Graduate Logic Seminar, previous semesters

This is an historic listing of the talks in the Graduate Logic Seminar.

Spring 2020

February 17 - James Hanson

Title: The Topology of Definable Sets in Continuous Logic

Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.

February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott

Tejas Bhojraj - Quantum Kolmogorov Complexity.

Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.

Josiah Jacobsen-Grocott - A Characterization of Strongly $\eta$-Representable Degrees.

Abstract: $\eta$-representations are a way of coding sets in computable linear orders that were first introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to characterize the sets with $\eta$-representations as well as the sets with subclasses of $\eta$-representations except for the case of sets with strong $\eta$-representations, the only class where the order type of the representation is unique.

We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$ approximations. We use connected approximations to give a characterization of the degrees with strong $\eta$-representations as well new characterizations of the subclasses of $\eta$-representations with known characterizations.

March 2 - Patrick Nicodemus

Title: A Sheaf-theoretic generalization of Los's theorem

Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.

March 9 - Noah Schweber

Title: Algebraic logic and algebraizable logics

Abstract: Arguably the oldest theme in what we would recognize as "mathematical logic" is the algebraic interpretation of logic, the most famous example of this being the connection between (classical) propositional logic and Boolean algebras. But underlying the subject of algebraic logic is the implicit assumption that many logical systems are "satisfyingly" interpreted as algebraic structures. This naturally hints at a question, which to my knowledge went unasked for a surprisingly long time: when does a logic admit a "nice algebraic interpretation?"

Perhaps surprisingly, this is actually a question which can be made precise enough to treat with interesting results. I'll sketch what is probably the first serious result along these lines, due to Blok and Pigozzi, and then say a bit about where this aspect of algebraic logic has gone from there.

Fall 2019

September 16 - Daniel Belin

Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic

Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.

September 23 - Daniel Belin

Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued

September 30 - Josiah Jacobsen-Grocott

Title: Scott Rank of Computable Models

Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.

October 7 - Josiah Jacobsen-Grocott

Title: Scott Rank of Computable Codels - Continued

October 14 - Tejas Bhojraj

Title: Solovay and Schnorr randomness for infinite sequences of qubits.

Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.

October 23 - Tejas Bhojraj

Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued

Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.

October 28 - Two short talks - Iván Ongay Valverde and James Earnest Hanson

Iván Ongay Valverde - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)

In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):

- Is the axiom weaker if we demand that $W$ is clopen? - If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$? - Can we expand this axiom to spaces that are not second countable and metric?

These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.

James Earnest Hanson - Strongly minimal sets in continuous logic

The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.

November 4 - Two short talks - Manlio Valenti and Patrick Nicodemus

Manlio Valenti - The complexity of closed Salem sets (20 minutes version)

A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets.
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.

Patrick Nicodemus - Proof theory of Second Order Arithmetic and System F

A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.

November 11 - Manlio Valenti

Title: The complexity of closed Salem sets (full length)

Abstract: A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets.
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.

November 18 - Iván Ongay Valverde

Title: A couple of summer results

Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.

In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.

Spring 2018

January 29, Organizational meeting

This day we decided the schedule for the semester.

February 5, Uri Andrews

Title: Building Models of Strongly Minimal Theories - Part 1

Abstract: Since I'm talking in the Tuesday seminar as well, I'll use the Monday seminar talk to do some background on the topic and some lemmas that will go into the proofs in Tuesday's talk. There will be (I hope) some theorems of interest to see on both days, and both on the general topic of answering the following question: What do you need to know about a strongly minimal theory in order to compute copies of all of its countable models. I'll start with a definition for strongly minimal theories and build up from there.

February 12, James Hanson

Title: Finding Definable Sets in Continuous Logic

Abstract: In order to be useful the notion of a 'definable set' in continuous logic is stricter than a naive comparison to discrete logic would suggest. As a consequence, even in relatively tame theories there can be very few definable sets. For example, there is a superstable theory with no non-trivial definable sets. As we'll see, however, there are many definable sets in omega-stable, omega-categorical, and other small theories.

February 19, Noah Schweber

Title: Proper forcing

Abstract: Although a given forcing notion may have nice properties on its own, those properties might vanish when we apply it repeatedly. Early preservation results (that is, theorems saying that the iteration of forcings with a nice property retains that nice property) were fairly limited, and things really got off the ground with Shelah's invention of "proper forcing." Roughly speaking, a forcing is proper if it can be approximated by elementary submodels of the universe in a particularly nice way. I'll define proper forcing and sketch some applications.

