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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:''' TBA 
−  * '''Where:''' Van Vleck B235 (unless otherwise announced).  +  * '''Where:''' on line (ask for code). 
−  * '''Organizers:''' [https://www.math.wisc.edu/~schweber/ Noah Schweber]  +  * '''Organizers:''' [https://www.math.wisc.edu/~jgoh/ Jun Le Goh] 
   
−  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: joingradlogicsem@lists.wisc.edu 
   
−  === January 29, Organizational meeting ===  +  == Spring 2021  Tentative schedule == 
   
−  This day we decided the schedule for the semester.
 +  === February 16 3:30PM  Short talk by Sarah Reitzes (University of Chicago) === 
   
−  === February 5, [http://www.math.wisc.edu/~andrews/ Uri Andrews] ===
 +  Title: Reduction games over $\mathrm{RCA}_0$ 
   
−  Title: Building Models of Strongly Minimal Theories  Part 1
 +  Abstract: In this talk, I will discuss joint work with Damir D. Dzhafarov and Denis R. Hirschfeldt. Our work centers on the characterization of problems P and Q such that P $\leq_{\omega}$ Q, as well as problems P and Q such that $\mathrm{RCA}_0 \vdash$ Q $\to$ P, in terms of winning strategies in certain games. These characterizations were originally introduced by Hirschfeldt and Jockusch. I will discuss extensions and generalizations of these characterizations, including a certain notion of compactness that allows us, for strategies satisfying particular conditions, to bound the number of moves it takes to win. This bound is independent of the instance of the problem P being considered. This allows us to develop the idea of Weihrauch and generalized Weihrauch reduction over some base theory. Here, we will focus on the base theory $\mathrm{RCA}_0$. In this talk, I will explore these notions of reduction among various principles, including bounding and induction principles. 
   
−  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
 +  === March 23 4:15PM  Steffen Lempp === 
−  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: Degree structures and their finite substructures 
   
−  Title: Finding Definable Sets in Continuous Logic
 +  Abstract: Many problems in mathematics can be viewed as being coded by sets of natural numbers (as indices). 
 +  One can then define the relative computability of sets of natural numbers in various ways, each leading to a precise notion of “degree” of a problem (or set). 
 +  In each case, these degrees form partial orders, which can be studied as algebraic structures. 
 +  The study of their finite substructures leads to a better understanding of the partial order as a whole. 
   
−  Abstract: In order to be useful the notion of a 'definable set' in continuous logic is stricter than a naive comparison to discrete logic
 +  === March 30 4PM  Alice Vidrine === 
−  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 nontrivial definable sets. As we'll see, however, there are many definable sets in omegastable,
 
−  omegacategorical, and other small theories.
 
   
−  === February 19, [https://sites.google.com/a/wisc.edu/schweber/ Noah Schweber] ===
 +  Title: Categorical logic for realizability, part I: Categories and the Yoneda Lemma 
   
−  Title: Proper forcing
 +  Abstract: An interesting strand of modern research on realizabilitya semantics for nonclassical logic based on a notion of computationuses the language of toposes and Grothendieck fibrations to study mathematical universes whose internal notion of truth is similarly structured by computation. The purpose of this talk is to establish the basic notions of category theory required to understand the tools of categorical logic developed in the sequel, with the end goal of understanding the realizability toposes developed by Hyland, Johnstone, and Pitts. The talk will cover the definitions of category, functor, natural transformation, adjunctions, and limits/colimits, with a heavy emphasis on the ubiquitous notion of representability. 
   
−  Abstract: Although a given forcing notion may have nice properties on its own, those properties might vanish when we apply it repeatedly.
 +  [https://hilbert.math.wisc.edu/wiki/images/Catslides1.pdf Link to slides] 
−  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 ===  +  === April 27 4PM  Alice Vidrine === 
   
−  Title: A survey of computable and constructive mathematics in economic history  +  Title: Categorical logic for realizability, part II 
   
−  === March 5, [http://www.math.wisc.edu/~makuluni/ Tamvana Makulumi] ===
 +  Abstract: Realizability is an approach to semantics for nonclassical logic that interprets propositions by sets of abstract computational data. One modern approach to realizability makes heavy use of the notion of a topos, a type of category that behaves like a universe of nonstandard sets. In preparation for introducing realizability toposes, the present talk will be a brisk introduction to the notion of a topos, with an emphasis on their logical aspects. In particular, we will look at the notion of a subobject classifier and the internal logic to which it gives rise. 
−   
−  Title: Convexly Orderable Groups
 
−   
−  === March 12, [https://math.nd.edu/people/visitingfaculty/danielturetsky/ Dan Turetsky] (University of Notre Dame) ===
 
−   
−  Title: Structural Jump
 
−   
−  === March 19, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] ===
 
−   
−  Title: Networks and degrees of points in nonsecond 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, [http://www.math.wisc.edu/~ongay/ Iván OngayValverde] ===
 
−   
−  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 antichain complements on <math>\omega^{<\omega}</math>, and as the degrees of enumerationpointed trees on <math>2^{<\omega}</math>, and we will remark on some additional applications of these characterizations.
 
−   
−  === April 30, [http://www.math.wisc.edu/~ongay/ Iván OngayValverde]===
 
−   
−  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, [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 modeltheoretic 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 Ramseytype 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 GromovHausdorff metric on type space in continuous logic
 
−   
−  Abstract: The GromovHausdorff 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 noncompact metric spaces, but in that context it is only a pseudometric: there are nonisomorphic metric spaces with GromovHausdorff distance 0. This gives rise to an equivalence relation that is slightly coarser than isomorphism. There are continuous firstorder 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 RyllNardzewski 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 GromovHausdorff 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 CantorBendixson analysis on type spaces in continuous firstorder 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 GromovHausdorff (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 GHcategorical theories have 'GHatomic' models over arbitrary sets, but GHatomic models fail to be GHunique in general.
 
−   
−  === October 23, [http://www.math.wisc.edu/~makuluni/ Tamvana Makulumi] ===
 
−   
−  Title: Boxy sets in ordered convexlyorderable structures.
 
−   
−  === October 30, [http://www.math.wisc.edu/~ongay/ Iván OngayValverde] ===
 
−   
−  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 MartinLöf 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 MartinLöf randoms.
 
−   
−  === December 4, Tejas Bhojraj ===
 
−   
−  Title: Quantum Algorithmic Randomness
 
−   
−  Abstract: I will discuss the recent paper by Nies and Scholz where they define quantum MartinLof randomness (qMLR) for infinite sequences of qubits. If time permits, I will introduce the notion of quantum Solovay randomness and show that it is equivalent to qMLR in some special cases.
 
−   
−  === December 11, Grigory Terlov ===
 
−   
−  Title: The Logic of Erdős–Rényi Graphs
 
   
 ==Previous Years==   ==Previous Years== 
   
−  The schedule of talks from past semesters can be found [[Logic Graduate Seminar, previous semestershere]].  +  The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semestershere]]. 