Logic Seminar
John Baldwin
UIC
On strongly minimal Steiner systems: Zilber's Conjecture, Universal Algebra, and Combinatorics
Abstract: Those who heard a version of this talk in early September will be interested to see unintelligible questions replaced by theorems.
With Gianluca Paolini (in preparation), we constructed, using a variant
on the Hrushovski dimension function,
for every $k \geq 3$, $2^\mu$ families of
strongly minimal Steiner $k$-systems. We study the mathematical properties
of these counterexamples to Zilber's trichotomy conjecture rather than thinking of them as merely exotic examples. In particular the long study of finite Steiner systems in reflected in results that depend on the block size $k$.
A quasigroup is a structure with a binary operation such that for each equation $xy=z$ the values of two of the variables determines a unique value for the third.
The
new Steiner $3$-systems are bi-interpretable with strongly
minimal Steiner quasigroups. For $k >3$, we show the pure $k$-Steiner systems have `essentially unary definable closure' and do not interpret a quasigroup. But we show that for $q$ a prime power the Steiner $q$-systems can be interpreted into specific sorts of quasigroups, block algebras.
We extend the notion of an $(a,b)$-cycle graph arising in the study of finite and infinite Stein triple systems (e.g Cameron-Webb) by introducing what we call the $(a,b)$-path graph of a block algebra. We exhibit theories of strongly minimal block algebras where all $(a,b)$-paths are infinite and others in which all are finite only in the prime model. We show how to obtain combinatorial properties (e.g. 2-transitivity) by either varying the basic collection of finite partial Steiner systems or modifying the $\mu$ function which ensures strong minimality.
Tuesday February 19, 2019 at 3:30 PM in 427 SEO