# Math 144: Model Theory and Algebra (Fall 2013)

## Monday, Wednesday, Friday 11 am - 12 pm

 Faculty: Nate Ackerman TA: ---- E-mail: nate AT math.harvard.edu Office: Office Hours: Thursday 1-2pm

## Final Projects:

Below is list potential topics for final projects. In addition to brief descriptions of the topic, there is also a list of resources (books, articles, etc.) which might be helpful and can serve as a starting point for your research. I have also included a list of questions which might help you begin to think about the area (note you are not not required to answer all of these questions).
• Abstract Model Theory:
This deals with the notion of an "abstract logic". In particular, with the notion of an abstract logic in hand, one is able to compare different logics as well as define properties of logics from which various other theorems can be proved. For example, Lindstrom's theorem gives a characterization of first order logic. There is also a characterization in terms of ultraproducts.

Potential Questions

• What is Lindstrom's characterization of first order logic?
• What are Lindstrom quantifiers?

Resources

• Abstract Elementary Classes:
Abstract Elementary Classes (or AEC) is a collection of models which satisfies properties which hold in most common logics (e.g. first order, L_{omega_1,omega}). AEC's are meant to capture the notion of "models of a formula" without any specific syntax. In particular this is a notion which allows (some of) classical model theory to be developed without any reference to specific formulas. As such results which are proven for AEC's hold for a wide variety of logics.

Potential Questions

• What is Shelah's eventual categoricity conjecture?
• What is the connection between accessible categories and AEC's

Resources

Because first order formulas are finite objects there is a fundamental difference between the study of first order finite model theory and first order infinite model theory. Admissible model theory (i.e. the model theory of L_{omega_1,omega} on an admissible set) generalizes the notion of "finite" in such a way as to allow many results of first order infinite model theory to have analogs.

Note Admissible Model Theory is at the intersection between set theory, model theory, and recursion theory. As such it would be helpful if the person choosing this project had taken a class in at least one of recursion theory or set theory.

Potential Questions

• What is Barwise compactness?
• What is the "model existence theorem"? Why doesn't the model existence theorem work if the admissible set is uncountable?

Resources

• Applications to Set Theory:
Over the years there have been many techniques from model theory which have found applications in set theory.

Note it is highly recommended that whoever does this project has a firm background in set theory. It is also important that this project not just deal with the set theory but also discuss the techniques/ideas in the model theory context.

Potential Questions

• What are measurable cardinals?
• What are compact cardinals?
• What is the relationship between indiscernible sequences and 0#?

Resources

• Borel Sym(N)-Spaces and the Logic Action:
Suppose L is a countable language. Let be the Str(L) of all L-structures with underlying set Naturals and let Sym(N) be the collection of permutations of the Naturals. Each element of Sym(N) induces an automorphism of each element of Str(L). When both Str(L) and Sym(N) are given an appropriate Borel structure, this map Sym(N) x Str(L) -> Str(L) is called the "Logic Action". Viewing models in this way allows us to bring in the tools of descriptive set theory to answer model theoretic questions as well as to bring in model theoretic tools into descriptive set theory.

It is recommended that if you choose this topic you have some familiarity with set theory. However, while there is a huge amount of mathematics surrounding descriptive set theory, it is important that the focus be on the relationship to model theory.

Potential Questions

• What is the topological Vaught's Conjecture?
• What is the relativized logic action?
• What is Borel reducibility?
• How can Borel reducibility be used to describe how complicated a theory is? What is an example of a theory which is maximally complicated in this sense? What about minimally complicated?

Resources

• Categorical Logic:
There are many structures in mathematics which can be combined with structures that seemingly have little to do with them. For example one often sees variants of groups which come up in areas which seem on their surface to be unrelated to group theory, e.g. topological groups, algebraic groups, Lie groups, etc. One would like to think of such objects as "groups in different categories", with topological groups being groups in the category topological spaces, algebraic groups being groups in the category of varieties, and Lie groups as being groups in the category of smooth manifolds. Categorical logic is a way to make this intuition precise. In particular categorical logic gives a way to interpret any first order theory in a a category with enough structure.

Potential Questions

• What is a classifying topos?
• What is an algebra for monad?
• What is a general definition for a free algebra?
• What is a coalgebra?

Resources

• Computable Model Theory:
Given a language L we say an L-structure with underlying set Naturals is "computable" if all functions and relations are uniformly computable. Computable model theory studies computable structures as well as the effectiveness of results from first order model theory.

Potential Questions

• What is the difference between a computable structure and a decidable structure?
• When can you effectively omit types?
• What are automatic structures?

