Frontiers in Fundamental Mathematics Research Nexus, 24 Nov

Date/Time: Thursday 24 November 2022; 1:30 pm
Location: RC-4082 (Red Centre Building, Centre Wing, UNSW Sydney)

Don’t miss this special event featuring two leading International Speakers followed by a Catered Reception.

• 1:30pm – 2:30pm The Arithmetic of Power Series, Frank Calegari
• 2:30pm – 3:00pm  Coffee Break
• 3:00pm – 4:00pm Topology and Complexity of Polynomial System Solving, Gregorio Malajovich
• 4:00pm – 5:00pm  Reception

Mahler & Nexus Lectures

1st talk

  Speaker: Frank Calegari (University of Chicago)
  Title: The Arithmetic of Power Series

Abstract

A holomorphic function of a complex variable P(z) around z = 0 has a power series expansion P(z) = ∑ a_n zⁿ. 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.

Bio

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.

2nd talk

  Speaker: Gregorio Malajovich (Universidade Federal do Rio de Janeiro)
  Title: Topology and Complexity of Polynomial System Solving

Abstract

Bézout´s theorem states that the number of roots of a “generic” system of n polynomials in n variables is the product of the total degrees. This theorem induces a canonical topology and geometry in solution space (projective space) and an inner product structure in coefficient space. This leads to a very specific definition of the condition number. Smale’s 17^th problem, now solved, was stated in terms of those structures. One was asked for an algorithm in average polynomial time to find one root of a random system.

Sharper theorems for root counting are known. For instance, Bernstein’s Theorem in terms of Minkowski’s mixed volume can be exponentially better than Bézout’s bound, albeit it counts only the solutions with no vanishing coordinate. I shall argue that this theorem induces a sharper topology for the solution space (certain toric variety) and a better notion of condition number. The widely used homotopy algorithms may fail where a toric variety based algorithm succeeds.

This suggests a program to develop efficient algorithms for polynomial systems that are not constrained to be “dense” and “random”. For instance, it is possible to obtain non-uniform complexity bounds that are polynomial in the condition number of the system one wants to solve. Uniform complexity bounds under some extra assumptions are also available, and involve another classic convex geometry invariant associated to the mixed volume, that is Aleksandrov’s mixed area.

Short Bio

Gregorio Malajovich graduated in Mathematics at the Universidade Federal do Rio de Janeiro (UFRJ) in 1989, where he also obtained a Master degree. In 1993, he completed the PhD program at the University of California at Berkeley. He is in the faculty of UFRJ since 1991, and is currently serving as Chair of the Department of Applied Mathematics.

He is author of a linear algebra textbook (in Portuguese) and a book on polynomial system solving.

His main research interests are complexity of numerical algorithms, algorithms in manifolds and numerical algebraic geometry.