

(38 intermediate revisions by 4 users not shown) 
Line 1: 
Line 1: 
−  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 4p5p  +  * '''When:''' Tuesdays 45 PM 
−  * '''Where:''' on line (ask for code).  +  * '''Where:''' Van Vleck 901 
 * '''Organizers:''' [https://www.math.wisc.edu/~jgoh/ Jun Le Goh]   * '''Organizers:''' [https://www.math.wisc.edu/~jgoh/ Jun Le Goh] 
   
Line 9: 
Line 9: 
 Sign up for the graduate logic seminar mailing list: joingradlogicsem@lists.wisc.edu   Sign up for the graduate logic seminar mailing list: joingradlogicsem@lists.wisc.edu 
   
−  == Fall 2020  Tentative schedule ==  +  == Fall 2021 tentative schedule == 
   
−  === January 28  Talk by visitor  No seminar ===
 +  To see what's happening in the Logic qual preparation sessions click [[Logic Qual Prephere]]. 
−  === February 3  Talk by visitor  No seminar ===
 
−  === February 10  No seminar (speaker was sick) ===
 
   
−  === February 17  James Hanson ===  +  === September 14  organizational meeting === 
   
−  Title: The Topology of Definable Sets in Continuous Logic
 +  We met to discuss the schedule. 
   
−  Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.
 +  === September 28  Ouyang Xiating === 
   
−  === February 24  Two short talks  Tejas Bhojraj and Josiah JacobsenGrocott ===
 +  Title: Firstorder logic, database and consistent query answering 
   
−  '''Tejas Bhojraj'''  Quantum Kolmogorov Complexity.
 +  Abstract: Databases are a crucial component of many (if not all) modern 
 +  applications. In reality, the data stored are often dirty and contain 
 +  duplicated/missing entries, and it is a natural practice to clean the data 
 +  first before executing the query. However, the same query might return 
 +  different answers on different cleaned versions of the dataset. It is then 
 +  helpful to compute the consistent answers: the query answers that will always 
 +  be returned, regardless of how the dirty data is cleaned. In this talk, we 
 +  first introduce the connection between firstorder logic and query languages 
 +  on databases, and then discuss the problem of Consistent Query Answering 
 +  (CQA): How to compute consistent answers on dirty data? Finally, we show 
 +  when the CQA problem can be solved using firstorder logic for path queries. 
   
−  Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.
 +  === October 12  Karthik Ravishankar === 
   
−  '''Josiah JacobsenGrocott'''  A Characterization of Strongly $\eta$Representable Degrees.
 +  Title: Notions of randomness for subsets of the Natural Numbers 
   
−  Abstract:  +  Abstract: There are a number of notions of randomness of sets of natural numbers. These notions have been defined based on what a 'random object' should behave like such as being 'incompressible' or being 'hard to predict' etc. There is often a interplay between computability and randomness aspects of subsets of natural numbers. In this talk we motivate and present a few different notions of randomness and compare their relative strength. 
−  $\eta$representations are a way of coding sets in computable linear orders that were first
 
−  introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to
 
−  characterize the sets with $\eta$representations as well as the sets with subclasses of
 
−  $\eta$representations except for the case of sets with strong $\eta$representations, the only
 
−  class where the order type of the representation is unique.
 
   
−  We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$
 +  === October 26  Alice Vidrine === 
−  approximations. We use connected approximations to
 
−  give a characterization of the degrees with strong $\eta$representations as well new
 
−  characterizations of the subclasses of $\eta$representations with known characterizations.
 
   
−  === March 2  Patrick Nicodemus ===
 +  Title: Categorical logic for realizability, part III: Actual realizability 
   
−  Title: A Sheaftheoretic generalization of Los's theorem
 +  Abstract: Realizability is an approach to semantics for nonclassical logic that interprets propositions by sets of abstract computational data. In the present talk we describe the notion of a Schonfinkel algebra (also called a partial combinatory algebra), which gives us a very general notion of computation. We then describe the construction of a topos whose notions of morphism and subobject must respect the computational structure, and describe the unusual features of these toposes, closing with some discussion of LawvereTierney topologies on such toposes. 
   
