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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 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:''' | * '''When:''' Mondays 3:30-4:30 PM | ||
* '''Where:''' Van Vleck | * '''Where:''' Van Vleck B235 | ||
* ''' | * '''Organizer:''' Mariya Soskova | ||
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact | The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact 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 | == Spring 2025 == | ||
The | The seminar will be run as a 1-credit seminar Math 975. In Spring 2025, we will finish last semester's topic on Higher Computability Theory.Once we are done students will present a logic topic of their choice (it could be original work, but does not have to be). If you are not enrolled but would like to audit it, please contact [mailto:soskova@wisc.edu Mariya Soskova]. | ||
Presentation Schedule: [https://docs.google.com/spreadsheets/d/1uRSaI1edJ5sepz57NV07ohIfBSKL9FgkvJvMAewk1ms/edit?usp=sharing Sign up here.] | |||
Notes on Higher Computability Theory: [https://uwmadison.box.com/s/j3xftdj1i70d4lblxhzswhg9e25ajcpq Download the notes here.] You will need your UW-login. Please, do not distribute these notes without permission from the author. | |||
<!--Zoom link for remote attendance: https://uwmadison.zoom.us/j/96168027763?pwd=bGdvL3lpOGl6QndQcG5RTFUzY3JXQT09 (Meeting ID: 961 6802 7763, Password: 975f23)--> | |||
=== | === '''January 27 - Organizational Meeting and Sapir Ben-Shahar''' === | ||
Mariya Soskova will call for volunteers to sign up for presentations. | |||
1 | Sapir Ben-Shahar will wrap up Section 5.1 | ||
=== '''February 3 - Taeyoung Em''' === | |||
3. | Taeyoung Em will present Section 5.3. | ||
=== '''February 10 - Hongyu Zhu''' === | |||
5. | Hongyu Zhu will present Section 5.3 | ||
=== '''February 17 - Karthik Ravishankar''' === | |||
'''Title:''' Strong minimal covers and the cupping property | |||
'''Abstract:''' A longstanding question in degree theory has been whether every minimal Turing degree has a strong minimal cover. Meanwhile a strong example of degrees without SMC's are those which have the cupping property. It is known that PA degrees have the cupping property, as do degrees with a certain amount of escaping power. On the other hand, it is known that being weak in the sense of being non DNC and Hyperimmune-free lets you have a SMC. Degrees with the cupping property are closed upwards while it is not known if degrees with SMC are closed downwards. It is also not known if every degree either has the cupping property or a SMC. In this talk we will review several of these results and present techniques used to build SMCs. | |||
=== '''February 24 - Hongyu Zhu''' === | |||
'''Title:''' Seeing the forest does not account for the trees | |||
'''Abstract:''' Say a first-order theory (or a type) has bounded axiomatization if it has an axiomatization by <math>\forall_n</math>-formulas for some finite n. In this talk, we will discuss basic properties of theories and types with (or without) bounded axiomatizations, and in particular whether boundedness of theories implies that of types. (The meaning of the title will be explained in due time.) | |||
=== '''March 3 - Uri Andrews''' === | |||
'''Title:''' On the spectra of computable models of disintegrated strongly minimal theories with bounded ranks | |||
'''Abstract:''' The spectrum of a strongly minimal theory characterizes which of its countable models have computable copies (indexed by their dimensions). We will focus on the disintegrated strongly minimal theories, i.e., where the algebraic closure of a set is the union of the algebraic closures of the elements of the set. | |||
Somewhere in the late aughts, Alice Medvedev and I proved that if a theory is disintegrated strongly minimal and has a finite signature, then either all models are computable, no models are computable, or only the prime model is computable. Steffen Lempp and I tried to push this sort of analysis past finite signatures and we have results about theories which are disintegrated strongly minimal and every symbol in the (infinite) signature has rank less than or equal to 1 in the theory (i.e., you cannot have R(a,b,\bar{z}) if a and b are algebraically independent). Over this past winter break, I found a strategy to bring (some of) this analysis to strongly minimal theories in infinite languages as long as there is some finite N so that every symbol has rank less than or equal to N. I'll describe this strategy, and depending on time, I might even present something that loosely resembles a proof. | |||
