February 15: Joerg Brendle: Almost disjoint families
16.00--17.00
Abstract:
We present a survey on some classical and some recent results
on (maximal) almost disjoint families. We shall focus on a. d. families
on a countable set, but we will also touch on the uncountable case.
One main topic will be cardinals defined in terms of m. a. d. families and
their relation to other cardinal invariants.
To main page
March 15: André Nies: Demuth randomness and computational complexity
17.00--18.00
Abstract:
Demuth tests generalize Martin-Löf tests in that one can
exchange the m-th component for a computably bounded number of
times. The interest in this notion comes from the fact that a Demuth
random set can still be Delta_2. In this case it bounds a simple set
by a result of Kucera. With Kucera, we show that each c.e. set below
a Demuth random is strongly jump traceable. This is yet another
instance of the principle that among ML-random sets, more random is
equivalent to being closer to computable.
To main page
March 30: Pantelis Eleftheriou: On groups definable in o-minimal structures
17.00--18.00
Abstract:
An o(rder)-minimal structure is a dense linearly ordered
structure such that every definable subset of the universe is a finite
union of points and intervals; equivalently, it can be defined just
using the order. O-minimal structures have proved to be the right
context where model theory for ordered structures can be developed.
More specifically, the geometry of definable objects in an o-minimal
structure resembles the geometry of real semi-algebraic sets. In this
talk, we will exemplify this resemblance in the category of definable
groups, culminating to Pillay's Principle, which establishes a rigid
connection between groups definable in o-minimal structures and real
Lie groups. Time permitting, we will provide examples that illustrate
the limits of this resemblance.
To main page
April 13: Vaughan Pratt: Presketches: Algebra without algebras via categories without functors
15.00--16.00
Abstract:
Bypassing the traditional separation of theory and model, we introduce the notion of presketch as a pointed category, one with a set of distinguished objects as its points or types. Algebras and homomorphisms arise simply as the objects and morphisms of a presketch. As a generalization of the completion of the rationals to the reals, a presketch is full when it densely embeds its points, and complete when it is full and maximal up to equivalence. Every complete presketch is a topos by virtue of being equivalent to a presheaf category, and every presheaf category arises as a complete presketch. The category of models of an Ehresmann sketch arises as a full subcategory of a presketch consisting of those algebras respecting specified limits and colimits; as such the models of a sketch in general do not form a topos.
The passage to a disketch as a category with two sets of distinguished objects, positive and negative, or types and properties, generalizes the passage from sets (more generally the objects of the ambient enriching category V) to Chu spaces by interpreting the morphisms from a type to an algebra A as its individuals of that type, and those from A to a property as the local states of observation in A of that property. C.I. Lewis's problematic qualia (1924) are accounted for in this framework simply as those entities that are ambiguously an individual and a state. As often happens, the previous absence of any mathematically plausible account of qualia might explain the strongly partisan division of philosophers into qualiaphiles and qualiaphobes.
Presketches exploit the Yoneda Lemma to move functors and natural transformations out of the passenger compartment and under the bonnet where they can be accessed as needed without intruding unnecessarily on the working mathematician's day-to-day use of algebra.
To main page
May 21: Gido Scharfenberger-Fabian: Advanced Souslin tree constructions for algebraic issues
Abstract:
I will describe a method for Souslin tree constructions which uses Baire category arguments and some classical descriptive set theory.
This will be applied to obtain Souslin algebras (complete Boolean algebras associated to Souslin trees) with highly remarkable features such as, e.g., chain homogeneity (the property that all maximal chains of the algebra are mutually isomorphic as linear orders).
To main page