S4-algebras (or interior algebras) provide semantics for the well-known modal logic S4, and there is a syntactic criterion characterizing when a variety of S4-algebras is locally finite in terms of its "depth" (a classical result of Segerberg and Maksimova). Since the logic MS4 (monadic S4) axiomatizes the one-variable fragment of predicate S4, it is natural to try to generalize the Segerberg--Maksimova theorem to this setting. We discuss several results in this direction. We establish that this theorem naturally extends to a family of subvarieties of MS4 containing, in particular, S4_u (S4 with a universal modality). On the other hand, we provide a translation of varieties of S5_2-algebras into varieties of MS4-algebras of depth 2 which preserves and reflects local finiteness, demonstrating that the problem of characterizing locally finite varieties of MS4-algebras is at least as hard as the corresponding problem for S5_2 (the bimodal logic of two unrelated S5 modalities), a wide-open problem. Finally, we discuss another natural subvariety of MS4 obtained by asserting a monadic analogue of Casari's predicate formula; this subvariety plays a role in obtaining a faithful provability interpretation of monadic intuitionistic predicate logic, and is expected to have a more manageable characterization of local finiteness.

Local finiteness in varieties of MS4-algebras

Feb. 27, 2024 2:30pm (MATH 3…

Lie Theory

Farid Aliniaeifard (UBC)

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We define vertex-colourings for edge-coloured digraphs, which unify the theory of $P$-partitions and proper vertex-colourings of graphs. Furthermore, we use our vertex-colourings to define generalized chromatic functions, which merge the chromatic symmetric and quasisymmetric functions and generating functions of $P$-partitions. We also discuss the relations between generalized chromatic functions, Schur functions in noncommuting variables, and the well-known Stanley-Stembridge (3+1)-free conjecture.

Generalized Chromatic Functions Sponsored by the Meyer Fund

Feb. 27, 2024 3:30pm (MATH 3…

Topology

Courtney Hauf

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This is look at operads and the definition of a permutative category.

Operads and Permutative Categories

Feb. 27, 2024 4:50pm (MATH 3…

Grad Student Seminar

Jon Kim (University of Colorado)

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Hodge theory provides us with powerful tools for studying the cohomology of smooth manifolds through PDEs. Surprisingly, it has deep connections to complex geometry in studying bidgree differential forms on complex manifolds, specifically complex projective varieties. In this talk, I will introduce Hodge theory on complex manifolds and discuss how these connections from PDEs to complex geometry are made. We will conclude by giving a short proof of the Hodge decomposition via the existence of a Green's operator and showcase some useful consequences.