Mathematical Research Seminar - Archive

Ken Brown asked whether there is any group G such that the coset lattice of G is contractible.  I'll start by telling you what this question means.  I'll then describe some recent progress of myself and John Shareshian towards a negative answer.
An essential tool of our work is Smith Theory, which examines the action of a p-group on a topological space, and examines its fixed point set.  As time allows, I will survey some results from this theory.

Graph theory has changed completely from the late-19th century to the late 20th century, from a collection of mainly recreational problems to a well-developed mainstream area of mathematics. In this talk I outline its development over this period, both chronologically and thematically.

Euler was the most prolific mathematician of all time – but what did he do?  In this talk I survey his life in Basel, St Petersburg and Berlin, and outline some of the contributions he made to the many areas in which he worked – for the very pure to the very applied.

Exchange of information occurs on all levels of living matter, and it is the basis without which life could not be developed, especially on the level of multi cell systems. In society of animals information is communicated in a
specific way, and as far it is known it is always reliable, because it is the basis for survival of species. In human societies information as the rule is often not reliable because it is used for acquiring power and privileges, a very specific attribute of this species. Among humans information is transmitted by the word of mouth and written word.
Reliability of information is therefore at the heart of problems in organized society and the only way tool to verify is science. Central role in modern communication is played by internet, a very powerful tool, to which everybody has access to transmit any information and therefore it is of paramount importance to select reliable information. In that respect the network GEOSET was set up about which will be given description in this talk.

In the talk, I will outline several approaches to non-commutative generalizations of Stone duality that have been developed in recent years. The general idea can be very briefly explained as follows:   we consider an algebra that generalizes a Boolean algebra (or a distributive lattice, or a frame) and enquire how the dual topological (localic) object of the commutative structure can be upgraded to dualize the whole algebra. We present dualities for Boolean inverse semigroups, pseudogroups and their non-involutive analogues called complete restriction semigroups. These objects are dualized by some topological (or, more generally, localic) categories or groupoids. I am also going to explain the natural role of quantales in these dualities. The talk is based  on several papers authored by to Mark Lawson, Daniel Lenz, Pedro Resende and the speaker. 

In equity-linked life insurance contracts, the benefits are directly linked to the value of an investment portfolio. The financial risk can be totally charged to the policyholder in pure equity-linked constracts, or it can be shared between the policyholder and the insurance company, in guaranteed equity-linked contracts (minimum guarantees are offered to the policyholder). In this talk, we will mainly focus on the latter case. Whenever the market value drops below the guaranteed sum, so called additional policy reserves (APR) are required. Since these APR are stochastic and cause additional costs for the insurance company, it is important to quantify the distribution of the APR accurantely. Results obtained by Nonnenmacher and Russ will be presented and also some possibilities for further research.

Classical covering theory ensures each connected metric space with reasonable local properties ( locally path connected, and semi locally simply connected) admits a universal covering (essentially uniquely).  In this case the preimages of the natural covering map are discrete.

How might one reasonably generalize the latter facts, sacrificing discreteness of fibres and nice local properties of the base, and obtain a fibre bundle rather than a traditional covering map, whose total space is simply connected, so that all paths in the base space lift uniquely, and so that all fibres are totally disconnected?

There is no universally agreed upon definition of generalized universal cover.

However we will discuss one of the most natural definitions, see why fibre bundles are often hopeless to obtain, but present a substantial class of nontrivial examples where fibre bundles are indeed achieved. along with the other mentioned properties.

In particular, certain planar sets generate bundles whose fibres are the classical free topological groups over countably many generators in the sense of Graev or Markov.

This talk is motivated by a question of Andrej Bauer.
 http://mathoverflow.net/questions/165684/a-generalization-of-covering-spaces-to-fiber-bundles-with-totally-path-disconnec [1]
 

Open abstract.