### Topics in relational Hrushovski constructions

Omer Mermelstein, Fall 2018

The focus of this course will be the "classic" Hrushovski ab-initio constructions introduced in "A new strongly minimal set" - sparse strongly minimal hypergraphs constructed by amalgamation, foreign in nature to the then-known strongly minimal structures, which were all of an algebraic nature.

The spine of the course will consist of an exposition of the original constructions and their background, an exploration of their associated geometries (finitary matroids), and reduction relations (in the sense of 'reduct') between variations on the constructions. Other isolated topics/results (within the course) we may touch upon include: gammoids and flatness, fusion, coefficients in (0,1), aleph-zero categorical construction, specific counter-examples constructed using Fraïssé-Hrushovski amalgamation, ampleness.

Presentation and choice of topics will be heavily biased towards combinatorics. The lion's share of proofs will be elementary and hands-on, at the cost of getting our hands dirty with technical complexity. One should expect to emerge a resourceful engineer rather than a visionary geometer.

Time permitting, during the final weeks of the course, the instructor may indulge in getting everyone's hands even dirtier with some results in ordinal Ramsey theory

The spine of the course will consist of an exposition of the original constructions and their background, an exploration of their associated geometries (finitary matroids), and reduction relations (in the sense of 'reduct') between variations on the constructions. Other isolated topics/results (within the course) we may touch upon include: gammoids and flatness, fusion, coefficients in (0,1), aleph-zero categorical construction, specific counter-examples constructed using Fraïssé-Hrushovski amalgamation, ampleness.

Presentation and choice of topics will be heavily biased towards combinatorics. The lion's share of proofs will be elementary and hands-on, at the cost of getting our hands dirty with technical complexity. One should expect to emerge a resourceful engineer rather than a visionary geometer.

Time permitting, during the final weeks of the course, the instructor may indulge in getting everyone's hands even dirtier with some results in ordinal Ramsey theory

Familiarity with mathematical logic will be helpful. However, in our setting, abstract model-theoretic notions will have a concrete manifestation, making the course fairly self-contained.

credits:

3

semester:

Spring

prereqs:

Math 770, and Math 773 or concurrent registration in Math 773
For fall 2018: Familiarity with mathematical logic will be helpful. However, in our setting, abstract model-theoretic notions will have a concrete manifestation, making the course fairly self-contained.