Prof. Charlie Elliott, Prof. Helen Byrne, Dr Sheehan Olver, Dr Hamza Fawzi, Prof. Dominic Vella
10:40 Snap, Crack and Pop: Dynamic Elastic Instabilities (Prof Dominic Vella, MR2)
11:25 Semidefinite Approximations of the Matrix Logarithm (Dr Hamza Fawzi, MR2)
12:20 ABC for Surface PDE (Prof Charlie Elliott, MR2)
14:00 Approaches to Modelling and Remodelling Biological Tissues (Prof Helen Byrne, MR2)
14:45 Multivariate Orthogonal Polynomials on Triangles and Spheres, for Solving PDEs (Dr Sheehan Olver, MR2)
15:30 Tea break
16:00 Panayiota Katsamba (MR4)
16:15 Mahed Abroshan (MR4)
16:30 James Munro (MR4)
16:45 Samuel Power (MR4)
17:00 End Book now!
PLENARY TALK ABSTRACTS
Speaker: Prof Dominic Vella Title: Snap, Crack and Pop: Dynamic elastic instabilities Abstract: Everyday life throws up many examples of elastic instabilities: from the snap-through inversion of an umbrella in a strong wind to the ‘crack’ of a whip. Traditionally, the focus in understanding such instabilities has been ‘when’ they happen (so that they can be avoided in building structures) but there is now increasing interest in ‘how’ they happen, i.e. the dynamics of instability. Beginning with the (apparently elementary) problem of a string moving around a pulley, I will show that such problems can lead to some surprises: elastic objects may adopt unexpected shapes during instability, and their motion can be significantly slower than expected.
————————————————————- Speaker: Dr Hamza Fawzi Title: Semidefinite Approximations of the Matrix Logarithm Abstract: The matrix logarithm, when applied to symmetric positive definite matrices, is known to be concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix functions, many of which are of importance in quantum information theory. In this talk I will show that certain rational approximations of the matrix logarithm remarkably preserve this concavity property and moreover, are amenable to semidefinite programming. Such approximations allow us to use off-the-shelf semidefinite programming solvers for convex optimization problems involving the matrix logarithm. I will conclude by showing some applications to problems in quantum information theory.
Joint work with James Saunderson (Monash University) and Pablo Parrilo (MIT).
————————————————————- Speaker: Prof Helen Byrne Title: Approaches to modelling and remodelling biological tissues Abstract: Biological tissues are complex, evolving structures, characterised by multiple interactions that act across diverse space and time scales. In this talk I will illustrate how mathematical modelling can provide new mechanistic insight into their behaviours while also acting as a source of mathematically challenging problems. I will take my inspiration from recent studies of degenerative diseases, wound healing and tissue engineering.
————————————————————- Speaker: Prof Charlie Elliot Title: ABC for Surface PDE Abstract: In this talk, we discuss the role of analysis with respect to some models in cell biology which lead to surface PDEs. The developments in large scale computation together with the theory of well posedness of nonlinear PDEs and their rigorous numerical analysis support the importance of ABM (Analysis Based Modelling) and ABC (Analysis Based Computation). In this talk we develop these ideas in the context of (a) the morphology of two phase biomembranes (b) cell motility (c) receptor ligand dynamics and (d) deformations of cell membranes induced by the cytoskeleton. In each of these we are concerned with the well posedness of coupled bulk/surface PDE systems and their numerical analysis.
————————————————————- Speaker: Dr Sheehan Olver Title: Multivariate orthogonal polynomials on triangles and spheres, for solving PDEs Abstract: Univariate orthogonal polynomials have a long history in applied and computational mathematics, playing a fundamental role in quadrature, spectral theory and solving differential equations with spectral methods. Unfortunately, while numerous theoretical results concerning multivariate orthogonal polynomials exist, they have an unfair reputation of being unwieldy on non-tensor product domains, and their use in applications has been limited. In reality, many of the powerful computational aspects of univariate orthogonal polynomials translate naturally to multivariate orthogonal polynomials, including the existence of Jacobi operators, fast evaluation of expansions using Clenshaw’s algorithm and the ability to construct sparse partial differential operators, a la the ultraspherical spectral method [Olver & Townsend 2012]. We demonstrate these computational aspects using multivariate orthogonal polynomials on a triangle and spheres, including the fast solution of general partial differential equations.
STUDENT/POSTDOC TALK ABSTRACTS
Speaker: Panayiota Katsamba Title: Hydrodynamics of bacteriophages Abstract: Bacteriophage viruses have the striking appearance of microscopic spaceships. One of the most abundant entities in our planet, they crowd fluid environments in anticipation of a random encounter with bacteria, and use a remarkable nanometre-size machinery for infection: fibres that recognise and attach to specific receptors on their victim’s surface and a hollow tube through which their genetic material is ejected inside the host cell cytoplasm for replication. Flagellotropic phages first attach to the flagella of bacteria and find a way to reach the cell body for infection since they lack the ability to move independently. The means by which they move up the flagellum has intrigued the scientific community for over 30 years. In 1973 Berg and Anderson proposed the nut-and-bolt mechanism and 26 years later, Berg’s group provided supporting evidence for it. Just like a nut being rotated will move along a bolt, under this scenario the phage wraps itself around a flagellum possessing helical grooves (due to the helical rows of flagellin molecules) and exploits the rotation of the flagellum in order to passively travel along it. In this work, we provide a first-principle theoretical model for this nut-and-bolt mechanism and show that it is able to predict experiment observations.
————————————————————- Speaker: Mahed Abroshan Title: File Synchronization and Edit Channel problems Abstract: Two problems of updating a sequence after some edits and designing good codes for edit channels are in a sense dual problems. File synchronization problem has application in file sharing platforms like Dropbox. Edit channel appears in some storage systems like DNA storage. In this talk, I will define VT codes which are a modular summation of the input sequence. VT codes can address the problem when we have single edits, surprisingly we do not know any code with similar performance for even two edits. I will explain some combinatorial aspects of VT codes and then explain our work on segmented edit channels and also Multilayer codes, a new class of codes which can recover multiple edits in a sequence.
————————————————————- Speaker: James Munro Title: Capillary retraction of a stretched fluid edge Abstract: Surface tension always causes the edge of a falling sheet of viscous fluid to approach the same curved shape. Moreover, this shape is independent of any stretching along the sheet edge, and so the same solution can be applied to a wide variety of problems with stretched sheets or bursting films. In this talk, I’ll find the similarity shape for the edge, explain why it’s universal and give some examples of retracting fluid edges.
————————————————————- Speaker: Samuel Power Title: Uniform Sampling of Polytopes and Concentration of Measure Abstract: Consider the task of generating a sequences of points, uniformly at random, from the interior of a generic convex body K. In a low-dimensional setting, this is fairly straightforward: find a simpler body X which you can draw samples from, translate and rescale it so that it encloses K, and use an accept-reject procedure. However, as we move into higher dimensions, the subtleties of concentration of measure cause this (and other naive methods) to fail dramatically.
In this talk, I will review some existing Markov chain-based procedures for tackling this task, and describe a novel algorithm which makes use of asymptotic geometric structure to obtain improved mixing properties in high dimension. No background in convex analysis will be required to follow the talk, and I will make ample use of visualisations and animation throughout.