# Applied Mathematics Seminar

Seminars are held at 2:00 pm on Wednesdays in the Access Grid Room ( Carslaw Building, 8th floor), unless otherwise noted.

For more information and to be added to the mailing list, please contact Eduardo G. Altmann .

## Upcoming Seminars

Wednesday March 22

Sheehan Olver (School of Mathematics and Statistics, University of Sydney)

** Title:** Solving PDEs on triangles using multivariate orthogonal polynomials

** Abstract: ** Univariate orthogonal polynomials have 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. In reality, many of the powerful computational aspects of univariate orthogonal polynomials translate naturally to multivariate orthogonal polynomials, including the existence of Jacobi operators and the ability to construct sparse partial differential operators, a la the ultrapsherical spectral method [Olver & Townsend 2012]. We demonstrate these computational aspects using multivariate orthogonal polynomials on a triangle, including the fast solution of general partial differential equations.

Wednesday April 19

Prof. Nihat Ay (Max-Planck-Institute for the Mathematics in the Sciences, Leipzig, Germany)

** Title:**Information Geometry and its Application to Complexity Theory

** Abstract: ** In the first part of my talk, I will review information-geometric structures and highlight the important role of divergences. I will present a novel approach to canonical divergences which extends the classical definition and recovers, in particular, the well-known Kullback-Leibler divergence and its relation to the Fisher-Rao metric and the Amari-Chentsov tensor.

Divergences also play an important role within a geometric approach to complexity. This approach is based on the general understanding that the complexity of a system can be quantified as the extent to which it is more than the sum of its parts. In the second part of my talk, I will motivate this approach and review corresponding work.

References:

1. N. Ay, S.I. Amari. A Novel Approach to Canonical Divergences within Information Geometry. Entropy (2015) 17: 8111-8129.

2. N. Ay, J. Jost, H. V. Le, L. Schwachhöfer. Information geometry and sufficient statistics. Probability Theory and Related Fields (2015) 162: 327-364.

3. N. Ay, J. Jost, H. V. Le, L. Schwachhöfer. Parametrized measure models. Bernoulli (2016) accepted. arXiv:1510.07305.

4. N. Ay, J. Jost, H. V. Le, L. Schwachhöfer. Information geometry. Ergebnisse der Mathematik und Ihrer Grenzgebiete/A Series of Modern Surveys in Mathematics, Springer 2017, forthcoming book.

5. N. Ay. Information Geometry on Complexity and Stochastic Interaction. Entropy (2015) 17(4): 2432-2458.

## Previous seminars

### 2017

Wednesday January 25

Dr. Paul Griffiths (Oxford Brookes University, UK)

** Title:** Shear-thinning: A stabilising effect? Yes, no, maybe?

** Abstract: ** In this talk we will investigate how viscosity effects the stability of a fluid flow. By assuming a shear-thinning viscosity relationship, where an increase in shear-rate results in a decrease in fluid viscosity, we show that flows can be both stabilised or destabilised, depending on (i) the fluid model in question and (ii) the ‘amount’ of shear-thinning the fluid exhibits. Using a two-dimensional boundary-layer flow as our ‘toy model’ we are able to show equivalence between different shear-thinning models. The effect shear-thinning has on important parameters such as the critical Reynolds number, and the maximum frequency of the disturbances will be discussed and interpreted in the wider context.

Wednesday February 22

Dr. Maria Vlassiou (Eindhoven University of Technology, Netherlands)

** Title:** Heavy-traffic limits for layered queueing networks

** Abstract: ** Heavy-traffic limits for queueing networks are a topic of continuing interest. Presently, the class of networks for which these limits have been rigorously derived is restricted. An important ingredient in such work is the demonstration of state space collapse (SSC), which loosely speaking shows that in diffusion scale the queuing process for the stochastic model can be approximately recovered as a continuous lifting of the workload process. This often results in a reduction of the dimensions of the original system in the limit, leading to improved tractability. In this talk, we discuss diffusion approximations of layered queuing networks through two examples.

In the first example, we establish a heavy-traffic limit through SSC for a computer network model. For this model, SSC is related to an intriguing separation of time scales in heavy traffic. The main source of randomness occurs at the top layer; the interactions at the other layer are shown to converge to a fixed point at a faster time scale.

The second example focuses on a network of parallel single-server queues, where the speeds of the servers are varying over time and governed by a single continuous-time Markov chain. We obtain heavy-traffic limits for the distributions of the joint workload, waiting-time and queue length processes. We do so by using a functional central limit theorem approach, which requires the interchange of steady-state and heavy-traffic limits. For this model, we show that the SSC property does not hold.

Wednesday March 1

Dr. Daniel Lecoanet (Princeton University, Princeton, USA)

** Title:** Measuring Core Stellar Magnetic Field using Wave Conversion

** Abstract: **

By studying oscillation modes at the surface of stars, astrophysicists are able to infer characteristics of their deep interior structure. This was first applied to observations of the Sun, but recently space-based telescopes have measured oscillations in many other stars, leading to many new mysteries in stellar structure and evolution. Recent work has suggested that low dipole oscillation amplitudes in evolved red giant branch stars may indicate strong core magnetic fields. Here we present both numerical simulations and analytic calculations of the interactions of waves with a strong magnetic field. We can solve the problem very accurately by using the WKB approximation to reduce the 2D PDE into a series of ODEs for different heights. We find that magnetic fields convert the buoyancy-driven waves observable at the surface of the star to magnetic waves, which are not present at the surface, in agreement with observations.