- Wednesday 11/01/2017, 17:00 hrs, Week 1,
S17#04-05.

**No Talk**.

- Wednesday 18/01/2017, 17:00 hrs, Week 2,
S17#04-05.

**Dilip Raghavan**.*Michael's Problem*.

I will present some joint work in progress with Juris Steprans on Michael's problem.

- Wednesday 25/01/2017, 17:00 hrs, Week 3,
S17#04-05.

**Chong Chi Tat**.*Ramsey's Theorem for trees*.

We sketch a proof of**Π**conservation of Ramsey's theorem of finite coloring on trees. As a corollary,^{1}_{1}**TT**does not imply^{1}**Σ**induction over_{2}**RCA**._{0}

This is joint work with Li Wei, Wang Wei and Yang Yue.

- Wednesday 01/02/2017, 17:00 hrs, Week 4,
S17#04-05.

**Li Wei**.*Ramsey's theorem for trees (Part II)*. We sketch the internal forcing part of the proof of Ramsey's theorem of finite couloring on trees.

This is joint work with Chong Chit Tat, Wang Wei and Yang Yue.

- Wednesday 08/02/2017, 17:00 hrs, Week 5,
S17#04-05.

**Wang Wei**.*Two constructions of cofinal extensions of countable non-standard models*.

A model**N**of arithmetic is a cofinal extension of**M**iff every element of**N**is below some of**M**. Schmerl proved that every countable recursively saturated model of**IΣ**admits a cofinal elementary extension which is also recursively saturated. A proof of Schmerl's theorem will be sketched. We will also present another result which has a similar flavor: if the standard cut is undefinable in a countable_{1}**M**then in some elementary cofinal extension**N**of**M**the standard cut is still undefinable.

- Wednesday 15/02/2017, 17:00 hrs, Week 6,
S17#04-05.

**David Belanger**.*Pinpointing Fraisse's Conjecture*.

Fraisse's Order-Type Conjecture has to do with sequences of countable total orders embedding into one another. This conjecture was made a Theorem by Laver in the early seventies; we examine some results of Marcone, Montalban and Shore concerning the proof-theoretic strength of this theorem, and highlight some possible areas for future work.

- Wednesday 01/03/2017, 17:00 hrs, Week 7,
S17#04-05.

**Yu Hongyuan**.*Equivalence relations below*.**Δ**^{0}_{2}

We show the existence of universal for**n**-c.e. equvivalence relations and**ω**-c.e. equivalence relations. Furthermore, we study the density of**n**-c.e.,**ω**-c.e. and**Π**equivalence relations. Our conclusion is that all of the above are upwards dense but not downwards dense.^{0}_{1}

- Wednesday 08/03/2017, 17:00 hrs, Week 8,
S17#04-05.

**Steffen Lempp**.*On cototality and the skip operator in the enumeration degrees*.

In the enumeration degrees, we study the notion of cototality (first defined by Pankratov and Solon fifteen years ago (cf. the Pankratov abstract for the 2000 Mal'cev meeting as well as Solon's 2005 and 2006 papers), a notion which has recently been shown to be relevant to other fields like ergodic theory and group theory (cf. Jeandel, in preparation): An enumeration degree is called*cototal*if it contains a set**A**with**A ≤ N-A**. We present several more examples of naturally occurring cototal sets and separate cototality from a number of related notions, like totality, weak cototality and graph cototality.

Closely related to this investigation is the notion of the*skip*operator, which we define by letting**A**be the complement of the set^{◊}**K**. The skip is a weak version of the jump_{A}= {e | e ∈ Φ_{e}(A)}**J**; indeed the following three sets are equivalent in the enumeration degrees:_{e}(A)**J**, the join of_{e}(A)**K**and its complement, the join of_{A}**A**and**A**. However,^{◊}**A**is strictly below**J**in the enumeration degrees, whereas in general that_{e}(A)**A**is incomparable to**A**in the enumeration degrees. We will present a skip inversion theorem and a number of results that the skip operator can exhibit some bizarre behavior.^{◊}

This is joint work with Uri Andrews, Hristo Ganchev, Rutger Kuyper, Joseph Miller, Alexandra Soskova and Mariya Soskova.

- Wednesday 15/03/2017, 17:00 hrs, Week 9,
S17#04-05.

**Wu Guohua**.*Constructions of d-r.e. degrees: hard questions, easy proofs*.

In this talk, I will give a brief introduction of d-r.e. degree constructions, and show how to use easy constructions to obtain known results, which are usually considered to have difficult constructions.

- Wednesday 22/03/2017, 17:00 hrs, Week 10,
S17#04-05.

**Frank Stephan**.*Finitely generated semiautomatic groups*.

The present work shows that Cayley automatic groups are semiautomatic and exhibits some further constructions of semiautomatic groups. Furthermore, the present work establishes that every finitely generated group of nilpotency class 3 is semiautomatic.

This is joint work with Sanjay Jain and Bakhadyr Khoussainov. The paper was presented at CiE 2016; the paper and the slides are available at the speaker's homepage.

- Wednesday 29/03/2017, 17:00 hrs, Week 11,
S17#04-05.

**Juris Steprans**.*Universal structures without saturation*.

A very early result of model theory tell us that saturated structures are also universal. While the negation does not hold, there are some very subtle counterexamples that make the study of universal structures in the absence of saturated ones a fascinating study. Some of the history of this subject and recent results will be examined.

- Wednesday 05/04/2017, 17:00 hrs, Week 12,
S17#04-05.

**Teoh Zu Yao**.*Ramsey algebras, origin & open problems*.

Some 30 years ago, Carlson introduced the notion of a (topological) Ramsey space as a generalization to the Ellentuck space. This generalization allowed him to derive a wide array of classically known combinatorial results such as Ellentuck's theorem, Hindman's theorem, and the Hales-Jewett theorem. When a space in question is induced by an algebra, Carlson suggested that a purely combinatorial approach to the subject should be attempted. This motivates the notion of a Ramsey algebra. In this talk, we will first give a more detailed account of the history of the subject. We will introduce the notion of a Ramsey algebra and explain how it is related to Ramsey spaces. We will also mention a few results that might be of interest to the audience. Finally, we will highlight two open problems whose solutions have remained elusive.

- Wednesday 12/04/2017, 17:00 hrs, Week 13,
S17#04-05.

**Michael McInerney**.*Multiple genericity*.

We introduce a hierarchy of genericity notions between 1-genericity and 2-genericity. We investigate the interaction between these notions and some domination properties related to Downey and Greenberg's hierarchy of totally α-c.a. degrees. Downward density of these generics holds at some levels of the hierarchy, but fails at others. We separate each of these notions in degree.

This is joint work with Ng Keng Meng.