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## Upcoming talks and events:

# The Mitchell order for Ramsey cardinals

The usual Mitchell relation on normal measures on a measurable cardinal $\kappa$ orders the measures based on the degree of measurability that $\kappa$ retains in their respective ultrapowers. We shall examine the analogous ordering of appropriate witnessing objects for Ramsey (and Ramsey-like) cardinals. It turns out that the resulting order is well-behaved and its degrees neatly stratify the large cardinal hierarchy between Ramsey, strongly Ramsey, and super Ramsey cardinals. We also give a soft killing argument for this notion of Mitchell rank.

This is joint work with Victoria Gitman and Erin Carmody.

# Euclidean domains of arbitrarily high Euclidean rank

We will construct Euclidean domains of arbitrarily high Euclidean rank.

# Theories where any definable infinite set has interior

We consider the case of a theory $T$ which expands that of divisible ordered Abelian groups

and has the property that in any model of $T$ any infinite definable subset has non-empty interior

(in the order topology). We will call such theories visceral. Visceral theories arise naturally in the study theories with a strong form of the

independence property and generalize the class of o-minimal and weakly o-minimal theories. We consider the

structure of definable sets and definable functions in visceral theories, giving some weak structural results. We

also consider how to build general classes of examples visceral theories, and relate these example back to questions

about strong forms of the independence property.

# Layered partial orders

If $\kappa$ is a regular uncountable cardinal and $\mathbb{P}$ is a partial order, we say that $\mathbb{P}$ is $\kappa$ stationarily layered iff the set of regular suborders of $\mathbb{P}$ is stationary in $[\mathbb{P}]^{<\kappa}$. This is a strong form of the $\kappa$-chain condition, and in fact implies that $\mathbb{P}$ is $\kappa$-Knaster. I will discuss two recent applications involving layered posets:

(1) a new characterization of weak compactness: a regular $\kappa$ is weakly compact iff every $\kappa$-cc poset is

$\kappa$ stationarily layered. This is joint work with Philipp Luecke.

(2) a general theorem about preservation of $\kappa$-cc under universal Kunen-style iterations.

# Transfer of the Ramsey Property between Classes

In this talk, we investigate some ways in which the property of being Ramsey may be transferred between classes of finite structures. We look at some category-theoretic and model-theoretic approaches.

# The Hanf number for amalgamation

In a joint work with Chris Lambie-Hanson, we study a family of abstract elementary classes (AEC) that we call coloring classes. Each coloring class is an AEC in a relational language $L$ containing exactly the $L$-structures whose finite substructures are isomorphic to one of the “allowed” finite structures. The work takes advantage of the fact that model-theoretic properties (e.g., existence of models and amalgamation) can be rephrased as properties of certain coloring functions. This allows us to improve the results of Baldwin, Kolesnikov, and Shelah: we show in ZFC that disjoint amalgamation can hold up to beth_{alpha}, alpha less than omega_1 (previously, only consistency results were known). We also give a partial answer to the question of Grossberg about the Hanf number for amalgamation property (not just disjoint amalgamation).

# NERDS October 17, 2015 at Assumption College

The 2015 autumn meeting of the New England Recursion & Definability Seminar will be held on the campus of Assumption College, in Worcester Massachusetts, on Saturday, October 17, from 10:00 until 4:15. Titles and abstracts are now posted at the link for NERDS, under the Seminars tab. Directions and visitor information are available here.

# A-Computable Graphs

A locally finite computable graph *G* is called *A*-computable if *A* can compute the neighbors of the vertices of *G*. William Gasarch and Andrew Lee introduced this notion to study graphs that are “between” computable and highly computable (i.e., ∅-computable). For any noncomputable c.e. set *A*, they proved that the *A*-computable graphs behave just like computable graphs when it comes to colorings. In this talk, we will see that their result also works for Euler paths and domatic partitions. Although it does not extend to arbitrary sets *A* (that are not necessarily c.e.), we will classify the sets for which it does.

