The Arithmetic of Power Series

The Mahler lectures are a biennial activity organised by the Australian Mathematical Society, and supported by the Australian Mathematical Sciences Institute. The tour invites a prominent international mathematician to travel to Australian universities to deliver lectures at a variety of levels, including several public lectures.

Abstract

A holomorphic function $P(z)$ of a complex variable around $z = 0$ has a power series expansion $P(z) = \sum a_n z^n$. What constraints are imposed on $P(z)$ by assuming that all the coefficients $a_n$ are integers? We discuss some variations on this problem starting with some very elementary observations and leading up to a resolution of a 50 year old conjecture, as well as the surprising links to differential equations and group theory.

About the author

Born in Melbourne, Frank Calegari attended Melbourne University as an undergraduate and completed his graduate studies at the University of California at Berkeley and a postdoctoral fellowship at Harvard University. He joined the Faculty of Northwestern University in 2006 and has since been a Fellow of the American Mathematical Institute and a von Neumann Fellow of Mathematics at the Institute for Advanced Study. Frank has been a Professor of Mathematics at the University of Chicago since 2015. His numerous awards include a Sloan Fellowship (2009) and in 2013 he become a fellow of the American Mathematical Society.

His research is in the area of algebraic number theory. Frank is particularly interested in the Langlands programme, especially, the notion of reciprocity linking Galois representations and motives to automorphic forms. For reprints and preprints, please visit the research page on his website. Frank is a former American Institute of Mathematics 5-year fellow.

Frank’s other interests include coffee, cooking, cricket, and classical piano, and he has even performed live with Zubin Mehta and the Israeli Philharmonic Orchestra.

 

Zoom Details:

Topic: The Arithmetic of Power Series

Time: Nov 28, 2022 04:00 PM Canberra, Melbourne, Sydney

 

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