=== | === '''March 10 - Logan Heath''' === | ||
Title: | '''Title:''' Degree Spectra of Theories | ||
'''Abstract:''' I will discuss the notion of the degree spectrum of a theory, introduce a class of questions one might ask about such a thing, point to a few of the answers to such questions, and look a little more closely at one such spectrum to highlight the sorts of techniques that arise in the area. | |||
=== '''March 17 - Yiqing Wang''' === | |||
'''Title:''' The compactness theorem is overrated | |||
'''Abstract:''' Elementary classes, or first-order logic in general, are limited in their ability to capture many natural mathematical classes, such as locally finite groups and Archimedean ordered fields. Conversely, obtaining meaningful results in the generality of non-elementary classes can be impossible. In 1978, Shelah introduced the notion of Abstract Elementary Classes (AECs), providing a framework for studying classes that are not first-order axiomatizable yet still possess rich model-theoretic properties and carry the same 'test question'. | |||
In this talk, I will try to give an overview of AECs, prove Shelah’s Presentation Theorem, and highlight some open problems in this area. | |||
=== '''March 31 - Chiara Travesset''' === | |||
'''Title:''' The Sacks Density Theorem | |||
'''Abstract:''' The Sacks Density Theorem states that between any two c.e. degrees, there are two incomparable c.e. degrees. I will present a detailed proof of this theorem. | |||
=== '''April 7th - Taeyoung Em''' === | |||
'''Title:''' Sets that encode themselves | |||
'''Abstract:''' Introreducible sets were introduced by Dekker and Myhill. Mansfield proved that complementary retraceable sets are computable and Seetapun and Slaman proved that complementary introreducible sets are computable. In this talk, I will present some results of Appel and McLaughlin on regressive sets, and maybe some other results. | |||
=== '''April 14th - Lucas Duckworth''' === | |||
'''Title:''' The Paris-Harrington Theorem and Computability Results for Ramsey Theory | |||
'''Abstract:''' The Paris-Harrington Theorem, a slight extension of the famous Finite Ramsey Theorem, was shown to be not provable in PA. This talk will explore this original proof, which proves this result using an interesting variety of arguments combining combinatorics and logic. If time permits, I will also speak about a few results of Jokusch, Specker, Yates, and others on various results using computability to prove properties about homogeneous sets from basic partitions in the original Finite Ramsey Theorem. | |||
=== '''April 21 - Ang Li's defense''' === | |||
=== | '''Title:''' Computability-Theoretic Analysis of Ordered Groups, Logical Depth, and Introenumerability. | ||
'''Abstract:''' The first part of this talk continues the study of connections between reverse mathematics and Weihrauch reducibility. In particular, we analyze the uniform computational power of problems formed from Maltsev’s classification of the order types of countable ordered groups. Several non-reducibility results are obtained via the first-order part—an interior algebraic operator on Weihrauch degrees that has attracted recent attention. | |||
In the second part, we turn to two measure and category questions: knowing that some classes of enumeration degrees have measure zero, what level of randomness can they have or must avoid as reasonable classes of randomness have measure one; knowing that the class of shallow sets is comeager, what level of genericity can deep sets have or must avoid as the classes of generic sets are comeager. | |||
=== '''April 28th - Sapir Ben-Shahar''' === | |||
'''Title:''' Monadic definability and Matroids | |||
