Invited Speaker
Frank Garvan
University of Florida
Gainesville, FL

Paper: Approximations to Pi via the Dedekind Eta function
Talk: Approximations to Pi via the Dedekind Eta function
(QuickTime movie - 1.5M)
Abstract: Arguably the most efficient algorithm currently known for the extended precision calculation of Pi is a quartic iteration due to J.M. and P.B. Borwein. In their paper, the Borweins show how this iteration and others are intimately connected to the work of Ramanujan. This connection is shown utilizing their alpha-function which is defined in terms of theta-functions. They are able to find the p-th order iterations based on this function using modular equations for the theta-functions. In this paper we construct an infinite family of alpha functions. Each one gives rise to a p-th order iteration. For p=4 we obtain an iteration related to the Borwein quartic iteration. For p=3 we obtain a cubic iteration due to the Borweins but does not come from the alpha-function. For p=7 we find a septic iteration that is analogous to the cubic iteration. Our method depends on using the computer and a symbolic algebra package to find and solve certain modular equations.

About Myself

Born: Australia
1973-1976: Undergraduate study in Pure Mathematics, at the University of New South Wales, Kensington, Australia. 1977: Dip. Ed. 1981-1982: Masters degree in Mathematics at the University of New South Wales under the supervision of Mike Hirschhorn. 1983-1986: PhD in Mathematics at the Pennsylvania State University under the supervision of George Andrews. 1987: Postdoc at University of Wisconsin, under the supervision of Richard Askey.1988: Postdoc at the IMA, University of Minnesota.1988-1989: Postdoc at Macquarie University, North Ryde, Australia under supervision of John Loxton. 1990: Postdoc at Dalhousie University, Halifax, Nova Scotia under the supervision of Jon and Peter Borwein.
Family: Frank. Cyndi - charming roomate. Jeffrey - likes to play all electric appliances very loudly. Gerard - likes to weave home-made hammocks in the backyard. Maggie - keeps everyone in line. Morton, Zac and Summer - our corgies. Simba - our cat that thinks she's a dog. Fluffster - deceased (pet rabbit). Ziggy - deceased (pet hamster).
Interests: Frank's hobbies include banging his eighty-eights, backyard pit digging and tunnel construction. Also, he does a little story-telling on the side:
  • 1973-1976: Long-haired Frank was often seen riding his bicycle with umbrella attached.
  • 1977: Frank's bike is stolen.
  • 1978: Frank was often seen riding his new bicycle to work come rain or shine (in shorts) even though the "Math-Master" thought it unbecoming.
  • 1979-1980: Frank's bike was often seen parked in front of the Royal Mail Hotel.
  • 1981-1982: Frank's bike was often seen parked in front of the Lord Robert's Hotel (Darlinghurst).
  • 1983-1986: Frank accumulates a new bike, wife and two kids.
  • 1986: Frank was often seen riding his bicycle through the back blocks of Pennsylvania.
  • 1987: Frank was often seen riding his bike to work.
  • 1988: Frank and his frozen beard were often see riding to work from St Paul to Minneapolis.
  • 1988-1989: Frank was often seen riding his bicycle across Sydney until he came a cropper on the Lane Cove bridge.
  • 1990: Frank was often seen riding his bike through rain, and slush.
  • 1991-present: Frank's bike now has a garage-door opener installed.

Comments: A sucessful day: (1) Getting home safely, (2) Eating a good meal, (3) Watching "Dumb and Dumber", (4) Taking the dogs for a walk and not getting yelled at by the neighbors, (5) Finding that 2x4 to lock the bedroom door.