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 a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.

The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.

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Fall 2019 - Tentative schedule

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.

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


November 4 - Two short talks

Manlio Valenti and Patrick Nicodemus

November 11 - Manlio Valenti I

November 18 - Manlio Valenti II

November 25 - Two short talks

Speakers TBD

December 2 - Iván Ongay Valverde I

December 9 - Iván Ongay Valverde II

Previous Years

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