Jon Borwein s Remembrance Day

Transcription

1 Jon Borwein s Remembrance Day Institut Henri Poincaré - Paris February, 10, 2017 Speaker: Francisco Javier Aragón Artacho, Department of Mathematics University of Alicante Title: Walking guided by Dr Pi Abstract: I will share my experience as a postdoc of Jon Borwein at the University of Newcastle. I will mainly focus on our joint project Walking on real numbers, where we analyzed the randomness of the digits of various mathematical constants by representing them as walks in the plane. Thanks to Jon s influence, this work had an important media coverage. I will also briefly discuss about the Douglas Rachford algorithm, focusing on some very recent results that I believe Jon would have enjoyed hearing about. 1

2 Speaker: David H. Bailey, Lawrence Berkeley National Lab (retired) and University of California, Davis, Department of Computer Science Title: Computer discovery and analysis of large Poisson polynomials using 64, 000-digit arithmetic Abstract: In two earlier studies of lattice sums arising from the Poisson equation of mathematical physics, we established that a certain simple class of two-argument lattice sums φ(x, y) always reduce to an algebraic number when x and y are rational, and we also computed the explicit minimal polynomials associated with these algebraic numbers for a few specific rational arguments x and y. Based on these results, Jason Kimberley of the University of Newcastle conjectured a number-theoretic formula for the degree of the algebraic number in the case x = y = 1/s for some integer s. These earlier studies were hampered by the enormous cost and complexity of the requisite computations. In this study, we address the Poisson polynomial problem with significantly more capable computational tools. As a result of this improved capability, we have confirmed that Kimberley s formula holds for all integers s up to 52 (except for s = 41, 43, 47, 49, 51, which are still too costly to test), and also for s = 60 and s = 64. As far as we are aware, these computations, which employed up to 64, 000-digit precision, producing polynomials with degrees up to 512 and integer coefficients up to , constitute the largest successful integer relation computations performed to date. By examining the computed results, we found connections to a sequence of polynomials defined in a 2010 paper by Savin and Quarfoot. These investigations subsequently led to a proof, by Watson Ladd of U.C. Berkeley, of Kimberley s formula and also the fact that when s is even, the polynomial is palindromic. 2

3 Speaker: Patrick L. Combettes, Department of Mathematics, North Carolina State University, Raleigh Title: Perspectives Functions: Constructions and Applications Abstract: Many functions encountered in applied mathematics can be expressed in terms of perspective functions. We analyze various algebraic and convex-analytical properties of these functions and provide general schemes to construct lower semicontinuous convex functions from them. Several new examples are presented and applications are discussed. Connections with the work of Jon Borwein will be made. Speaker: Ivar Ekeland, Université Paris-Dauphine Title: Perturbed minimization principles and differentiability Abstract: If a real function f(x) attains its minimum at a point x, then f (x) = 0. This equation is then used to locate the point x. If the function f is bounded from below but does not attain its minimum, then a suitable perturbation f + g of this function may attain its minimum, and this will yield interesting information (Ekeland, 1972). I will describe the genesis and early history of this idea. 3

4 Speaker: Martin Grötschel, President of the Berlin-Brandenburgische Akademie der Wissenschaften (BBAW) Title: Jon Borwein: Electronic Information and Communication Abstract: I will briefly indicated Jon Borwein s activities concerning information and communication, open access and related issues. I worked?with Jon Borwein for a significant period of time on these matters in the Committee of Electronic Information and Communication (CEIC) of the International Mathematical Union. The documents produced by CEIC had significant influence on the development on these issues worldwide. Speaker: Alexander Ioffe, Technion, Israel Institute of Technology Title: On Some Problems of Variational Analysis Abstract: I plan to discuss several groups of problems in which contribution of Jon Borwein led to substantial and sometimes decisive developments. In particular, I shall speak about (a) the role of Borwein-Preiss variational principle and Borwein-Fitzpatrick weak-star sequential compactness theorem in the theory of nonconvex subdifferentials; (b) the Borwein-Moors idea of rich families of subspaces and its role in developments of separable reduction philosophy for various aspects of variational analysis in Banach spaces; (c) developments of nonsmooth transversality theory for which the Bauschke-Borwein theorem on linear convergence of alternating projections was one of the principal promoters. 4

5 Speaker: Adrian S. Lewis, School of Operations Research and Information Engineering, Cornell University Title: Alternating projections, transversality, and metric regularity Abstract: Among Jon Borwein s many contributions to variational analysis and optimization, his early highlighting of the idea of metric regularity and his broad exploration (with Heinz Bauschke) of the Method of Alternating Projections (MAP) combine particularly elegantly. In fact, MAP a rudimentary but enduring algorithm for exploring the intersection of two arbitrary closed sets concisely illustrates several far-reaching and interdependent variational ideas. A transversality measure, intuitively an angle and generically nonzero, controls several key properties: MAP s linear convergence rate, a posteriori error bounds, sensitivity to data perturbations, and robustness relative to problem description. These linked ideas emerge in a wide variety of computational problems. Speaker: Matthew K. Tam, Institut fu r Numerische und Angewandte Mathematik, Universita t Go ttingen Title: Jon Borwein: The graduate advisor Abstract: In this talk, I will share my experiences from my time as a honours student, and later as a PhD student, under Jon s supervisor at the University of Newcastle as well as some of the work we did together. 5

6 Speaker: Luc Trouche, École Normale Supérieure de Lyon, Institut Français de l Éducation Title: Exploring, experiencing and experimenting an inspiring journey in mathematics (education) Abstract: I will tell a story of a productive meeting between a mathematician, Jon Borwein, and two researchers in the field of mathematics education, resulting in a book (Monaghan, Trouche and Borwein, 2016). This story was brutally interrupted by Jon s death, but the questions asked remain as potential research programs. Monaghan, J., Trouche, L., & Borwein, J. (2016). Tools and Mathematics: Instruments for Learning, New York, Springer. Speaker: Qiji Jim Zhu, Department of Mathematics, Western Michigan University, Kalamazoo Title: Entropy Maximization in Finance Abstract: We highlight the role of entropy maximization in several fundamental results in financial mathematics. They are the two fund theorem for Markowitz efficient portfolios, the existence and uniqueness of a market portfolio in the capital asset pricing model, the fundamental theorem of asset pricing, the selection of a martingale measure for pricing contingent claims in an incomplete market and the calculation of super/sub-hedging bounds and portfolios. The connection of diverse important results in finance with the method of entropy maximization indicates the significant influence of methodology of physical science in financial research. This is a joint research project with Jon Borwein started from his comment that the two fund theorem in Markowitz portfolio theory can be viewed as exploring the structure of an entropy maximization problem and similar phenomena should exist in other financial problems. Sadly, he could not see its completion. 6