I’m young and there is not much here yet… but hopefully there will be more soon!
Given a finite commutative monoid , we show that submonoids of — where is equipped with the max operation — may be enumerated via the transfer matrix method. When is also idempotent, we show that there are finitely many integers and rational numbers (only depending on ) such that the number of submonoids of is . This answers a question of Knuth regarding ternary (and higher order) max-closed relations, and has applications to the enumeration of saturated transfer systems in equivariant infinite loop space theory.
Transfer systems are relatively simple combinatorial objects with deep connections to the theory of operads, an equivariant analogue of operads. In this talk, we informally present this homotopy-theoretic story so as to motivate the combinatorial study of saturated transfer systems, which in good cases classify linear isometries operads. We then give a novel method for enumerating saturated transfer systems on cylindrical modular lattices, for instance the subgroup lattices of finite cyclic groups, and tightly characterize the asymptotic behavior in the height of the cylinder. This talk presents joint work with several Reed College students along with Angélica Osorno and Kyle Ormsby.
Sometimes spaces seem to come with a natural sense of direction—consider a manifold equipped with a vector field, or a spacetime manifold, or a configuration space in concurrency theory, or just the ordinary interval. Directed topological spaces are models for this kind of geometry. The primary goal of this talk is to introduce the two most popular models—d-spaces and streams—and discuss the sense in which they are equivalent. We will then introduce the core ideas of directed homotopy theory.
This talk is about a very playful sequence, originally studied by John Conway, whose first terms are displayed in the title. We’ll define the sequence, prove some of its basic properties, and then discuss Conway’s “cosmological theorem,” which determines the limiting behavior of the sequence. A surprising role is played by a particular endomorphism of the free monoid on 92 generators.
We introduce Lawvere theories and then discuss Bohmann-Szymik’s work on their K-theory. From the K-theoretic perspective, Lawvere theories are a convenient generalization of rings in which Quillen’s +=Q theorem still holds, enabling a more computationally tractable K-theory than more general settings.
We give a high-level intuition for why spaces with highly-coherent homotopy-commutative and associative multiplications should model infinite loop spaces. We discuss two machines for identifying such space spaces: May’s operads and Segal’s gamma spaces. We conclude with a discussion of categories of operators and the May-Thomason uniqueness theorem for infinite loop space machines.
This talk will elaborate on the connectivity arguments that were swept under the rug during pre-talk 1. In particular, we will focus on tools and techniques coming from discrete Morse theory. After introducing the basics of discrete Morse theory, we will use it to prove the Solomon-Tits Theorem.
We motivate and define the classifying space of a category and discuss Quillen’s Theorems A and B, which give powerful tools for computing homotopical information about a functor.
A standard approach to proving meta-theoretic properties of a proof system or type theory is to first define a stronger dependent predicate R—often called a logical relation—by induction on syntax, and prove the desired property by showing that every well-formed term satisfies this stronger property. In categorical semantics, this technique looks like an Artin gluing of the syntactic category along a hom-functor. This talk will introduce this technique, at least in the special case of so-called “sconing,” and utilize it for a categorical proof of the existence property of intuitionistic logic.
Our standard intuition for abstract topology is geometric: the notion of open set axiomatized the properties of open balls in metric spaces. Maybe surprisingly, there is an equally applicable, and in some sense formally dual, logical interpretation of topological spaces. In this interpretation, open sets behave like “propositions subject to finite proof.” This talk—which assumes no knowledge of mathematical logic—will explore this perspective, in particular as it relates to point-free topology and sheaves.
You might have heard of “categorical semantics” or “internal logics,” or of the relationship between topoi and intuitionistic logic, perhaps in Jin and Carmine’s pizza seminar talks from last semester. You also might not have! Regardless, this talk aims to explain the big picture of what these terms mean, how they relate, and why non-logicians might care.
Girard’s paradox is an analogue of Russell’s paradox which demonstrates the inconsistency of a sufficiently rich type theory with a “type of all types.” In this talk, I will present a simplified version of the paradox due to Hurkens (1995), which in particular has a nice analogue in naive set theory. After treating this classical intuition and the type-theoretic argument, if we have time we will implement the paradox in Lean.
Topos theory plays a central role in geometry and categorical logic. For instance, topoi yield a natural semantic theory of intuitionistic dependent type theories. One important step in this theory is the observation that topoi are Heyting categories, i.e. that propositions in the internal logic of a topos can be refined along intuitionistic predicates. In this talk, I will attempt to give an accessible outline of this classical story and motivate the study of “linear topoi”, categories which exhibit topos-like behavior with respect to a linear internal logic. I will conclude with a discussion of some progress towards an axiomatic presentation of these categories.