Logic Seminar in Semester I AY 2015/2016
Talks are Wednesday afternoon in Seminar Room S17#04-06
and start at 17:00 hrs. Here an overview over the topics.
- 12/08/2015, 17:00 hrs, Week 1, S17#04-05.
Borisa Kuzeljevic.
Isomorphic Substructures of Fraïssé Limits.
We will present some results on embedding linear orders into the posets of
isomorphic substructures of Fraïssé limits. For a relational
structure X we denote this poset by P(X).
Because each chain in a poset can be extended to a maximal one, it is
enough to characterize the class M(X) of order types of all
maximal chains in P(X). For example, if the structure
X is either the rational line, the random k-uniform
hypergraph, the random poset or some of the Henson graphs, then a linear
order belongs to the class M(X) if and only if it is isomorphic
to the order type of a compact set of reals whose minimum is not isolated.
We will also show that our construction is applicable to any
Fraïssé structure whose age satisfies the strong amalgamation
property.
This is a joint work with Milos Kurilic.
- 19/08/2015, 17:00 hrs, Week 2, S17#04-06.
Ding Longyun.
On equivalence relations generated by Schauder bases.
In this talk, a notion of Schauder equivalence relation
RN/L is introduced, where L is a linear
subspace of RN and the unit vectors en
form a Schauder basis of L, where each vector in
RN can be written as
Σn an×en.
The main theorem shows that the following conditions are equivalent:
(1) The unit vector basis is boundedly complete;
(2) L is Fσ in RN;
(3) RN/L is Borel reducible to
RN/Ɩ∞.
We show that a Schauder equivalence relation generalised by any basis of
RN/Ɩ2 itself, but it is not true
for bases of c0 or Ɩ1.
Furthermore, among all Schauder equivalence relations generated by
sequences in c0, we find the minimum and the maximum
elements with respect to Borel reducibility.
We also schow that RN/Ɩp is
Borel reducible to RN/J iff p≤2
where J is James' space.
A pdf abstract is also available.
- 26/08/2015, 17:00 hrs, Week 3, S17#04-06.
Gao Su. Borel complete sections in Bernoulli shifts.
In this talk I will present some results about Borel complete
sections in Bernoulli shifts which are proved by a forcing construction.
These results reveal regularity properties of Borel complete sections for
countable group actions, and at the same time rule out certain Borel
constructions that turn out to be too regular to exist.
This is joint work with Steve Jackson, Brandon Seward and Edward Krohne.
- 02/09/2015, 17:00 hrs, Week 4, S17#04-06.
Alexander Kreuzer.
On the Uniform Computational Content of Computability Theory.
We demonstrate that the Weihrauch lattice can be used to study the
uniform computational content of computability theoretic properties
and theorems in one common setting.
The properties that we study include diagonal non-computability,
hyperimmunity, complete extensions of Peano arithmetic, 1-genericity,
Martin-Löf randomness and cohesiveness. The theorems that we include
in our case study are the Low Basis Theorem of Jockusch and Soare, the
Kleene-Post Theorem and Friedman's Jump Inversion Theorem. It turns
out that all the aforementioned properties and many theorems in
computability theory, including all theorems that claim the existence
of some Turing degree, have very little uniform computational content.
They are all located outside of the upper cone of binary choice (also
known as LLPO) and we call problems with this property
indiscriminative. Since practically all theorems from classical
analysis whose computational content has been classified are
discriminative, our observation could yield an explanation for why
theorems and results in computability theory typically have very
little direct consequences in other disciplines such as analysis.
Two notable exceptions to this are the Low Basis Theorem which is
discriminative, this is perhaps why it is considered to be one of the
most applicable theorems in computability theory, and the Baire
category theorem (or to be precise certain formulations of it) which
is an indiscriminative principle occurring in mathematical
analysis. We will see that the Baire category theorem is related to
1-genericity.
In some cases a bridge between the indiscriminative world and the
discriminative world of classical mathematics can be established via a
suitable residual operation and we demonstrate this in case of the
cohesiveness problem, which turns out to be the quotient of two
discriminative problems, namely the limit operation and the jump of
Weak König's Lemma.
This is joint work with Vasco Brattka and Matthew Hendtlass.
- 09/09/2015, 17:00 hrs, Week 5, S17#04-06.
David Belanger. New conservation theorems for COH.
We prove a series of conservation results for COH over various axiom
systems, extending earlier results of Cholak-Jockusch-Slaman and of
Chong-Slaman-Yue. Our method is new, however, and carries a number of
advantages. Along the way we revisit and formalize several well-known
theorems from recursion theory, including some from Post, from
Friedberg and from Jockusch-Stephan.
- 16/09/2015, 17:00 hrs, Week 6, S17#04-06.
Dilip Raghavan.
Real games and strategically selective coideals, Part 1.
A classical theorem of Mathias says that the Axiom of
Determinacy for Real games (ADR) implies that
there are no tall selective coideals on omega. Farah introduced the
notion of a semiselective coideal and proved in ZF that these
are precisely those ideals on omega whose quotient adds a selective
ultrafilter. It is unkown whether ADR implies the
non-existence of tall semiselective coideals. We introduce the notion
of strategically selective coideal, and show that tall strategically
selective coideals do not exist under ADR. The
Axiom of Choice implies that the notions of semiselective and
strategically selective coideal coincide.
This is joint work with Paul Larson.
- 23/09/2015, 17:00 hrs, Recess Week, S17#04-06.
