Transseries

Working Group

Winter 2008

organized by Professor Edgar ( edgar@math.ohio-state.edu)

Other faculty will participate; graduate students are invited.

Graduate student participants may obtain 2 hours credit by enrolling for Math 693 ( call number 12998-8).
Interested undergraduates should contact me, perhaps by email.
Non-credit participation is also possible.

The Working Group on transseries will meet Mondays and Wednesdays at 12:30 in MA 317. (MA Math Building, not MW Math Tower) The first meeting is Monday, January 7.

The differential field of transseries was discovered independently in various parts of mathematics: asymptotic analysis, model theory, computer algebra, surreal numbers. Some feel it was surprisingly recent for something so natural. Roots of the subject go back to Écalle working in asymptotic analysis, Dahn and Göring working in model theory, Geddes & Gonnet working in computer algebra, Kruskal working in surreal numbers. They arrived at eerily similar mathematical structures.

But in fact no knowledge of model theory or asymptotic analysis or computer algebra or surreal numbers is required in order to understand this new, beautiful, complex object.

The Working Group will be (at least for my part) not on the applications. But on the algebraic, analytic, combinatorial structure of the set of transseries. Now that I understand it, I am convinced it is elementary but very rich, and worth study in its own right.

For some idea about the subject... Here is a 5-page introduction: PDF FILE

Documents Available On-Line

PDF FILE 5-page introduction

PDF FILE G. Edgar, "Transseries for beginners"

PDF FILE G. Edgar, "Transseries speculation: compositiion, recursion, ..."

PDF FILE M. C. Schmeling, "Corps de transéries" Ph.D. thesis (2001)

PS FILE J. van der Hoeven, "Operators on generalized power series" Illinois J. Math. 45 (2001), no. 4, 1161--1190.

PS FILE J. van der Hoeven, "Generalized power series solutions to linear partial differential equations" J. Symbolic Comput. 42 (2007), no. 8, 771--791.

PS FILE J. van der Hoeven, "Efficient accelero-summation of holonomic functions" J. Symbolic Comput. 42 (2007), no. 4, 389--428

PDF FILE J. van der Hoeven, "Integral transseries" preprint (2005)

PS FILE J. van der Hoeven, "Transserial Hardy fields" preprint (2006)

PS FILE J. van der Hoeven, "On effective analytic continuation" preprint

PDF FILE O. Costin, "Topological construction of transseries and introduction to generalized Borel summability", Analyzable functions and applications, 137--175, Contemp. Math., 373, Amer. Math. Soc., Providence, RI, 2005.

PDF FILE O. Costin, Spring 2007 course notes

PDF FILE D. Gruntz, "A new algorithm for computing asymptotic series", Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation, 239-244

PDF FILE B. Salvy, J. R. Shackell, "Asymptotic expansions of functional inverses." Papers from the International Symposium on Symbolic and Algebraic Computation, 1992 130-137

PDF FILE J. Shackell, "Nested expansions and Hardy fields." Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation, 234-238

PDF FILE L. van den Dries, A. Macintyre, Angus, D. Marker, "Logarithmic-exponential series". Ann. Pure Appl. Logic 111 (2001), no. 1-2, 61--113.