L1

Universal fluctuations are shown to exist when well-known and widely used numerical algorithms are applied with random data. Similar universal behavior is shown in stochastic algorithms and algorithms that model neural computation. The question of whether universality is present in all, or nearly all, computation is raised. (Joint work with G.Menon, S.Olver and T. Trogdon.)

A fast method for computing eigenpairs of positive definite matrices using GPUs is presented. The method uses Chebyshev polynomial spectral transformations to map the desired eigenvalues of the original matrix $A$ to exterior eigenvalues of the transformed matrix $p(A)$, making them easily computable using existing Krylov methods. The construction of the transforming polynomial $p(z)$ can be done efficiently and only requires knowledge of the spectral radius of $A$. Computing $p(A)v$ can be done using only the action of $Av$. This requires no extra memory and is typically easy to parallelize. The method is implemented using the highly parallel GPU architecture and for specific problems, has a factor of 10 speedup over current GPU methods and a factor of 100 speedup over traditional shift and invert strategies on a CPU.

Evolution by natural selection has resulted in a remarkable diversity of organism morphologies. But is it possible for developmental processes to create “any possible shape?” Or are there intrinsic constraints? I will discuss our recent exploration into the shapes of bird beaks. Initially, inspired by the discovery of genes controlling the shapes of beaks of Darwin's finches, we showed that the morphological diversity in the beaks of Darwin’s Finches is quantitatively accounted for by the mathematical group of affine transformations. We have extended this to show that the space of shapes of bird beaks is not large, and that a large phylogeny (including finches, cardinals, sparrows, etc.) are accurately spanned by only three independent parameters -- the shapes of these bird beaks are all pieces of conic sections. After summarizing the evidence for these conclusions, I will delve into our efforts to create mathematical models that connect these patterns to the developmental mechanism leading to a beak. It turns out that there are simple (but precise) constraints on any mathematical model that leads to the observed phenomenology, leading to explicit predictions for the time dynamics of beak development in song birds. Experiments testing these predictions for the development of zebra finch beaks will be presented.

Based on the following papers:

http://www.pnas.org/content/107/8/3356.short

http://www.nature.com/ncomms/2014/140416/ncomms4700/full/ncomms4700.html

 “Understanding the generation and control of pattern and form is still a challenging and major problem in the biomedical sciences. I shall describe three very different problems. First I shall briefly describe the development and application of the mechanical theory of morphogenesis and the discovery of morphogenetic laws in limb development and how it was used to move evolution backwards. I shall then describe a surprisingly informative model, now used clinically, for quantifying the growth of brain tumours, enhancing imaging techniques and quantifying individual patient treatment protocols prior to their use.  Among other things, it is used to estimate patient life expectancy and explain why some patients live longer than others with the same treatment protocols. Finally I shall describe an example from the social sciences which quantifies marital interaction that is used to predict marital stability and divorce.  From a large study of newly married couples it had a 94% accuracy. I shall show how it has helped design a new scientific marital therapy which is currently used in clinical practice.”

I will explain how, given a crepant morphism with one-dimensional fibres between 3-folds, it is possible to use noncommutative deformations to run the MMP in a satisfyingly algorithmic fashion.  As part of this, a flop is viewed homologically as the solution to a universal property, and so is constructed not by changing GIT, but instead by changing the algebra. Carrying this extra information of the new algebra allows us to continue to flop, and thus continue the MMP, without having to calculate everything from scratch. Proving things in this manner does in fact have other consequences too, and I will explain some them, both theoretical and computational.

The surface subgroup problem asks whether a given group contains a subgroup that is isomorphic to the fundamental group of a closed surface. In this talk I will survey the role that the surface subgroup problem plays in some important solved and unsolved problems in the theory of 3-manifolds, the geometric group theory, and the theory of arithmetic manifolds.

We propose a model where an algorithmic trader takes a view on the distribution of prices at a future date and then decides how to trade in the direction of her predictions using the optimal mix of market and limit orders. As time goes by, the trader learns from changes in prices and updates her predictions to tweak her strategy. Compared to a trader that cannot learn from market dynamics or form a view of the market, the algorithmic trader's profits are higher and more certain. Even though the trader executes a strategy based on a directional view, the sources of profits are both from making the spread as well as capital appreciation of inventories. Higher volatility of prices considerably impairs the trader's ability to learn from price innovations, but this adverse effect can be circumvented by learning from a collection of assets that co-move.

The synthesis of complex-shaped colloids and nanoparticles has recently undergone unprecedented advancements. It is now possible to manufacture particles shaped as dumbbells, cubes, stars, triangles, and cylinders, with exquisite control over the particle shape. How can particle geometry be exploited in the context of capillarity and surface-tension phenomena? This talk examines this question by exploring the case of complex-shaped particles adsorbed at the interface between two immiscible fluids, in the small Bond number limit in which gravity is not important. In this limit, the "Cheerio's effect" is unimportant, but interface deformations do emerge. This drives configuration dependent capillary forces that can be exploited in a variety of contexts, from emulsion stabilisation to the manufacturing of new materials. It is an opportunity for the mathematics community to get involved in this field, which offers ample opportunities for careful mathematical analysis. For instance, we find that the mathematical toolbox provided by 2D potential theory lead to remarkably good predictions of the forces and torques measured experimentally by tracking particle pairs of cylinders and ellipsoids. New research directions will also be mentioned during the talk, including elasto-capillary interactions and the simulation of multiphase composites.