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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 a space focused principally on practicing presentation skills or learning materials that are not usually presented in 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 4p-5p
* '''When:''' Mondays 3:30-4:30 PM
* '''Where:''' Van Vleck B223.
* '''Where:''' Van Vleck B223
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]
* '''Organizers:''' [https://people.math.wisc.edu/~slempp/ Steffen Lempp] and [https://sites.google.com/view/hongyu-zhu/ Hongyu Zhu]


The talk schedule is arranged 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.


Sign up for the graduate logic seminar mailing list:  join-grad-logic-sem@lists.wisc.edu
Sign up for the graduate logic seminar mailing list:  [mailto:join-grad-logic-sem@lists.wisc.edu 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 [https://people.math.wisc.edu/~slempp/ Steffen Lempp]  and [mailto:hongyu@math.wisc.edu Hongyu Zhu].


== Fall 2019 - Tentative schedule ==
Presentation Schedule: https://docs.google.com/spreadsheets/d/1JC6glG_soNLtaMQWaAuADlUu8dh2eJ0NL-MaUr7-nOk/edit?usp=sharing


=== September 5 - Organizational meeting ===
Zoom link for remote attendance: https://uwmadison.zoom.us/j/96168027763?pwd=bGdvL3lpOGl6QndQcG5RTFUzY3JXQT09 (Meeting ID: 961 6802 7763, Password: 975f23)


=== September 9 - No seminar ===
=== January 29 - Organizational Meeting ===


=== September 16 - Daniel Belin ===
Steffen Lempp will give an overview and present some very basic forcing construction.
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.
We will then assign speakers to dates and topics.


=== September 23 - Daniel Belin ===
=== '''February 5 - Taeyoung Em''' ===
'''Title:''' Introduction to forcing


Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued
'''Abstract:''' We introduce new definitions and properties regarding forcing.


=== September 30 - Josiah Jacobsen-Grocott ===
=== '''February 12 - Hongyu Zhu''' ===
'''Title:''' Slaman-Woodin Forcing and the Theory of Turing Degrees


Title: Scott Rank of Computable Models
'''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.


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.
=== '''February 19 - John Spoerl''' ===
'''Title:''' Forcing with Trees - Spector's and Sack's Minimal Degrees


=== October 7 - Josiah Jacobsen-Grocott ===
'''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'.


Title: Scott Rank of Computable Codels - Continued
=== '''February 26 - Karthik Ravishankar''' ===
'''Title:''' The 3 element chain as an initial segment of the Turing Degrees


=== October 14 - Tejas Bhojraj ===
'''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.


Title: Solovay and Schnorr randomness for infinite sequences of qubits.
=== '''March 4 - Karthik Ravishankar''' ===
'''Title:''' Bushy Tree forcing and constructing a minimal degree which is DNC


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.
'''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.


=== October 23 - Tejas Bhojraj ===
=== '''March 11 - Josiah Jacobsen-Grocott''' ===
'''Title:''' A uniformly e-pointed tree on Baire space without dead ends that is not of cototal degree


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


Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.
=== '''March 18 - Alice Vidrine''' ===
'''Title:''' There is no non-computable bi-introreducible set


=== October 28 - Two short talks ===
'''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.


'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)
=== '''April 1 - Hongyu Zhu''' ===
'''Title:''' The Conservativeness of WKL_0 over RCA_0 for <math>\Pi_1^1</math>-formulas


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):
'''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.


- 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.
<!-- Template


'''James Earnest Hanson''' - Strongly minimal sets in continuous logic
=== '''September 18 - xxx''' ===
'''Title:''' TBA ([https://wiki.math.wisc.edu/images/***.pdf Slides])


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.
'''Abstract:''' TBA


=== November 4 - Two short talks ===
-->


'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)
== Previous Years ==
 
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''' - TBD
 
=== 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.
<br/>
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.
 
=== November 18 - Manlio Valenti II ===
 
=== November 25 - Two short talks ===
Speakers TBD
 
=== December 2 - Iván Ongay Valverde I ===
 
=== December 9 - Iván Ongay Valverde II ===
 
==Previous Years==


The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].
The schedule of talks from past semesters can be found [[Graduate Logic 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.