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The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is an space focus principally in  practicing presentation skills or learning materials that are not usually presented on a class.
The Graduate Logic Seminar is an informal space where graduate students and professors present topics related to logic which are not necessarily original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.


* '''When:''' Mondays, 4:00 PM – 5:00 PM (unless otherwise announced).
* '''When:''' Mondays 3:30-4:30 PM
* '''Where:''' Van Vleck B235 (unless otherwise announced).
* '''Where:''' Van Vleck B223
* '''Organizers:''' [https://www.math.wisc.edu/~msoskova/ Mariya Soskava]
* '''Organizers:''' [https://people.math.wisc.edu/~slempp/ Steffen Lempp] and [https://sites.google.com/view/hongyu-zhu/ Hongyu Zhu]


Talks schedule are arrange and decide at the beginning of each semester. If you would like to participate, please contact one of the organizers.
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.


== Spring 2018 ==
Sign up for the graduate logic seminar mailing list:  [mailto:join-grad-logic-sem@lists.wisc.edu join-grad-logic-sem@lists.wisc.edu]


=== January 29, Organizational meeting ===
== Spring 2024 ==


This day we decided the schedule for the semester.
The seminar will be run as a 1-credit seminar Math 975 . In Spring 2024, the topic will be forcing constructions in computability theory. If you are not enrolled but would like to audit it, please contact [https://people.math.wisc.edu/~slempp/ Steffen Lempp]  and [mailto:hongyu@math.wisc.edu Hongyu Zhu].


=== February 5, [http://www.math.wisc.edu/~andrews/ Uri Andrews] ===
Presentation Schedule: https://docs.google.com/spreadsheets/d/1JC6glG_soNLtaMQWaAuADlUu8dh2eJ0NL-MaUr7-nOk/edit?usp=sharing


Title: Building Models of Strongly Minimal Theories - Part 1
Zoom link for remote attendance: https://uwmadison.zoom.us/j/96168027763?pwd=bGdvL3lpOGl6QndQcG5RTFUzY3JXQT09 (Meeting ID: 961 6802 7763, Password: 975f23)


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
=== January 29 - Organizational Meeting ===
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 ===
Steffen Lempp will give an overview and present some very basic forcing construction.


Title: Finding Definable Sets in Continuous Logic
We will then assign speakers to dates and topics.


Abstract: In order to be useful the notion of a 'definable set' in continuous logic is stricter than a naive comparison to discrete logic
=== '''February 5 - Taeyoung Em''' ===
would suggest. As a consequence, even in relatively tame theories there can be very few definable sets. For example, there is a
'''Title:''' Introduction to forcing
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, [https://sites.google.com/a/wisc.edu/schweber/ Noah Schweber] ===
'''Abstract:''' We introduce new definitions and properties regarding forcing.  


Title: Proper forcing
=== '''February 12 - Hongyu Zhu''' ===
'''Title:''' Slaman-Woodin Forcing and the Theory of Turing Degrees


Abstract: Although a given forcing notion may have nice properties on its own, those properties might vanish when we apply it repeatedly.
'''Abstract:''' We will discuss how to use Slaman-Woodin forcing to interpret true second(first, resp.)-order arithmetic in the Turing degrees (Turing degrees below 0', resp.), thereby showing they have the same Turing degree.
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 ===
=== '''February 19 - John Spoerl''' ===
'''Title:''' Forcing with Trees - Spector's and Sack's Minimal Degrees


Title: A survey of computable and constructive mathematics in economic history
'''Abstract:''' We'll take a look at Spector's forcing which uses perfect trees as conditions.  Then we'll see where we might make some improvements which leads to Sack's sharpening of Spector's theorem: there is a minimal degree below 0'.


=== March 5, [http://www.math.wisc.edu/~makuluni/ Tamvana Makulumi] ===
=== '''February 26 - Karthik Ravishankar''' ===
'''Title:''' The 3 element chain as an initial segment of the Turing Degrees


Title: Convexly Orderable Groups
'''Abstract:''' In this talk, we'll look at the construction of a minimal degree with a strong minimal cover which shows that the three-element chain can be embedded as an initial segment of the Turing Degrees. The construction builds off ideas of Spector's minimal degree with stronger assumptions on the forcing conditions used. If time permits, we'll also talk about Copper's Jump Inversion building off Sack's construction.