−  Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit  the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory  this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.
 +  (The abstracts for parts I and II, which were given in spring 2021, can be found [https://hilbert.math.wisc.edu/wiki/index.php/Graduate_Logic_Seminar,_previous_semesters#March_30_4PM__Alice_Vidrine here].) 
   
−  === March 9  Noah Schweber ===  +  === November 9  Antonio Nákid Cordero === 
   
−  Title: Algebraic logic and algebraizable logics
 +  === November 23  Antonio Nákid Cordero? === 
   
−  Abstract: Arguably the oldest theme in what we would recognize as "mathematical logic" is the algebraic interpretation of logic, the most famous example of this being the connection between (classical) propositional logic and Boolean algebras. But underlying the subject of algebraic logic is the implicit assumption that many logical systems are "satisfyingly" interpreted as algebraic structures. This naturally hints at a question, which to my knowledge went unasked for a surprisingly long time: when does a logic admit a "nice algebraic interpretation?"
 +  === December 7  John Spoerl === 
   
−  Perhaps surprisingly, this is actually a question which can be made precise enough to treat with interesting results. I'll sketch what is probably the first serious result along these lines, due to Blok and Pigozzi, and then say a bit about where this aspect of algebraic logic has gone from there.
 +  == Previous Years == 
−   
−  === March 16  Spring break  No seminar ===
 
−   
−  === '''Due to the cancellation of facetoface instruction in UWMadison through at least April 10, the seminar is suspended until further notice''' ===
 
−   
−   
−   
−  == Fall 2019 ==
 
−   
−  === September 5  Organizational meeting ===
 
−   
−  === September 9  No seminar ===
 
−   
−  === September 16  Daniel Belin ===
 
−  Title: Lattice Embeddings of the mDegrees and Second Order Arithmetic
 
−   
−  Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the mdegrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the manyone degrees codes satisfiability in secondorder 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 ordertheoretic properties of manyone reducibility.
 
−   
−  === September 23  Daniel Belin ===
 
−   
−  Title: Lattice Embeddings of the mDegrees and Second Order Arithmetic  Continued
 
−   
−  === September 30  Josiah JacobsenGrocott ===
 
−   
−  Title: Scott Rank of Computable Models
 
−   
−  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.
 
−   
−  === October 7  Josiah JacobsenGrocott ===
 
−   
−  Title: Scott Rank of Computable Codels  Continued
 
−   
−  === October 14  Tejas Bhojraj ===
 
−   
−  Title: Solovay and Schnorr randomness for infinite sequences of qubits.
 
−   
−  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.
 
−   
−  === October 23  Tejas Bhojraj ===
 
−   
−  Title: Solovay and Schnorr randomness for infinite sequences of qubits  continued
 
−   
−  Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.
 
−   
−  === October 28  Two short talks ===
 
−   
−  '''Iván Ongay Valverde'''  Exploring different versions of the SemiOpen Coloring Axiom (SOCA)
 
−   
−  In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the SemiOpen 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):
 
−   
−   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.
 
−   
−  '''James Earnest Hanson'''  Strongly minimal sets in continuous logic
 
−   
−  The precise structural understanding of uncountably categorical theories given by the proof of the BaldwinLachlan 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 BaldwinLachlan style characterization in the presence of strongly minimal sets.
 
−   
−  === November 4  Two short talks ===
 
−   
−  '''Manlio Valenti'''  The complexity of closed Salem sets (20 minutes version)
 
−   
−  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 ndimensional 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.
 
−   
−  '''Patrick Nicodemus'''  Proof theory of Second Order Arithmetic and System F
 
−   
−  A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form"  a direct proof without any detours. Such results have numerous and immediate consequences  often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.
 
−   
−  === 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 ndimensional 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  Iván Ongay Valverde ===
 
−   
−  Title: A couple of summer results
 
−   
−  Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.
 
−   
−  In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$vector spaces of $\mathbb{R}$. We will show that there is a nonmeasurable set whose Turing closure becomes measurable (and one that stays nonmeasurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.
 
−   
−  === November 25  Anniversary of the signing of the Treaty of Granada  No seminar ===
 
−   
−  === December 2  Anniversary of the Battle of Austerlitz  No seminar ===
 
−   
−  === December 9  Anniversary of the death of Pope Pius IV  No seminar ===
 
−   
−  ==Previous Years==  
   
 The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semestershere]].   The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semestershere]]. 