Difference between revisions of "Graduate Logic Seminar"
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=== March 9 - Noah Schweber === | === March 9 - Noah Schweber === | ||
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+ | Title: Algebraic logic and algebraizable logics | ||
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+ | 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?" | ||
+ | |||
+ | 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. | ||
=== March 16 - Spring break - No seminar === | === March 16 - Spring break - No seminar === |
Revision as of 14:17, 6 March 2020
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.
- When: Mondays 4p-5p
- Where: Van Vleck B215.
- Organizers: Omer Mermelstein
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
Contents
- 1 Spring 2020 - Tentative schedule
- 1.1 January 28 - Talk by visitor - No seminar
- 1.2 February 3 - Talk by visitor - No seminar
- 1.3 February 10 - No seminar (speaker was sick)
- 1.4 February 17 - James Hanson
- 1.5 February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott
- 1.6 March 2 - Patrick Nicodemus
- 1.7 March 9 - Noah Schweber
- 1.8 March 16 - Spring break - No seminar
- 1.9 March 23 - Two short talks - Harry Main-Luu and Daniel Belin
- 1.10 March 30 - TBD
- 1.11 April 6 - TBD
- 1.12 April 13 - Faculty at conference - No seminar
- 1.13 April 20 - Harry Main-Luu
- 1.14 April 27 - Harry Main-Luu
- 2 Fall 2019
- 2.1 September 5 - Organizational meeting
- 2.2 September 9 - No seminar
- 2.3 September 16 - Daniel Belin
- 2.4 September 23 - Daniel Belin
- 2.5 September 30 - Josiah Jacobsen-Grocott
- 2.6 October 7 - Josiah Jacobsen-Grocott
- 2.7 October 14 - Tejas Bhojraj
- 2.8 October 23 - Tejas Bhojraj
- 2.9 October 28 - Two short talks
- 2.10 November 4 - Two short talks
- 2.11 November 11 - Manlio Valenti
- 2.12 November 18 - Iván Ongay Valverde
- 2.13 November 25 - Anniversary of the signing of the Treaty of Granada - No seminar
- 2.14 December 2 - Anniversary of the Battle of Austerlitz - No seminar
- 2.15 December 9 - Anniversary of the death of Pope Pius IV - No seminar
- 3 Previous Years
Spring 2020 - Tentative schedule
January 28 - Talk by visitor - No seminar
February 3 - Talk by visitor - No seminar
February 10 - No seminar (speaker was sick)
February 17 - James Hanson
Title: The Topology of Definable Sets in Continuous Logic
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.
February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott
Tejas Bhojraj - Quantum Kolmogorov Complexity.
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.
Josiah Jacobsen-Grocott - A Characterization of Strongly $\eta$-Representable Degrees.
Abstract: $\eta$-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 $\Sigma^0_2$ 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: A Sheaf-theoretic generalization of Los's theorem
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.
March 9 - Noah Schweber
Title: Algebraic logic and algebraizable logics
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?"
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.
March 16 - Spring break - No seminar
March 23 - Two short talks - Harry Main-Luu and Daniel Belin
March 30 - TBD
April 6 - TBD
April 13 - Faculty at conference - No seminar
April 20 - Harry Main-Luu
April 27 - Harry Main-Luu
Fall 2019
September 5 - Organizational meeting
September 9 - No seminar
September 16 - Daniel Belin
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.
September 23 - Daniel Belin
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued
September 30 - Josiah Jacobsen-Grocott
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 Jacobsen-Grocott
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 Semi-Open Coloring Axiom (SOCA)
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):
- 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 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.
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 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 - 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 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.
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 non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) 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 here.