A $t-(\kappa ,k,s)$ block design is a set of $\kappa$ elements and a collection of $k$-element subsets $B$ of $P$ (called blocks) with the property that each $t$-element subset of $P$ occurs in exactly $s$ blocks. A $k$-Steiner system is a $2-(\kappa ,k,1)$ system.

Using variants of the Hrushovski method we construct infinite block designs and Steiner systems that are a) $\aleph}_{1$-categorical and with more work b) have $t$-transitive automorphism groups for prescribed $t$.

The strongly minimal Steiner $k$-Steiner system (M,R) from Baldwin and Pao (2021) can be `coordinatized' in the sense of Ganter and Werner (1975) by a quasigroup if $k$ is a prime-power. But for the basic construction this coordinatization is never definable in (M,R) by Baldwin (2023). Nevertheless, by refining the construction, if $k$ is a prime power there is a (2,k)-variety of quasigroups which is strongly minimal and definably coordinatizes a Steiner $k$-system.

We compare these structures with Steiner systems given by Fra{\"\i}ss\'{e} constructions by Barbina and Casanovas (2019) and compare versions of the small intersection property for $\aleph}_{0$-categorical structures and countably saturated structures by Lascar (2002).

Infinite combinatorics from finite structures

Tue, Feb. 20 3:30pm (MATH 3…

Topology

Elizabeth Tatum (Universität Bonn)

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In recent work, Guchuan Li, Sarah Petersen, and I have constructed models for $C}_{2$-equivariant analogues of the integral Brown-Gitler spectra. In this talk, I will start by introducing the classical Brown-Gitler spectra and some of their applications. After that, I will discuss the $C}_{2$-equivariant integral Brown-Gitler spectra, as well as the applications we are beginning to study.

Equivariant Brown-Gitler spectra

Tue, Feb. 20 4:50pm (MATH 3…

Grad Student Seminar

Alex LaJeunesse (University of Colorado)

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Algebraic K-theory is a booming field that interacts with many diverse areas of mathematics. It’s universality makes it difficult to understand, but much progress has been made in recent years using a variety of “trace” methods; that is, approximating K-theory by simpler algebraic invariants. In this talk I will introduce algebraic K-theory and the related Hochschild homology, with the ultimate goal of constructing a useful map from the former to the latter.