Difference between revisions of "Graduate Logic Seminar"

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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 B139
* '''Organizers:''' [https://www.math.wisc.edu/~msoskova/ Mariya Soskava]
+
* '''Organizers:''' Karthik Ravishankar and [https://sites.google.com/wisc.edu/antonio Antonio Nakid Cordero]
  
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:  join-grad-logic-sem@lists.wisc.edu
  
=== January 29, Organizational meeting ===
+
== Fall 2022 ==
  
This day we decided the schedule for the semester.
+
=== September 12 - Organizational Meeting ===
  
=== February 5, [http://www.math.wisc.edu/~andrews/ Uri Andrews] ===
+
We will meet to assign speakers to dates.
  
Title: Building Models of Strongly Minimal Theories - Part 1
+
=== '''September 19 - Karthik Ravishankar''' ===
 +
'''Title:''' Lowness for Isomorphism ([https://wiki.math.wisc.edu/images/Karthik_talk.pdf Slides])
  
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
+
'''Abstract:''' A Turing degree is said to be low for isomorphism if it can only compute an isomorphism between computable structures only when a computable isomorphism already exists. In this talk, we show that the measure of the class of low for isomorphism sets in Cantor space is 0 and that no Martin Lof random is low for isomorphism.
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 ===
+
=== '''September 26 - Antonio Nakid Cordero'''  ===
 +
'''Title:''' When Models became Polish: an introduction to the Topological Vaught Conjecture ([[Media:GradLogSem - Topological Vaught Conjecture.pdf|Slides]])
  
Title: Finding Definable Sets in Continuous Logic
+
'''Abstract:''' Vaught's Conjecture, originally asked by Vaught in 1961, is one of the most (in)famous open problems in mathematical logic. The conjecture is that a complete theory on a countable language must either have countably-many or continuum-many non-isomorphic models. In this talk, we will discuss some of the main ideas that surround this conjecture, with special emphasis on a topological generalization in terms of the continuous actions of Polish groups.
  
Abstract: In order to be useful the notion of a 'definable set' in continuous logic is stricter than a naive comparison to discrete logic
+
=== '''October 10 - Yunting Zhang''' ===
would suggest. As a consequence, even in relatively tame theories there can be very few definable sets. For example, there is a
+
'''Title:''' Some History of Logic ([[Media:Godel.pdf|Slides]])
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:''' The lives of great thinkers are sometimes overshadowed by their achievements-and there is perhaps no better illustration of this phenomenon than the life and work of Gödel. Take a look at Gödel's own timeline and see how wars and other mathematicians influenced him.
  
Title: Proper forcing
+
=== '''October 17 - Alice Vidrine''' ===
 +
'''Title:''' Local operators, bilayer Turing reducibility, and enumeration Weihrauch degrees ([[Media:LTeW-talk.pdf|Slides]])
  
Abstract: Although a given forcing notion may have nice properties on its own, those properties might vanish when we apply it repeatedly.
+
'''Abstract:'''  Realizability toposes have a rich variety of subtoposes, corresponding to their local operators. These local operators are somewhat difficult to study in their usual form, which seems far removed from the usual objects of computability theoretic study. Recent work by Takayuki Kihara has given a characterization of the local operators on the effective topos in computability theoretic terms related to Weihrauch reduction, and which generalizes to several other realizability toposes of possible interest to computability theorists. This narrative-focused talk outlines what a realizability topos looks like, what local operators are, what Kihara's bilayer Turing reduction looks like, and how this leads to preliminary questions about a relative of the Weihrauch degrees based on enumeration reduction.
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 ===
+
=== '''October 24 - Hongyu Zhu''' ===
 +
'''Title:''' Investigating Natural Theories through the Consistency Operator ([[Media:ConsistencyOperator.pdf|Slides]])
  
Title: A survey of computable and constructive mathematics in economic history
+
'''Abstract:''' The phenomenon that "natural" theories tend to be linearly ordered in terms of consistency strength is a long-standing mystery. One approach to solving the problem is Martin's Conjecture, which roughly claims that the only natural functions on the Turing degrees are transfinite iterates of the Turing jump. In this talk we will focus on a similar approach, working inside the Lindenbaum algebra of elementary arithmetic instead of the Turing degrees. Here, the consistency operator takes the role of the jump. We will see that while some nice analogous claims can be established, there are also counterexamples that prevent us from strengthening the results in various ways.
  
