Cora Cartis is among the newly elected Society for Industrial and Applied Mathematics (SIAM) Fellows for 2023.
Quasiconvexity and nonlinear Elasticity
Abstract
Quasiconvexity is the fundamental existence condition for variational problems, yet it is poorly understood. Two outstanding problems remain:
- 1) does rank-one convexity, a simple necessary condition, imply quasiconvexity in two dimensions?
- 2) can one prove existence theorems for quasiconvex energies in the context of nonlinear Elasticity?
In this talk we show that both problems have a positive answer in a special class of isotropic energies. Our proof combines complex analysis with the theory of gradient Young measures. On the way to the main result, we establish quasiconvexity inequalities for the Burkholder function which yield, in particular, many sharp higher integrability results.
The talk is based on joint work with Kari Astala, Daniel Faraco, Aleksis Koski and Jan Kristensen.
In Search of Euler Equilibria Via the MR Equations
Abstract
The subject of “geometric” fluid dynamics flourished following the seminal work of VI.
Arnold in the 1960s. A famous paper was published in 1970 by David Ebin and Jerrold
Marsden, who used the manifold structure of certain groups of diffeomorphisms to obtain
sharp existence and uniqueness results for the classical equations of fluid dynamics. Of
particular importance are the fixed points of the underlying dynamical system and the
“accessibility” of these Euler equilibria. In 1985 Keith Moffatt introduced a mechanism
for reaching these equilibria not through the Euler vortex dynamics itself but via a
topology-preserving diffusion process called “Magnetic Relaxation”. In this talk, we will
discuss some recent results for Moffatt’s MR equations which are mathematically
challenging not only because they are active vector equations but also because they have
a cubic nonlinearity.
This is joint work with Rajendra Beckie, Adam Larios, and Vlad Vicol.
Earlier this month, in Lecture Theatre 2 in the Andrew Wiles Building, a collection of talented Oxford Mathematics students, together with colleagues in STEM subjects and beyond, performed Fermat's Last Tango to sell-out crowds over five performances.
Written in 2000 by Joanne Sydney Lessner and Joshua Rosenblum, Fermat's Last Tango tells the story, in words and music, of a 300 hundred-year-old mathematical mystery and the man who spent seven years trying to solve it. Sound familiar?
You may be interested to learn that the University are launching a TikTok channel next month (Cambridge have one). Just in time for the ban, you might think. They have sought advice from Information Security (in light of e.g. the UK public sector moving to limit or ban TikTok's use).
Network Rail are responsible for managing a suite of whole lifecycle forecasting models for railway assets (such as track, signalling systems and bridges). These models are used to help understand the future required spend on their assets as well as estimate the impact of different funding scenarios.