February 26, Patrick Nicodemus

Title: A survey of computable and constructive mathematics in economic history

March 5, Tamvana Makulumi

Title: Convexly Orderable Groups

March 12, Dan Turetsky (University of Notre Dame)

Title: Structural Jump

March 19, Ethan McCarthy

Title: Networks and degrees of points in non-second countable spaces

April 2, Wil Cocke

Title: Characterizing Finite Nilpotent Groups via Word Maps

Abstract: In this talk, we will examine a novel characterization of finite nilpotent groups using the probability distributions induced by word maps. In particular we show that a finite group is nilpotent if and only if every surjective word map has fibers of uniform size.

April 9, Tejas Bhojraj

Title: Quantum Randomness

Abstract: I will read the paper by Nies and Scholz where they define a notion of algorithmic randomness for infinite sequences of quantum bits (qubits). This talk will cover the basic notions of quantum randomness on which my talk on Tuesday will be based.

April 16, Iván Ongay-Valverde

Title: What can we say about sets made by the union of Turing equivalence classes?

Abstract: It is well known that given a real number x (in the real line) the set of all reals that have the same Turing degree (we will call this a Turing equivalence class) have order type 'the rationals' and that, unless x is computable, the set is not a subfield of the reals. Nevertheless, what can we say about the order type or the algebraic structure of a set made by the uncountable union of Turing equivalence classes?

This topic hasn't been deeply studied. In this talk I will focus principally on famous order types and answer whether they can be achieved or not. Furthermore, I will explain some possible connections with the automorphism problem of the Turing degrees.

This is a work in progress, so this talk will have multiple open questions and opportunities for feedback and public participation.(hopefully).

April 23, Ethan McCarthy (Thesis Defense) Start 3:45 Room B231

Title: Cototal enumeration degrees and their applications to effective mathematics

Abstract: The enumeration degrees measure the relative computational difficulty of enumerating sets of natural numbers. Unlike the Turing degrees, the enumeration degrees of a set and its complement need not be comparable. A set is total if it is enumeration above its complement. Taken together, the enumeration degrees of total sets form an embedded copy of the Turing degrees within the enumeration degrees. A set of natural numbers is cototal if it is enumeration reducible to its complement. Surprisingly, the degrees of cototal sets, the cototal degrees, form an intermediate structure strictly between the total degrees and the enumeration degrees.

Jeandel observed that cototal sets appear in a wide class of structures: as the word problems of simple groups, as the languages of minimal subshifts, and more generally as the maximal points of any c.e. quasivariety. In the case of minimal subshifts, the enumeration degree of the subshift's language determines the subshift's Turing degree spectrum: the collection of Turing degrees obtained by the points of the subshift. We prove that cototality precisely characterizes the Turing degree spectra of minimal subshifts: the degree spectra of nontrivial minimal subshifts are precisely the cototal enumeration cones. On the way to this result, we will give several other characterizations of the cototal degrees, including as the degrees of maximal anti-chain complements on [math]\displaystyle{ \omega^{\lt \omega} }[/math], and as the degrees of enumeration-pointed trees on [math]\displaystyle{ 2^{\lt \omega} }[/math], and we will remark on some additional applications of these characterizations.

April 30, Iván Ongay-Valverde

Title: Definibility of the Frobenius orbits and an application to sets of rational distances.

Abstract: In this talk I'll present a paper by Hector Pastén. We will talk about how having a formula that identify a Frobenius orbits can help you show an analogue case of Hilbert's tenth problem (the one asking for an algorithm that tells you if a diophantine equation is solvable or not).

Finally, if time permits, we will do an application that solves the existence of a dense set in the plane with rational distances, assuming some form of the ABC conjecture. This last question was propose by Erdös and Ulam.

Fall 2017

September 11, Organizational meeting

This day we decided the schedule for the semester.

September 18, Noah Schweber

Title: The Kunen inconsistency

Abstract: While early large cardinal axioms were usually defined combinatorially - e.g., cardinals satisfying a version of Ramsey's theorem - later focus shifted to model-theoretic definitions, specifically definitions in terms of elementary embeddings of the whole universe of sets. At the lowest level, a measurable cardinal is one which is the least cardinal moved (= critical point) by a nontrivial elementary embedding from V into some inner model M.

