Mid-Atlantic Mathematical Logic Seminar

October 27, 2017

Room 6417

MAMLS Logic Friday is a one-day logic meeting taking place at the CUNY Graduate Center with the support of the National Science Foundation. It will feature talks in set theory, computability theory, and model theory.

While graduate students, young researchers, female mathematicians and members of underrepresented groups are particularly encouraged to apply for travel support, it should be stressed that any participants without their own sources of funding are eligible to apply. Requests will be handled on a case-by-case basis within the limits of the budget. To apply for travel support or to find out more information, please contact Victoria Gitman (vgitman@nylogic.org).

Information about hotels can be found here.

**Speakers**

- Stamatis Dimopoulos, Bristol University
- Chris Lambie-Hanson, Bar-Ilan University
- Tim McNicholl, Iowa State University
- Ivo Herzog, Ohio State University
- Matthew Harrison-Trainor, University of Waterloo

**Schedule**

**9:00-9:30**

**Breakfast** (Math Lounge 4214)

Morning Session

**9:30-10:15**

**Ivo Herzog**

**The Logic of Positive Primitive Formulae**

We will discuss the structure of positive primitive formulae in the language of modules over a ring R. This structure includes two cubical lattices related by the Prest dual. We place particular emphasis on the resemblance of positive primitive logic to quantum logic and von Neumann’s work on the coordinatization of complemented modular lattices. For a ring (R,*) with involution, this gives rise to a quantum logic coordinatized by a *-regular ring obtained by adjusting a classical construction of Olivier for commutative rings.

**10:30-11:15**

**Tim McNicholl**

**Computable metric structure theory**

I will begin by reviewing the evolution of computable structure theory beginning with its origins in the work of van der Waerden on constructive algebra. I will then discuss recent work on extending the computable structures program to metric structures by means of the framework of computable analysis. I will focus on Banach spaces, and in particular recent results on computable categoricity and degrees of categoricity of

spaces. The solutions of some of the resulting problems involve am interesting blend of methods from functional analysis and classical computability theory.

Lunch

**11:30-12:30**

Lunch Break

Afternoon Session

**12:45-1:30**

**Chris Lambie-Hanson**

**Reflections on graph coloring**

In 1951, de Bruijn and Erdős published a compactness theorem for graphs with finite chromatic number, proving that, if is a graph, is a natural number, and all finite subgraphs of have chromatic number at most , then has chromatic number at most . Since then, infinitary generalizations of this theorem, for the chromatic number as well as the coloring number of graphs, have attracted much attention. In this talk, we will briefly review some of the historical highlights in this area and then present some new work. These results show that the coloring number can exhibit only a limited amount of incompactness, while large amounts of incompactness for the chromatic number are implied by relatively weak hypotheses. This indicates that the coloring number and chromatic number behave quite differently with respect tocompactness and illustrates the difficulty involved in obtaining infinitary analogues of the de Bruijn-Erdős result at infinite, accessible cardinals. This is joint work with Assaf Rinot.

**1:45-2:30**

**Matthew Harrison-Trainor**

**Some Computable Structure Theory of Finitely Generated Structures**

Every countable structure has a sentence of infinitary logic, called a Scott sentence, which describes it up to isomorphism among countable structures. We can characterize the complexity of a structure by the complexity of the simplest description of that structure. A finitely generated structure always has a

description. We show that there is a finitely generated group which has no simpler description. The proof of this leads us to talk about notions of universality for finitely generated structures. Finitely generated groups are universal, but finitely generated fields are not. By this, we mean that for every finitely generated structure, there is a finitely generated group which has the same computability-theoretic properties; but the same is not true for finitely generated fields. We apply the results of this investigation to pseudo Scott sentences.

**2:45-3:30**

**Stamatis Dimopoulos**

**Woodin-for-strong-compactness cardinals, a new identity crisis**

Woodin and Vopěnka cardinals are established notions in the large cardinal hierarchy and despite being defined in different context, they proved to be very similar. In fact, Vopěnka cardinals are obtained by replacing a strongness clause in the definition of Woodinness by a supercompactness clause. Since strong compactness is an intermediate large cardinal notion between strongness and supercompactness, it is natural to consider a “Woodinised” version of it. In this talk, we give the definition of this new type of large cardinal, called Woodin for strong compactness, and will present some results about them. The highlight is that the analogue of Magidor’s “identity crisis” theorem for the first strongly compact holds for these cardinals too: the first Woodin for strong compactness cardinal can consistently be the first Woodin or the first Vopěnka cardinal.