Seminar Manifesto  [1] 

The first seminar was on Nov. 3. The next seminar is scheduled for Friday, December 1st at 3:30 in  CH 0232. Speaker: O Costin, Ohio State

Title: An introduction to analyzable functions.

Abstract:
Analyzable functions resemble in many ways analytic functions. They are described locally by summable expansions (in terms of power series, but also exponentials and logs). In specific problems  these expansions can often be determined algorithmically.   Though usually divergent classically, the expansions are summable in a natural and constructive way. Unlike analytic functions,  analyzable functions are rich enough to encompass solutions of a wide spectrum of problems. I will describe how these functions arise naturally, why they solve such a vast array of problems, which problems are now known to be solvable in terms of them, how  to determine their properties, and  also why the summation procedure is a natural extension of usual summation. 

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An abbreviated transcript of the first talk and a few of the items to be covered at the beginning of the next meeting are linked here:  Analyzability, part I  [2] 

Brief sketch of the rigorous construction of transseries: Sketch of transseries construction  [3] 

A complete, self contained self-contained exposition of transseries, with proofs and the definition of formally contractive mappings, and some simple results on their summability, including "medianization" which tackles singularities of the transforms (p.26) are linked here: Transseries, rigorous construction  [4]

Discussions based on the first meeting, posted by Harvey Friedman

Regarding O. Costin's 11/3/06 Connections Seminar Lecture [5]  posted by Harvey Friedman, November 7,2006; This discusses some topics form logic and descriptive set theory that are related to some very simple aspects of Costin's talk.
What is o-minimality [6] posted by Harvey Friedman, November 8, 2006; The notion of o-minimality is of fundamental interest in a wide variety of mathematical contexts, including transcendental and analyzable functions. This presents a new, and particularly simple, equivalent of this notion in basic mathematical terms, without involving logic.