No Talk.
- 30/09/2015, 17:00 hrs, Week 7, S17#04-06.
Chong Chi Tat.
Ramsey's Theorem on Trees.
Ramsey's theorem on trees concerns the existence of a
monochromatic tree isomorphic to the full binary tree, for a given finite
colouring of the latter. The existence of a monochromatic perfect tree is
immediate in the system of Peano arithmetic. The interesting question is
to identify the weakest system that is sufficient for it. We will give a
progress report of our study.
This is joint work with Li Wei and Wang Wei.
- 07/10/2015, 17:00 hrs, Week 8, S17#04-06.
Dilip Raghavan.
Real games and strategically selective coideals, Part 2.
A classical theorem of Mathias says that the Axiom of
Determinacy for Real games (ADR) implies that
there are no tall selective coideals on omega. Farah introduced the
notion of a semiselective coideal and proved in ZF that these
are precisely those ideals on omega whose quotient adds a selective
ultrafilter. It is unkown whether ADR implies the
non-existence of tall semiselective coideals. We introduce the notion
of strategically selective coideal, and show that tall strategically
selective coideals do not exist under ADR. The
Axiom of Choice implies that the notions of semiselective and
strategically selective coideal coincide.
This talk is a continuation of the talk from 16/09/2015 and will
provide more mathematical details and selected proofs.
This is joint work with Paul Larson.
- 14/10/2015, 17:00 hrs, Week 9, S17#04-06.
Frank Stephan.
Covering the recursive sets.
The talk gives an overview of recent work which solves two problems
of Brendle, Brooke-Taylor, Ng and Nies. One construction is based on
a construction of Khan and Miller from 2014, the other construction
is a direct construction utilising martingales. Furthermore, the
talk introduces the notions of infinitely often subuniform families
and shows first results for these; these concepts also permit to get
some result towards the solution of the third problem of Brendle,
Brooke-Taylor, Ng and Nies.
This talk is based on joint work with Bjørn Kjos-Hanssen
and Sebastiaan A. Terwijn.
The current version of the paper is available on the
homepage
of Frank Stephan and had been presented at CiE 2015;
the paper of Brendle, Brooke-Taylor, Ng and Nies is on
http://arxiv.org/abs/1404.2839.
- 21/10/2015, 17:00 hrs, Week 10, S17#04-06.
Frank Stephan. The complexity of semiautomatic structures.
This talk extends previous work on semiautomatic structures and in
particular investigates complexity issues with groups and semigroups;
furthermore, results from recursion theory are used to show that one
specific structure is not semiautomatic while a related one can be
shown to be semiautomatic.
This is joint work with Sanjay Jain, Bakhadyr Khoussainov,
Dan Teng and Siyuan Zou.
- 28/10/2015, 17:00 hrs, Week 11, S17#04-06.
Jason Teutsch.
When cryptocurrencies mine their own business.
Bitcoin and hundreds of other cryptocurrencies employ a consensus protocol
called Nakamoto consensus which reward miners for maintaining a public
blockchain. In this talk, we study the security of this protocol with
respect to rational miners and show how a minority of the computation
power can incentivize the rest of the network to accept a blockchain of
the minority's choice. By deviating from the mining protocol, a mining
pool which controls at least 38.2% of the network's total computational
power can, with modest financial capacity, gain mining advantage over
honest mining. Such an attack creates a longer valid blockchain by forking
the honest blockchain, and the attacker's blockchain need not disrupt any
"legitimate" non-mining transactions present on the honest blockchain. By
subverting the consensus protocol, the attacking pool can double-spend
money or simply create a blockchain that pays mining rewards to the
attacker's pool. We show that our attacks are easy to encode in any
Nakamoto-consensus-based cryptocurrency which supports a scripting
language that is sufficiently expressive to encode its own mining puzzles.
This is joint work with Sanjay Jain and Prateek Saxena.
The current version of the paper is available from the
homepage of Jason Teutsch.
- 04/11/2015, 17:00 hrs, Week 12, S17#04-06.
Peng Cheng.
The properties of d.r.e. degrees.
In this talk, we will present some classical important properties about
n-r.e. degrees, especially in the d.r.e. cases.
We will present some main ideas about the proof in these theorems, also we
will give some open problems about d.r.e. degress which we are considering.
- 11/11/2015, 17:00 hrs, Week 13, S17#04-06.
Liu Yong.
Maximal d.r.e. degrees and avoiding the upper cone.
We will review briefly the strategy for avoiding the upper
cone of a given nonrecursive r.e. set, and the strategy for building a
maximal d.r.e. set. And explain how to combine these two strategies.
Therefore we can build a maximal d.r.e. set avoiding the upper cone of
a given nonrecursive r.e. set.
- 02/12/2015, 17:00 hrs, S17#04-06.
Slawomir Solecki.
Menger compacta as projective Fraïssé limits with
emphasis on dimension one.
In each dimension d, there exists a canonical compact, second
countable space, called the d-dimensional
Menger space, with certain universality and homogeneity properties. For
d = 0, it is the Cantor set, for d = ∞, it
is the Hilbert cube. I will concentrate on the 1-dimensional Menger
space. I will prove that it is a quotient of a projective
Fraïssé limit. I will describe how a property of
projective Fraïssé
limits coming from Logic, called the projective extension
property, can be used to prove high homogeneity of the 1-dimensional
Menger space.
Talks from the
previous semesters.