'''Abstract:''' Matroids are combinatorial structures that generalise the idea of (in)dependence, such as linear independence of vectors in vector spaces over some field. Matroids arise in a number of different contexts, including from vectors in vector spaces, graphs, points in a geometry, as models of strongly minimal theories, in combinatorial optimization problems, in phylogenetic trees, and many more. This talk will focus on gain-graphic matroids, which are matroids that arise from group-labeled graphs. Gain-graphic matroids are important in the study of structural matroid theory. It turns out that for ``nice enough" classes of matroids, properties that are definable in monadic second-order logic can be recognized in polynomial time. Whether a class of gain-graphic matroids is definable or not depends on which group is chosen to label the graphs. I'll start with a brief introduction to matroids and monadic second order logic, and then describe recent progress on the definability question for gain-graphic matroids. | |||
== Fall 2024 == | |||
The seminar will be run as a 1-credit seminar Math 975 . In Fall 2024, the topic will be Higher Computability Theory. We will follow notes by Noam Greenberg. If you are not enrolled but would like to audit it, please contact [mailto:soskova@wisc.edu Mariya Soskova]. | |||
Presentation Schedule: [https://docs.google.com/spreadsheets/d/1ect-dgHdoHOgq4-5BGFiDh6pPThLfDg69Yg__-b_5RY/edit?usp=sharing Sign up here.] | |||
Notes: [https://uwmadison.box.com/s/j3xftdj1i70d4lblxhzswhg9e25ajcpq Download the notes here.] You will need your UW-login. Please, do not distribute these notes without permission from the author. | |||
<!--Zoom link for remote attendance: https://uwmadison.zoom.us/j/96168027763?pwd=bGdvL3lpOGl6QndQcG5RTFUzY3JXQT09 (Meeting ID: 961 6802 7763, Password: 975f23)--> | |||
=== '''September 9 - Organizational Meeting''' === | |||
Mariya Soskova will start with the first sections from the notes. | |||
We will then assign speakers to dates and topics. | |||
=== '''September 16 - Sections 1.2-1.4''' === | |||
Kanav Madhura will continue with Sections 1.2-1.4. | |||
=== '''September 23 - Sections 1.3-1.4 and 2.1-2.2''' === | |||
Kanav Madhura will continue with Sections 1.3-1.4. Lucas Duckworth will be ready with Sections 2.1 and 2.2 should there be time. | |||
=== '''September 30 - Sections 2.2 and 2.3-2.5''' === | |||
Lucas Duckworth will finish Section 2.2. Karthik Ravishankar will begin 2.3, 2.4, and 2.5. | |||
=== '''October 7th - Sections 2.4 and 2.5''' === | |||
Karthik Ravishankar will finish, 2.4, and 2.5. Liang Yu will give a talk at 4:00pm. | |||
=== '''October 14th - Sections 2.6 and 2.7''' === | |||
Bjarki Gunnarsson will present Sections 2.6 and 2.7 | |||
=== '''October 21th - Section 3.1''' === | |||
Karthik Ravishankar will present Section 3.1 | |||
=== '''October 28th - Sections 3.2 and 3.3''' === | |||
Karthik Ravishankar will finish Sections 3.2 and John Spoerl will begin Section 3.3 | |||
=== '''November 4th - Sections 3.3 and 3.4''' === | |||
John Spoerl will finish Sections 3.3 and 3.4 | |||
=== '''November 11th - Section 4.1''' === | |||
Antonion Nakid-Cordero will present Section 4.1 | |||
=== '''November 19th - Sections 4.1 and 4.2''' === | |||
Start 4:00PM in VV901! Antonion Nakid-Cordero will continue with Section 4.1, Ang Li will begin Section 4.2. | |||
=== '''November 25th - Sections 4.2 and 4.3''' === | |||
Back to the usual time and place. Ang Li will begin Section 4.2. | |||
=== '''December 2nd - Section 4.3''' === | |||
Ang Li will present Section 4.3. | |||
=== '''December 9nd - Section 5.1''' === | |||
Last seminar for this semester. Sapir Ben-Shahar will begin Section 5.1 | |||
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=== '''September 18 - xxx''' === | |||
'''Title:''' TBA ([https://wiki.math.wisc.edu/images/***.pdf Slides]) | |||
'''Abstract:''' TBA | |||
--> | |||
== Previous Years == | == 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 18:10, 25 April 2025
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 3:30-4:30 PM
- Where: Van Vleck B235
- Organizer: Mariya Soskova
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact the organizers.
Spring 2025
The seminar will be run as a 1-credit seminar Math 975. In Spring 2025, we will finish last semester's topic on Higher Computability Theory.Once we are done students will present a logic topic of their choice (it could be original work, but does not have to be). If you are not enrolled but would like to audit it, please contact Mariya Soskova.
Presentation Schedule: Sign up here.
Notes on Higher Computability Theory: Download the notes here. You will need your UW-login. Please, do not distribute these notes without permission from the author.
January 27 - Organizational Meeting and Sapir Ben-Shahar
Mariya Soskova will call for volunteers to sign up for presentations.