There are several variations on this theme yielding stronger and stronger large cardinal notions; one of the most important is the inclusion of *correctness properties* of the target model M. The strongest such correctness property is total correctness: M=V. The critical point of an elementary embedding from V to V is called a *Reinhardt cardinal*. Shortly after their introduction in Reinhardt's thesis, however, the existence of a Reinhardt cardinal was shown to be inconsistent with ZFC.

I'll present this argument, and talk a bit about the role of choice.

September 25, Noah Schweber

Title: Hindman's theorem via ultrafilters

Abstract: Hindman's theorem is a Ramsey-type theorem in additive combinatorics: if we color the natural numbers with two colors, there is an infinite set such that any *finite sum* from that set has the same color as any other finite sum. There are (to my knowledge) two proofs of Hindman's theorem: one of them is a complicated mess of combinatorics, and the other consists of cheating wildly. We'll do.

October 2, James Hanson

Title: The Gromov-Hausdorff metric on type space in continuous logic

Abstract: The Gromov-Hausdorff metric is a notion of the 'distance' between two metric spaces. Although it is typically studied in the context of compact or locally compact metric spaces, the definition is sensible even when applied to non-compact metric spaces, but in that context it is only a pseudo-metric: there are non-isomorphic metric spaces with Gromov-Hausdorff distance 0. This gives rise to an equivalence relation that is slightly coarser than isomorphism. There are continuous first-order theories which are categorical with regards to this equivalence relation while failing to be isometrically categorical, so it is natural to look for analogs of the Ryll-Nardzewski theorem and Morley's theorem, but before we can do any of that, it'll be necessary to learn about the "topometric" structure induced on type space by the Gromov-Hausdorff metric.

October 9, James Hanson

Title: Morley rank and stability in continuous logic

Abstract: There are various ways of counting the 'size' of subsets of metric spaces. Using these we can do a kind of Cantor-Bendixson analysis on type spaces in continuous first-order theories, and thereby define a notion of Morley rank. More directly we can define > the 'correct' notion of stability in the continuous setting. There are also natural Gromov-Hausdorff (GH) analogs of these notions. With this we'll prove that inseparably categorical theories have atomic models over arbitrary sets, which is an important step in the proof of Morley's theorem in this setting. The same proof with essentially cosmetic changes gives that inseparably GH-categorical theories have 'GH-atomic' models over arbitrary sets, but GH-atomic models fail to be GH-unique in general.

October 23, Tamvana Makulumi

Title: Boxy sets in ordered convexly-orderable structures.

October 30, Iván Ongay-Valverde

Title: Dancing SCCA and other Coloring Axioms

Abstract: In this talk I will talk about some axioms that are closely related to SOCA (Semi Open Coloring Axiom), being the main protagonist SCCA (Semi Clopen Coloring Axiom). I will give a motivation on the statements of both axioms, a little historic perspective and showing that both axioms coincide for separable Baire spaces. This is a work in progress, so I will share some open questions that I'm happy to discuss.

November 6, Wil Cocke

Title: Two new characterizations of nilpotent groups

Abstract: We will give two new characterizations of finite nilpotent groups. One using information about the order of products of elements of prime order and the other using the induced probability distribution from word maps.

Or...

Title: Centralizing Propagating Properties of Groups

Abstract: We will examine some sentences known to have finite spectrum when conjoined with the theory of groups. Hopefully we will be able to find new examples.

November 13, Steffen Lempp

Title: The computational complexity of properties of finitely presented groups

Abstract: I will survey index set complexity results on finitely presented groups.

November 20, Ethan McCarthy

Title: Strong Difference Randomness

Abstract: The difference randoms were introduced by Franklin and Ng to characterize the incomplete Martin-Löf randoms. More recently, Bienvenu and Porter introduced the strong difference randoms, obtained by imposing the Solovay condition over the class of difference tests. I will give a Demuth test characterization of the strong difference randoms, along with a lowness characterization of them among the Martin-Löf randoms.

December 4, Tejas Bhojraj

Title: Quantum Algorithmic Randomness

Abstract: I will discuss the recent paper by Nies and Scholz where they define quantum Martin-Lof randomness (q-MLR) for infinite sequences of qubits. If time permits, I will introduce the notion of quantum Solovay randomness and show that it is equivalent to q-MLR in some special cases.

December 11, Grigory Terlov

Title: The Logic of Erdős–Rényi Graphs