The House of Lords just published a report recommending that students in the UK study mathematics until the age of 18. This recommendation cropped up last year in the Vorderman review. The Lords report can be found online in html and in pdf format. It’s a fairly long report but my response is as simple as my response to the Vorderman review: Where are the teachers going to come from? We don’t have enough teachers at the moment to provide the best-quality maths education. Giving the current ones more to do would lead to catastrophe. The Lords’ report even states The Department of Education, recognising the role of teaching in increasing the progression of students to A level STEM subjects,… has introduced a number of initiatives to increase the number of specialist teachers (such as, golden handshakes and bursaries), but, by their own admission, “the targets set by the previous Government for numbers of specialists teaching physics and maths will not be met”. Is the best that the DoE can come up with is golden handshakes and bursaries? It is unsurprising that the targets won’t be met. There will be no single solution to the problem of mathematics education in the UK, it will require many areas to be tackled but the solution must surely include higher pay for maths and science teachers. The simple reality is that my students can finish their degrees and get a job that pays them an average starting salary of (according to HESA) 23,160 pounds (according to AGR figures 26,500 pounds) with rapid increases, or spend a year on a bursary of less than 20,000, and join a profession where after a number of years they reach top of scale at 31,552 pounds? And yes I know I should take into account holidays, etc, but even with shorter holidays the working conditions are better in many professions. Looking at the figures, which do you think is the better...

## Best of xkcd: Maths I

posted by Kevin Houston

I’m currently at a conference so have been working hard on research-y type things rather than thinking up exciting blogs posts. So here are some of my favourite maths strips from...

## Monge and Optimal Transport...

posted by Kevin Houston

Mathematicians are ignorant of their history. They know the names of the greats but generally can repeat only one or two (erroneous) stories about them. For example, if asked for a mathematical story from the French Revolution, then many would plump for a story about Galois, shot at dawn. However, there were many other revolutionary mathematicians. One such is Monge. His name is familiar to differential geometers through Monge form and to analysts of PDEs through the Monge-Ampere equation. Information about him is scant in English. All the important books are in French and so my attempts to study his life in more depth have been thwarted by my lack of fluency. I think I first became interested in him when I was a student. I read a book (the name escapes me) which stated that he had studied the concept of the optimal transport of soil when constructing fortifications. The fact that caught my imagination was that the answer was constrained by the observation that the paths of two particles should not cross. At the time I came up with a counter-example but that was because I didn’t really understand the parameters of the problem. I was greatly interested then when I heard that one of this year’s Hardy Lectures would be about Monge and the optimal transport problem. I was greatly disappointed when I discovered that it clashed with previous commitments. When I was at the conference in Liverpool to celebrate the birthdays of Bill Bruce and Terry Wall Andrew Ranicki told me that the lecture had been recorded. And I’m glad it was, it’s a great talk. It’s given by Etienne Ghys. He takes in the cutting of stones, including how Monge designed a never used plan for the ceiling of...

## Cool proof of the Fundamental Theorem of Algebra...

posted by Kevin Houston

Despite being on holiday I can’t resist looking for cool proofs. This one is not so much cool as interesting in a Why-didn’t-I-think-of-that way. The fundamental theorem of algebra — that any polynomial has a complex root — is well known to be a theorem of analysis rather than algebra and many proofs are known. The proof I use in my course relies on Liouville’s Theorem and the Maximum Modulus Theorem. However, there is a more direct route using only Cauchy’s Integral Formula and the Estimation Lemma. I found the basic idea in Complex Proofs of Real Theorems by Peter Lax and Lawrence Zalcman. Here’s my modified version: Every polynomial , , , has a root in . The proof is as follows. Suppose not and derive a contradiction. Since for all the function defined by is differentiable on all of . Now, for , So there exists an such that implies that . Thus for such . As is differentiable on all of we have for all , by Cauchy’s Integral Formula, (and where denotes a circle of radius centred at the origin), The right hand side can be made as small as we like by taking large enough. Thus . This is impossible so we have a contradiction and therefore has a...

## New Maths Journals

posted by Kevin Houston

I’m on holiday at the moment so don’t have much time to write. If you are interested in the recent fuss about academics fighting for reasonably-priced journals, then pop over to Tim Gower’s blog to see this announcement on new journals. There’s also a little bit on Terry Tao’s blog. It looks like this could be...