Resources

• Continuous Logic and Metric Structures:
Continuous logic studies structures whose underlying sets are complete metric spaces and where the collection of truth values is a bounded interval of non-negative real numbers. There are several types of structures, such as Hilbert spaces, Banach spaces, etc.) which are particularly well suited to being studied by continuous logic.

Potential Questions

• What is the difference between continuous logic and fuzzy logic?
• How do you take ultraproducts of metric structures

Resources

• Finite model theory:
There are many results which are true about finite structures which either are not true about infinite structures or which cannot even be asked (like the 0-1 law). The study of finite structures also has a strong connection to complexity theory (from computer science).

There are connections between finite model theory and recursion theory/computer science. As such this project might be especially interesting for someone who has an interest in these areas.

Potential Questions

• What are threshold functions? When is there a 0-1 law for the distribution G(n^{-alpha}, p) on graphs?
• What is the connection between non-deterministic polynomial time computable relations and second order logic (on finite structures)?
• What is the connection between Trakhtenbrot's theorem and the completeness theorem for first order logic?

Resources

• Infinitary Logic:
If one generalizes first order logic by allowing conjunctions of size < kappa and existential quantification over collections of variables of length < gamma one obtains the logic L_{kappa, gamma}.

Potential Questions

• What are compact cardinals?
• What are game quantifiers?

Resources

• Model Theoretic Logics Edited by Barwise and Feferman. Chapter IX and X.
• Models of Peano Arithmetic and Set Theory:
By compactness it is known that there are models which satisfy all first order properties of the natural numbers but which contain "non-standard" natural numbers. As the theory of Peano arithmetic and its extensions plays a special role in mathematics there has been a great deal of work done studying models of Peano arithmetic.

Similarly by compactness there are models of Zermelo-Frankel set theory where the ordinals are "non-standard".

Potential Questions

• What are omega-models of set theory? What are omega_1-models of set theory?
• What is Tennenbaum's Theorem?

Resources

• Non-Absoluteness in Model Theory:
Most properties of first order structures which are studied in model theory are absolute between models of set theory, i.e. whether they hold of a model or a theory does not depend on the background model of set theory. However, there are some properties which can hold or not depending on the background model of set theory.

It is highly recommended that the person doing this project has a good familiarity with set theory. Also, as most properties of model theory are absolute this will require a fair amount of background research.

Potential Questions

• Is being isomorphic absolute? What is a potential isomorphism?
• What is Chang's conjecture?
• What is Shoenfield absoluteness? What can it say about absoluteness in model theory?

Resources

• Non-Standard Analysis:
One of the greatest mathematical discoveries of the last 350 years was the invention of calculus. However, at the time of their invention, many of the techniques of calculus relied on the informal notion of infinitesimals. While mathematics of the time was done in a much more informal way than it is now, the notion of an "infinitesimal" was considered a non-rigours notion even by the standards of the day. It took over 150 years after the invention of calculus for the techniques to be put on a rigours footing with a rigours definition of "limit". However, in the last 50 years, the notion of an infinitesimal has finally been made rigours in non-standard analysis.

Potential Questions

• What is an infinitesimal?
• What is a Loeb measure?
• What are the hyperreals?

Resources

• O-Minimal Theories:
When studying the dichotomy between structure and non-structure a theory having a linear order often is a sign that the models of the theory lack structure (and in particular such a theory is not stable). The notion of O-minimality is meant to capture (one of) the situations when a theory has an order but despite this fact the models are not too complicated. Examples of O-minimal theories include real closed fields, the theory of dense linear orderings, etc.

Potential Questions

• What is the connection between being O-minimal and being strongly minimal?

Resources

• Second Order Logic and Higher Order Logic:
In first order logic formulas are only able to quantify over elements of the model. In second order logic we allow formulas to quantify over subsets and functions. In third order logic we allow formulas to quantify over subsets of subsets, etc.

Potential Questions

• What are the definable second order quantifiers?
• What properties are categorical in second order logic?
• What is monadic second order logic?

Resources

• Stability Theory:
Stability theory can be thought of as studying the dichotomy between structure and non-structure. In particular, the property of being stable (for a first order theory) ensures that the theory is very well behaved. Further, the property of being unstable ensures that, in some sense which can be made precise, the theory must be badly behaved. In the course of studying arbitrary theories many other important dividing lines have been discovered (such as simple theories, superstable theories, etc).

While this is a very interesting topic with a lot of potential material, it also deals with extremely advanced ideas which go beyond the material covered in the class (in particular if the class was two semesters long this would be the topic of the second semester)

Potential Questions

• What is the stability spectrum?
• Suppose I(kappa, T) is the number of models of a theory T of size kappa. What are the possible values of the function I( -, T), for first order T?
• What is a forking extension of a type?
Resources

This page was created by Nate Ackerman, and last revised on August 23, 2013.