## Cone eversion

Last year, around the time Chris Wendl was running the h-principle learning seminar at UCL, I set my second years an exercise from Eliashberg-Mishachev as a difficult challenge problem: to find an explicit cone eversion. In other words, find a path in the space of functions on connecting to such that none of the intermediate functions has a critical point. One of these students, Tom Steeples, got hooked on the problem, almost solved it, and afterwards used Mathematica to produce some beautiful computer animations of a solution given by Tabachnikov in American Mathematical Monthly (1995) Vol 102, Issue 1, pp 52–56. Here, below the fold because it’s quite a large file, is one of his images. Reproduced with Tom’s kind permission (the copyright is his).

## Gromoll filtration

In my latest preprint with Georgios Dimitroglou Rizell, we use the topology of diffeomorphism groups of high-dimensional spheres to produce interesting examples of nontrivial topology in symplectomorphism groups of cotangent bundles. Until we started thinking about this, I didn’t know much about the higher homotopy groups of so here is some interesting stuff I learned while we were writing this paper.

## E-learning project report

My final report on the e-learning project “Video lectures filmed by students” is now available to download in PDF form.

The purpose of this e-learning project was to test the effectiveness and viability of getting students to film mathematics lectures and the effect on student learning of making these videos available. The project was made possible by an E-Learning Development Grant (ELDG) and by the cooperation of a large number of people who I thank at the end.

**Disclaimer.** The project analysis is not scientific: there is no attempt made at comparison with a control group, the data sets are not large and the statistical methods used to analyse them are crude. This report is intended to be at best a rough guide to the UCL Mathematics Departmental Teaching Committee as to what action to take on filming of mathematics lectures.

## December: video project update

The video project has been progressing nicely: all of the videos have now been compressed and most have been uploaded to either Youtube or Lecturecast. Read on for some of the results.

## Geometry and undecidability

These are the notes from a talk I gave to the UCL Undergraduate Mathematics Colloquium in early October and I would like to thank them for being such an attentive audience with so many good questions. The talk is a gentle introduction to the work of Nabutovsky and Weinberger, on how logical complexity gives rise to complexity for sublevel sets of functionals in geometry.

## Video-lecture project weeks 1 and 2

## E-Learning: Video lectures filmed by students

I recently received a grant from the UCL e-learning team to run a project for filming maths lectures.

**The aim:** The aim is to provide UCL mathematics students with high-quality video coverage of some of their core lectures. This would be particularly useful in mathematics where material is hard to absorb on a first hearing. We’d hope this would be particularly useful for our many overseas students with English as an n^{th} language for n>1. Many of our courses are big ancillary courses for other departments and having lectures available online would help alleviate the possible clashes that might occur in timetabling.

**The problem:** Mathematical pedagogy focuses on board-based lectures, which are difficult to film under the current system used at UCL because the camera quality is not sufficient to capture extensive boardwork.

**The proposal:** Use a small, high-quality camera with a mount (borrowed from the e-learning team), operated by students from the front of the lecture theatre. We will train a group of students to act as camera-operators.

This is only a trial, so we will concentrate on two second-year courses (Mathematical Methods 3 and Complex Analysis), filmed by a team of four students. These students would preferably be from other classes so that the students in the target classes can focus on learning and not on filming. This would result in a workload of one or two hours/week for each student, which is hopefully not too much to distract from their own lectures, and they will be paid for their work (that’s where the grant is going).

The results will be posted to Lecturecast and the students will be able to give continuous feedback via Moodle forums, to that we can optimise our filming techniques. We will also assess the effectiveness of this method and the usefulness of the new resource using questionnaires.

From a lecturer’s perspective, I want my lectures to be videoed so that I can watch them back and know where I need to explain things better next time. While it is slightly terrifying for the unintentional verbal and notational errors one makes during lectures to be captured and viewable, it would surely be more useful for me to spot them so that I know why my students are getting confused.

While high-quality tracking cameras might be an optimal solution to this problem, they are expensive and we would first like to assess the usefulness of video lectures to our students. I also think that having a camera operated by someone who understands and appreciates the lectures, someone who can point the camera at what they think is most relevant, will lead to a more natural and useful video. I have seen the results of tracking cameras before, and I usually ended up standing on the far side of the board from the equation I was talking about. Having the camera placed at the front of the lecture theatre also seems like a more surefire way of being able to see what’s been written.