Sapir Ben-Shahar will wrap up Section 5.1
February 3 - Taeyoung Em
Taeyoung Em will present Section 5.3.
February 10 - Hongyu Zhu
Hongyu Zhu will present Section 5.3
February 17 - Karthik Ravishankar
Title: Strong minimal covers and the cupping property
Abstract: A longstanding question in degree theory has been whether every minimal Turing degree has a strong minimal cover. Meanwhile a strong example of degrees without SMC's are those which have the cupping property. It is known that PA degrees have the cupping property, as do degrees with a certain amount of escaping power. On the other hand, it is known that being weak in the sense of being non DNC and Hyperimmune-free lets you have a SMC. Degrees with the cupping property are closed upwards while it is not known if degrees with SMC are closed downwards. It is also not known if every degree either has the cupping property or a SMC. In this talk we will review several of these results and present techniques used to build SMCs.
February 24 - Hongyu Zhu
Title: Seeing the forest does not account for the trees
Abstract: Say a first-order theory (or a type) has bounded axiomatization if it has an axiomatization by [math]\displaystyle{ \forall_n }[/math]-formulas for some finite n. In this talk, we will discuss basic properties of theories and types with (or without) bounded axiomatizations, and in particular whether boundedness of theories implies that of types. (The meaning of the title will be explained in due time.)
March 3 - Uri Andrews
Title: On the spectra of computable models of disintegrated strongly minimal theories with bounded ranks
Abstract: The spectrum of a strongly minimal theory characterizes which of its countable models have computable copies (indexed by their dimensions). We will focus on the disintegrated strongly minimal theories, i.e., where the algebraic closure of a set is the union of the algebraic closures of the elements of the set.
Somewhere in the late aughts, Alice Medvedev and I proved that if a theory is disintegrated strongly minimal and has a finite signature, then either all models are computable, no models are computable, or only the prime model is computable. Steffen Lempp and I tried to push this sort of analysis past finite signatures and we have results about theories which are disintegrated strongly minimal and every symbol in the (infinite) signature has rank less than or equal to 1 in the theory (i.e., you cannot have R(a,b,\bar{z}) if a and b are algebraically independent). Over this past winter break, I found a strategy to bring (some of) this analysis to strongly minimal theories in infinite languages as long as there is some finite N so that every symbol has rank less than or equal to N. I'll describe this strategy, and depending on time, I might even present something that loosely resembles a proof.
March 10 - Logan Heath
Title: Degree Spectra of Theories
Abstract: I will discuss the notion of the degree spectrum of a theory, introduce a class of questions one might ask about such a thing, point to a few of the answers to such questions, and look a little more closely at one such spectrum to highlight the sorts of techniques that arise in the area.
March 17 - Yiqing Wang
Title: The compactness theorem is overrated
Abstract: Elementary classes, or first-order logic in general, are limited in their ability to capture many natural mathematical classes, such as locally finite groups and Archimedean ordered fields. Conversely, obtaining meaningful results in the generality of non-elementary classes can be impossible. In 1978, Shelah introduced the notion of Abstract Elementary Classes (AECs), providing a framework for studying classes that are not first-order axiomatizable yet still possess rich model-theoretic properties and carry the same 'test question'.
In this talk, I will try to give an overview of AECs, prove Shelah’s Presentation Theorem, and highlight some open problems in this area.
March 31 - Chiara Travesset
Title: The Sacks Density Theorem
Abstract: The Sacks Density Theorem states that between any two c.e. degrees, there are two incomparable c.e. degrees. I will present a detailed proof of this theorem.
April 7th - Taeyoung Em
Title: Sets that encode themselves
Abstract: Introreducible sets were introduced by Dekker and Myhill. Mansfield proved that complementary retraceable sets are computable and Seetapun and Slaman proved that complementary introreducible sets are computable. In this talk, I will present some results of Appel and McLaughlin on regressive sets, and maybe some other results.