=== March 5, [http://www.math.wisc.edu/~makuluni/ Tamvana Makulumi] ===
+
=== '''October 31 - Break for Halloween''' ===
  
Title: Convexly Orderable Groups
+
=== '''November 7 - John Spoerl''' ===
 +
'''Title:''' Universal Algorithms
  
=== March 12, [https://math.nd.edu/people/visiting-faculty/daniel-turetsky/ Dan Turetsky] (University of Notre Dame) ===
+
'''Abstract:''' Among the many ways we can flex Gödel’s Incompleteness theorems, there is one that feels especially strange: there is a partial computable function F, such that F gives no output in the standard model of arithmetic, but for any function G on natural numbers (computable or not) there is a non-standard model in which F behaves exactly as G. I’ll discuss some related arguments and some philosophical questions this may raise about our notions of finiteness and determinism.
  
Title: Structural Jump
+
=== '''November 14 - Josiah Jacobsen-Grocott''' ===
 +
'''Title:''' The Paris-Harrington Theorem
  
=== March 19, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] ===
+
'''Abstract:''' In this talk, I will present the proof of the Paris-Harrington theorem. The
 +
Paris-Harrington theorem states that over PA the consistency of Peano arithmetic is equivalent to
 +
a strengthening of the finite Ramsey theorem. This was the first example of a result from "ordinary
 +
mathematics" that can not be proven by PA. The aim of this talk is to cover the main logical
 +
steps in this proof and give some of the combinatorics.
  
Title: Networks and degrees of points in non-second countable spaces
+
=== '''November 21 - Karthik Ravishankar''' ===
 +
'''Title:''' The computing power of Baire space vs Cantor Space
  
=== April 2, Wil Cocke ===
+
'''Abstract:''' Generic Muchnik reducibility extends the notion of Muchnik reducibility to the uncountable setting. In this talk we'll look at some recent work which comes up with another technique of constructing a structure of degree strictly between Cantor space and Baire Space. We'll see that there is a generic copy of Cantor space which computes no escaping function while every copy of Baire space computes a dominating function. We then construct a structure of intermediate degree which always computes an escaping function but no dominating function.
  
Title: Characterizing Finite Nilpotent Groups via Word Maps
+
=== '''November 28 - Logan Heath''' ===
 +
'''Title''': A Mathematical Analysis of Theories of Generative Grammar: The Peters-Ritchie Theorem
  
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.
+
'''Abstract''': This is a preview of a talk intended primarily for undergraduate students in linguistics who have just completed a first course in syntax. The goal of the talk is to spark interest in applications of mathematical techniques to the study of theories of syntax. We will focus on the Peters-Ritchie Theorem which shows that transformational grammars can generate languages which are c.e., but not computable.
  
=== April 9, Tejas Bhojraj ===
+
=== '''December 5 - Logan Heath''' ===
  
Title: Quantum Randomness
+
=== '''December 12 - Yuxiao Fu''' ===
  
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.
+
== Previous Years ==
  
=== April 16, [http://www.math.wisc.edu/~ongay/ Iván Ongay-Valverde] ===
+
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].
 
 
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, [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 23:42, 27 November 2022

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 B139
  • Organizers: Karthik Ravishankar and Antonio Nakid Cordero

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

Fall 2022

September 12 - Organizational Meeting

We will meet to assign speakers to dates.

September 19 - Karthik Ravishankar

Title: Lowness for Isomorphism (Slides)

Abstract: A Turing degree is said to be low for isomorphism if it can only compute an isomorphism between computable structures only when a computable isomorphism already exists. In this talk, we show that the measure of the class of low for isomorphism sets in Cantor space is 0 and that no Martin Lof random is low for isomorphism.

