When an experiment is conducted for purposes which include fitting a particular model to the data, then the ’optimal’ experimental design is highly dependent upon the model assumptions - linearity of the response function, independence and homoscedasticity of the errors, etc. When these assumptions are violated the design can be far from optimal, and so a more robust approach is called for. We should seek a design which behaves reasonably well over a large class of plausible models. I will review the progress which has been made on such problems, in a variety of experimental and modelling scenarios - prediction, extrapolation, discrimination, survey sampling, dose-response, etc
In this talk, we will highlight the importance of measurement, discuss what can and cannot be measured. Focusing on the measurement of position, importance, and shape, we illustrate by discussing the mathematics behind, GPS, Google and laser surgery. The talk will be accessible to a wide audience.
Quadration regards a triangle as an orthocentric quadrangle. Twinning is an involution between orthocentres and circumcentres. Together with variations of Conway’s Extraversion, these give rise to symmetric sets of points, lines and circles. There are eight vertices, which are also both orthocentres and circumcentres. Twelve edges share six midpoints, which, with six diagonal points, lie on the 50-point circle, better known as the 9-point circle. There are 32 circles which touch three edges and also touch the 50-point circle. 32 Gergonne points, when joined to their respective touch-centres, give sets of four segments which concur in eight deLongchamps points, which, with the eight centroids, form two harmonic ranges with the ortho- and circum-centres on each of the four Euler lines. Corresponding points on the eight circumcircles generate pairs of parallel Simson-Wallace lines, each containing six feet of perpendiculars. In three symmetrical positions these coincide, with twelve feet on one line. In the three orthogonal positions they are pairs of parallel tangents to the 50-point circle, forming the Steiner Star of David. This three-symmetry is shared with the 144 Morley triangles which are all homothetic. Time does not allow investigation of the 256 Malfatti configurations, whose 256 radpoints probably lie in fours on 64 guylines, eight through each of the eight vertices.
As the Euler theory of hydrodynamics (1757), the Born-Infeld theory of electromagnetism (1934) enjoys a simple and beautiful geometric structure. Quite surprisingly, the BI model which is of relativistic nature, shares many features with classical hydro- and magnetohydro-dynamics. In particular, I will discuss its very close connection with Moffatt’s topological approach to Euler equations, through the concept of magnetic relaxation.
The Marsden Memorial Lecture Series is dedicated to the memory of Jerrold E Marsden (1942-2010), a world-renowned Canadian applied mathematician. Marsden was the Carl F Braun Professor of Control and Dynamical Systems at Caltech, and prior to that he was at the University of California (Berkeley) for many years. He did extensive research in the areas of geometric mechanics, dynamical systems and control theory. He was one of the original founders in the early 1970’s of reduction theory for mechanical systems with symmetry, which remains an active and much studied area of research today.
Many problems in science and engineering require the reconstruction of an object - an image or signal, for example - from a collection of measurements. Due to time, cost or other constraints, one is often severely limited by the amount of data that can be collected. Compressed sensing is a mathematical theory and set of techniques that aim to improve reconstruction quality from a given data set by leveraging the underlying structure of the unknown object; specifically, its sparsity.
In this talk I will commence with an overview of the fundamentals of compressed sensing and discuss some of its applications. However, I will next explain that, despite the large and growing body of literature on compressed sensing, many of these applications do not fit into the standard framework. I will then describe a more general framework for compressed sensing which aims to bridge this gap. Finally, I will show that this new framework is not just useful in explaining existing applications of compressed sensing. The new insight it brings leads to substantially better compressed sensing-based approaches than the current state-of-the-art in a number of applications.
Wavelets have been successfully applied to many areas. For high-dimensional problems such as image/video processing, separable wavelets are widely used but are known to have some shortcomings such as lack of directionality and translation invariance. These shortcomings limit the full potential of wavelets. In this talk, we first present a brief introduction to orthonormal wavelets and tight framelets as well as their fast transforms using filter banks. Next we discuss recent exciting developments on directional tensor product complex tight framelets (TP-CTFs) for problems in more than one dimension. For image/video denoising and inpainting, we show that directional complex tight framelets have superior performance compared with current state-of-the-art methods. Such TP-CTFs inherit almost all the advantages of traditional wavelets but with directionality for capturing edges, enjoy desired features of the popular discrete Fourier/Cosine transform for capturing oscillating textures, and are computationally efficient. Such TP-CTFs are also naturally linked to Gabor (or windowed Fourier) transform and can be further extended. We expect that our approach of TP-CTFs using directional complex framelets can be applied to many other high-dimensional problems.
Please note, portions of this video are unavailable due to internet connectivity errors during the recording.
Period integrals are geometrical objects which can be realized as special functions, or sections of certain bundles. Their origin goes back to Euler, Gauss and Legendre in the study of complex algebraic curves. In their modern version, period integrals naturally arise in Hodge theory, and more recently in mathematical physics, and the theory of hypergeometric functions. I will give an overview of a recent program to use differential equations and D-module theory to study period integrals. Connections to hypergeometric functions of Gel'fand-Kapranov-Zelevinsky (GKZ) will also be considered. We will see that the theory is intimately related to a particular infinite dimensional representation of a reductive Lie algebra, and the topology of certain affine varieties. I will describe how the theory could help calculate period integrals, and offers new insights into the GKZ theory, and mirror symmetry for toric and flag varieties. This talk is based on joint works with S. Bloch, B. Lian, V. Srinivas, S-T. Yau, and X. Zhu.
Motivated by subtle questions in Donaldson-Thomas theory, we study the spectrum of the inertia operator on the Grothendieck module of algebraic stacks. We hope to give an idea of what this statement means. Along the way, we encounter some elementary, but apparently new, questions about finite groups and matrix groups. Prerequisites for this talk: a little linear algebra, and a little group theory.