=== March 12, [https://math.nd.edu/people/visiting-faculty/daniel-turetsky/ Dan Turetsky] (University of Notre Dame) ===
=== '''March 4 - Karthik Ravishankar''' ===
'''Title:''' Bushy Tree forcing and constructing a minimal degree which is DNC


Title: Structural Jump
'''Abstract:''' We shall look at a forcing technique called Bushy Tree forcing using it to show that there is no uniform way to compute a DNC_2 from a DNC_3 function and that there is a DNC function that is weak in the sense that it does not compute a computably bounded DNC function. We present a few other results along these lines and sketch the construction of a minimal degree that is DNC relative to any given oracle using bushy tree forcing.


=== March 19, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] ===
=== '''March 11 - Josiah Jacobsen-Grocott''' ===
'''Title:''' A uniformly e-pointed tree on Baire space without dead ends that is not of cototal degree


Title: Networks and degrees of points in non-second countable spaces
'''Abstract:''' A set is cototal if it is enumeration reducible to its complement. A tree is e-point if every path on the tree can enumerate the tree. McCathy proved that these notions are equivalent up to e-degree when considering e-pointed trees on cantor space. This fails when considering trees on Baire space. We give an example of a simple forcing construction that produces e-pointed trees on Baire space. We carefully analyze this forcing partial order to prove that generic e-pointed trees without dead ends are not of cototal degree.


=== April 2, Wil Cocke ===
=== '''March 18 - Alice Vidrine''' ===
'''Title:''' There is no non-computable bi-introreducible set


Title: Characterizing Finite Nilpotent Groups via Word Maps
'''Abstract:''' A set is said to be bi-introreducible if it can be computed by any of its infinite subsets, or any infinite subset of its complement. This talk will detail a Matthias forcing construction used to prove a theorem by Seetapun which implies that the bi-introreducible sets are exactly the computable sets.


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 1 - Hongyu Zhu''' ===
'''Title:''' The Conservativeness of WKL_0 over RCA_0 for <math>\Pi_1^1</math>-formulas


=== April 9, Tejas Bhojraj ===
'''Abstract:''' We will see how to use forcing to construct models of WKL_0 from models of RCA_0 while preserving certain arithmetical truths, thereby showing that WKL_0 is <math>\Pi_1^1</math>-conservative over RCA_0.


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.
<!-- Template


=== April 16, [http://www.math.wisc.edu/~ongay/ Iván Ongay-Valverde] ===
=== '''September 18 - xxx''' ===
'''Title:''' TBA ([https://wiki.math.wisc.edu/images/***.pdf Slides])


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


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.
== Previous Years ==


This is a work in progress, so this talk will have multiple open questions and opportunities for feedback and public participation.(hopefully).
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].
 
=== April 23, [http://www.math.wisc.edu/~mccarthy/ 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>\omega^{<\omega}</math>, and as the degrees of enumeration-pointed trees on <math>2^{<\omega}</math>, and we will remark on some additional applications of these characterizations.
 
=== April 30, [http://www.math.uconn.edu/~westrick/ Linda Brown Westrick] (from University Of Connecticut) ===
 
Title: TBA
 
Abstract: TBA
 
== Fall 2017 ==
 
=== September 11, Organizational meeting ===
 
This day we decided the schedule for the semester.
 
=== September 18, [https://sites.google.com/a/wisc.edu/schweber/ 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, [https://sites.google.com/a/wisc.edu/schweber/ 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, [http://www.math.wisc.edu/~makuluni/ Tamvana Makulumi] ===
 
Title: Boxy sets in ordered convexly-orderable structures.
 
=== October 30, [http://www.math.wisc.edu/~ongay/ 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, [https://www.math.wisc.edu/~lempp/ 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, [http://www.math.wisc.edu/~mccarthy/ 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
 
==Previous Years==
 
The schedule of talks from past semesters can be found [[Logic Graduate Seminar, previous semesters|here]].