# Computing uniform (metastable) rates of convergence from the statement of the theorem alone

Consider a convergence theorem of the following form.

(T): If the property P holds, then the sequence (*x _{n}*) converges.

For example, the monotone convergence principle states that any monotone, bounded sequence of reals converges.

This talk concerns the notion of metastable rates of convergence. The advantage of this notion is that metastable rates are often uniform and computable. Kohlenbach, using proof theory, showed that if the property P is of a certain form and the theorem (T) is provable in a certain formal type theory, then the rate of metastable convergence is both uniform and computable. Avigad and Iovino, using model theory, showed that if the theorem (T) is true and the property P is preserved by ultraproducts, then the rate of metastable convergence is uniform (no mention of computability). In this result, using computable analysis and computable continuous model theory, we show that if (T) is true and P is axiomatizable in continuous logic, then the corresponding metastable bounds are both uniform and computable from P. This generalizes both of the previous results.

# Title TBA

# Title TBA

# Tree representations from very large cardinals

We will discuss the propagation of certain tree representations in the presence of very large cardinals. These tree representations give generic absoluteness results and have structural consequences in the area of generalized descriptive set theory. In fact these representations give us a method for producing models of strong determinacy axioms.

# Model theory of generalized Urysohn spaces

Many well known examples of homogeneous metric spaces and graphs can be viewed as analogs of the rational Urysohn space (for example, the random graph as the Urysohn space with distances {0,1,2}). In this talk, I consider the R-Urysohn space, where R is an arbitrary ordered commutative monoid. I will first construct an extension R* of R, such that any model of the theory of the R-Urysohn space (in a relational language) can be given the structure of an R*-metric space. I will then characterize quantifier elimination in this theory by continuity of addition in R*. Finally, I will characterize various model theoretic properties of the R-Urysohn space (e.g. stability, simplicity, weak elimination of imaginaries) using natural algebraic properties of R.

# On computing VC-density in VC-minimal theories

In model theory, theories are typically distinguished by the complexity of their definable families. One popular notion of complexity, Vapnik-Chervonenkis density, is borrowed from statistical learning theory. In this talk, I discuss the general notion of computing VC-density in NIP theories, a notion explored by Aschenbrenner, Dolich, Haskell, MacPherson, and Starchenko in recent work. In this work, they ask if there is a relationship between dp-rank and VC-density. I show a partial result pointing in that direction by studying VC-minimality (a condition stronger than having minimal dp-rank). Any formula in a VC-minimal theory with two parameter variables has VC-density at most two. I conclude by discussing the possibility of extending this result to higher dimensions.

# Generalized Laver tables

The Laver tables are finite self-distributive algebras generated by one element that approximate the free left-distributive algebra on one generator if a rank-into-rank cardinal exists. We shall generalize the notion of a Laver table to a class of locally finite self-distributive algebraic structures with an arbitrary number of generators. These generalized Laver tables emulate algebras of rank-into-rank embeddings with an arbitrary number of generators modulo some rank. Furthermore, if there exists a rank-into-rank cardinal, then the free left-distributive algebras on any number of generators can be embedded in a canonical way into inverse limits of generalized Laver tables. As with the classical Laver tables, the reduced generalized Laver tables can be given an associative operation that is analogous to the composition of elementary embeddings and satisfies the same identities that algebras of elementary embeddings are known to satisfy. Furthermore, the notion of the critical point also holds in these generalized Laver tables as well even though generalized Laver tables are locally finite or finite. While the only classical Laver tables are the tables of cardinality $2^{n}$, the finite generalized laver tables occur much more frequently and many generalized Laver tables can be constructed from the classical Laver tables. We shall give some results that allow one to quickly compute the self-distributive operation in a certain class of generalized Laver tables.

# On maximal immediate extensions of valued fields

# Title TBA

# Title TBA

This talk will be jointly sponsored by the Commutative Algebra & Algebraic Geometry Seminar and the CUNY Logic Workshop.

# No talks November 27

There will be no talks on November 27, the day after Thanksgiving.