September 26 - Antonio Nakid Cordero

Title: When Models became Polish: an introduction to the Topological Vaught Conjecture (Slides)

Abstract: Vaught's Conjecture, originally asked by Vaught in 1961, is one of the most (in)famous open problems in mathematical logic. The conjecture is that a complete theory on a countable language must either have countably-many or continuum-many non-isomorphic models. In this talk, we will discuss some of the main ideas that surround this conjecture, with special emphasis on a topological generalization in terms of the continuous actions of Polish groups.

October 10 - Yunting Zhang

Title: Some History of Logic (Slides)

Abstract: The lives of great thinkers are sometimes overshadowed by their achievements-and there is perhaps no better illustration of this phenomenon than the life and work of Gödel. Take a look at Gödel's own timeline and see how wars and other mathematicians influenced him.

October 17 - Alice Vidrine

Title: Local operators, bilayer Turing reducibility, and enumeration Weihrauch degrees (Slides)

Abstract: Realizability toposes have a rich variety of subtoposes, corresponding to their local operators. These local operators are somewhat difficult to study in their usual form, which seems far removed from the usual objects of computability theoretic study. Recent work by Takayuki Kihara has given a characterization of the local operators on the effective topos in computability theoretic terms related to Weihrauch reduction, and which generalizes to several other realizability toposes of possible interest to computability theorists. This narrative-focused talk outlines what a realizability topos looks like, what local operators are, what Kihara's bilayer Turing reduction looks like, and how this leads to preliminary questions about a relative of the Weihrauch degrees based on enumeration reduction.

October 24 - Hongyu Zhu

Title: Investigating Natural Theories through the Consistency Operator (Slides)

Abstract: The phenomenon that "natural" theories tend to be linearly ordered in terms of consistency strength is a long-standing mystery. One approach to solving the problem is Martin's Conjecture, which roughly claims that the only natural functions on the Turing degrees are transfinite iterates of the Turing jump. In this talk we will focus on a similar approach, working inside the Lindenbaum algebra of elementary arithmetic instead of the Turing degrees. Here, the consistency operator takes the role of the jump. We will see that while some nice analogous claims can be established, there are also counterexamples that prevent us from strengthening the results in various ways.

October 31 - Break for Halloween

November 7 - John Spoerl

Title: Universal Algorithms

Abstract: Among the many ways we can flex Gödel’s Incompleteness theorems, there is one that feels especially strange: there is a partial computable function F, such that F gives no output in the standard model of arithmetic, but for any function G on natural numbers (computable or not) there is a non-standard model in which F behaves exactly as G. I’ll discuss some related arguments and some philosophical questions this may raise about our notions of finiteness and determinism.

November 14 - Josiah Jacobsen-Grocott

Title: The Paris-Harrington Theorem

Abstract: In this talk, I will present the proof of the Paris-Harrington theorem. The Paris-Harrington theorem states that over PA the consistency of Peano arithmetic is equivalent to a strengthening of the finite Ramsey theorem. This was the first example of a result from "ordinary mathematics" that can not be proven by PA. The aim of this talk is to cover the main logical steps in this proof and give some of the combinatorics.

November 21 - Karthik Ravishankar

Title: The computing power of Baire space vs Cantor Space

Abstract: Generic Muchnik reducibility extends the notion of Muchnik reducibility to the uncountable setting. In this talk we'll look at some recent work which comes up with another technique of constructing a structure of degree strictly between Cantor space and Baire Space. We'll see that there is a generic copy of Cantor space which computes no escaping function while every copy of Baire space computes a dominating function. We then construct a structure of intermediate degree which always computes an escaping function but no dominating function.

November 28 - Logan Heath

Title: A Mathematical Analysis of Theories of Generative Grammar: The Peters-Ritchie Theorem

Abstract: This is a preview of a talk intended primarily for undergraduate students in linguistics who have just completed a first course in syntax. The goal of the talk is to spark interest in applications of mathematical techniques to the study of theories of syntax. We will focus on the Peters-Ritchie Theorem which shows that transformational grammars can generate languages which are c.e., but not computable.

December 5 - Logan Heath

December 12 - Yuxiao Fu

Previous Years

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