April 14th - Lucas Duckworth
Title: The Paris-Harrington Theorem and Computability Results for Ramsey Theory
Abstract: The Paris-Harrington Theorem, a slight extension of the famous Finite Ramsey Theorem, was shown to be not provable in PA. This talk will explore this original proof, which proves this result using an interesting variety of arguments combining combinatorics and logic. If time permits, I will also speak about a few results of Jokusch, Specker, Yates, and others on various results using computability to prove properties about homogeneous sets from basic partitions in the original Finite Ramsey Theorem.
April 21 - Ang Li's defense
Title: Computability-Theoretic Analysis of Ordered Groups, Logical Depth, and Introenumerability.
Abstract: The first part of this talk continues the study of connections between reverse mathematics and Weihrauch reducibility. In particular, we analyze the uniform computational power of problems formed from Maltsev’s classification of the order types of countable ordered groups. Several non-reducibility results are obtained via the first-order part—an interior algebraic operator on Weihrauch degrees that has attracted recent attention.
In the second part, we turn to two measure and category questions: knowing that some classes of enumeration degrees have measure zero, what level of randomness can they have or must avoid as reasonable classes of randomness have measure one; knowing that the class of shallow sets is comeager, what level of genericity can deep sets have or must avoid as the classes of generic sets are comeager.
April 28th - Sapir Ben-Shahar
Title: Monadic definability and Matroids
Abstract: Matroids are combinatorial structures that generalise the idea of (in)dependence, such as linear independence of vectors in vector spaces over some field. Matroids arise in a number of different contexts, including from vectors in vector spaces, graphs, points in a geometry, as models of strongly minimal theories, in combinatorial optimization problems, in phylogenetic trees, and many more. This talk will focus on gain-graphic matroids, which are matroids that arise from group-labeled graphs. Gain-graphic matroids are important in the study of structural matroid theory. It turns out that for ``nice enough" classes of matroids, properties that are definable in monadic second-order logic can be recognized in polynomial time. Whether a class of gain-graphic matroids is definable or not depends on which group is chosen to label the graphs. I'll start with a brief introduction to matroids and monadic second order logic, and then describe recent progress on the definability question for gain-graphic matroids.
Fall 2024
The seminar will be run as a 1-credit seminar Math 975 . In Fall 2024, the topic will be Higher Computability Theory. We will follow notes by Noam Greenberg. If you are not enrolled but would like to audit it, please contact Mariya Soskova.
Presentation Schedule: Sign up here.
Notes: Download the notes here. You will need your UW-login. Please, do not distribute these notes without permission from the author.
September 9 - Organizational Meeting
Mariya Soskova will start with the first sections from the notes.
We will then assign speakers to dates and topics.
September 16 - Sections 1.2-1.4
Kanav Madhura will continue with Sections 1.2-1.4.
September 23 - Sections 1.3-1.4 and 2.1-2.2
Kanav Madhura will continue with Sections 1.3-1.4. Lucas Duckworth will be ready with Sections 2.1 and 2.2 should there be time.
September 30 - Sections 2.2 and 2.3-2.5
Lucas Duckworth will finish Section 2.2. Karthik Ravishankar will begin 2.3, 2.4, and 2.5.
October 7th - Sections 2.4 and 2.5
Karthik Ravishankar will finish, 2.4, and 2.5. Liang Yu will give a talk at 4:00pm.
October 14th - Sections 2.6 and 2.7
Bjarki Gunnarsson will present Sections 2.6 and 2.7
October 21th - Section 3.1
Karthik Ravishankar will present Section 3.1
October 28th - Sections 3.2 and 3.3
Karthik Ravishankar will finish Sections 3.2 and John Spoerl will begin Section 3.3
November 4th - Sections 3.3 and 3.4
John Spoerl will finish Sections 3.3 and 3.4
November 11th - Section 4.1
Antonion Nakid-Cordero will present Section 4.1
November 19th - Sections 4.1 and 4.2
Start 4:00PM in VV901! Antonion Nakid-Cordero will continue with Section 4.1, Ang Li will begin Section 4.2.
November 25th - Sections 4.2 and 4.3
Back to the usual time and place. Ang Li will begin Section 4.2.
December 2nd - Section 4.3
Ang Li will present Section 4.3.
December 9nd - Section 5.1
Last seminar for this semester. Sapir Ben-Shahar will begin Section 5.1
Previous Years
The schedule of talks from past semesters can be found here.