möbius function.
Time: Monday, September 21, 2–3 PM (CDT, UTC -5)
Speaker: Bogdan Chornomaz
Title: SSP ?= RC
Abstract: We will talk about SSP ?= RC conjecture, which states that a finite lattice satisfies an analogue of Sauer-Shelah-Perles lemma iff it is relatively complemented. In particular, we discuss one strategy of attacking the conjecture using certain colored bipartite graphs. We will hit the wall by constructing a “counterexample” in the language of those graphs. Then we will try to lift it to the counterexample to the general conjecture using some “pumping” constructions and will hit the wall again.
Time: Monday, September 14, 2–3 PM (CDT, UTC -5)
Speaker: Adam Prenosil
Title: Four-valued logics of truth, non-falsity, and material equivalence
Abstract: the purpose of this talk is to demonstrate how to work with quasivarieties (universal Horn classes) in a relational signature which consists of more than one predicate symbol. While universal algebraists work with the binary equality predicate, and algebraic logicians work with the unary truth predicate, the four-valued Belnap–Dunn logic provides a natural setting where two unary predicates arise (namely the truth and non-falsity predicates), as well as an equality predicate. We show how to axiomatize the logic of truth and non-falsity, as well as the logic of truth and equality, determined by the four-valued algebraic semantics of Belnap and Dunn.