Latest revision as of 22:43, 27 March 2024

The Graduate Logic Seminar is an informal space where graduate students and professors present topics related to logic which are not necessarily original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.

The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.

Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu

Spring 2024

The seminar will be run as a 1-credit seminar Math 975 . In Spring 2024, the topic will be forcing constructions in computability theory. If you are not enrolled but would like to audit it, please contact Steffen Lempp and Hongyu Zhu.

Presentation Schedule: https://docs.google.com/spreadsheets/d/1JC6glG_soNLtaMQWaAuADlUu8dh2eJ0NL-MaUr7-nOk/edit?usp=sharing

Zoom link for remote attendance: https://uwmadison.zoom.us/j/96168027763?pwd=bGdvL3lpOGl6QndQcG5RTFUzY3JXQT09 (Meeting ID: 961 6802 7763, Password: 975f23)

January 29 - Organizational Meeting

Steffen Lempp will give an overview and present some very basic forcing construction.

We will then assign speakers to dates and topics.

February 5 - Taeyoung Em

Title: Introduction to forcing

Abstract: We introduce new definitions and properties regarding forcing.

February 12 - Hongyu Zhu

Title: Slaman-Woodin Forcing and the Theory of Turing Degrees

Abstract: We will discuss how to use Slaman-Woodin forcing to interpret true second(first, resp.)-order arithmetic in the Turing degrees (Turing degrees below 0', resp.), thereby showing they have the same Turing degree.

February 19 - John Spoerl

Title: Forcing with Trees - Spector's and Sack's Minimal Degrees

Abstract: We'll take a look at Spector's forcing which uses perfect trees as conditions. Then we'll see where we might make some improvements which leads to Sack's sharpening of Spector's theorem: there is a minimal degree below 0'.

February 26 - Karthik Ravishankar

Title: The 3 element chain as an initial segment of the Turing Degrees

Abstract: In this talk, we'll look at the construction of a minimal degree with a strong minimal cover which shows that the three-element chain can be embedded as an initial segment of the Turing Degrees. The construction builds off ideas of Spector's minimal degree with stronger assumptions on the forcing conditions used. If time permits, we'll also talk about Copper's Jump Inversion building off Sack's construction.

March 4 - Karthik Ravishankar

Title: Bushy Tree forcing and constructing a minimal degree which is DNC

Abstract: We shall look at a forcing technique called Bushy Tree forcing using it to show that there is no uniform way to compute a DNC_2 from a DNC_3 function and that there is a DNC function that is weak in the sense that it does not compute a computably bounded DNC function. We present a few other results along these lines and sketch the construction of a minimal degree that is DNC relative to any given oracle using bushy tree forcing.

March 11 - Josiah Jacobsen-Grocott

Title: A uniformly e-pointed tree on Baire space without dead ends that is not of cototal degree

Abstract: A set is cototal if it is enumeration reducible to its complement. A tree is e-point if every path on the tree can enumerate the tree. McCathy proved that these notions are equivalent up to e-degree when considering e-pointed trees on cantor space. This fails when considering trees on Baire space. We give an example of a simple forcing construction that produces e-pointed trees on Baire space. We carefully analyze this forcing partial order to prove that generic e-pointed trees without dead ends are not of cototal degree.

March 18 - Alice Vidrine

Title: There is no non-computable bi-introreducible set

Abstract: A set is said to be bi-introreducible if it can be computed by any of its infinite subsets, or any infinite subset of its complement. This talk will detail a Matthias forcing construction used to prove a theorem by Seetapun which implies that the bi-introreducible sets are exactly the computable sets.

April 1 - Hongyu Zhu

Title: The Conservativeness of WKL_0 over RCA_0 for [math]\displaystyle{ \Pi_1^1 }[/math]-formulas

Abstract: We will see how to use forcing to construct models of WKL_0 from models of RCA_0 while preserving certain arithmetical truths, thereby showing that WKL_0 is [math]\displaystyle{ \Pi_1^1 }[/math]-conservative over RCA_0.


Previous Years

The schedule of talks from past semesters can be found here.