L1

When your lecturers say that they expect you to study your notes between lectures, what do they really mean?  There is research on how mathematicians go about reading maths effectively.  We'll look at a technique (self-explanation training) that has been shown to improve students' comprehension of proofs, and in this interactive workshop we'll practise together on some examples.  Please bring a pen/pencil and paper!

Prime numbers have been looked at for centuries, but some of the most basic questions about them are still major unsolved problems. These problems began as idle curiosities, but have grown to become hugely important not only in pure mathematics, but also have many applications to the real world. I'll talk about some of these quests to find patterns in the sequence of prime numbers.

Mathematics has provided us with several extremely useful tools to apply in the world beyond mathematics.  But it also provides us with mathematicians -- individuals who have trained habits of careful thinking in domains where that is the only way to make progress. This talk will explore some other domains -- such as saying sensible things about the long-term future, or how to identify good actions in the world -- where this style of thinking seems particularly desirable as progress can otherwise be elusive or illusory.  It will also consider how a mathematician's curiosity can help to identify important questions.

Taught courses offer a range of distinctive learning opportunities from lectures to tutorials/supervisions through to individual study. Orchestration refers to the combining and sequencing of these opportunities for maximum effect. This raises a question about who does the orchestration. In school, there is a good case for suggesting that it is teachers who take responsibility for orchestration of students’ learning opportunities. Moving to university, do students take on more responsibility for orchestration?

In this session there will be a chance to look back on the learning opportunities you experienced last term and to reflect on how (or even if) they were orchestrated. What could be different in the term ahead if you pay more attention to how distinctive learning opportunities are orchestrated?

In this talk, we will provide an overview of results in the setting of granular crystals, consisting of spherical beads interacting through nonlinear elastic spring-like forces. These crystals are used in numerous engineering applications including, e.g., for the production of "sound bullets'' or the examination of bone quality. In one dimension we show that there exist three prototypical types of coherent nonlinear waveforms: shock waves, traveling solitary waves and discrete breathers. The latter are time-periodic, spatially localized structures. For each one, we will analyze the existence theory, presenting connections to prototypical models of nonlinear wave theory, such as the Burgers equation, the Korteweg-de Vries equation and the nonlinear Schrodinger (NLS) equation, respectively. We will also explore the stability of such structures, presenting some explicit stability criteria for traveling waves in lattices. Finally, for each one of these structures, we will complement the mathematical theory and numerical computations with state-of-the-art experiments, allowing their quantitative identification and visualization. Finally, time permitting, ongoing extensions of these themes will be briefly touched upon, most notably in higher dimensions, in heterogeneous or disordered chains and in the presence of damping and driving; associated open questions will also be outlined.