Archive for the mathematics Category

The Next Semester

Posted in Artificial Intelligence, Education, mathematics, Maynooth with tags , , , , , , , on January 26, 2026 by telescoper

There’s just a week to go before the next Semester at Maynooth University so I’ve been looking at my calendar for the weeks ahead. Actually, I won’t start teaching again until Tuesday 3rd February, because Monday 2nd February is a national holiday. As it turns out, however, I don’t have any lectures, labs or tutorials on Mondays anyway so I won’t be missing a session either on February 2nd or on May 4th, another holiday. I will have to miss one on Friday 3rd April (Good Friday), though.

The Timetable has given me two 9 o’clock lectures a week for the forthcoming Semester, one on Tuesdays and the other on Thursdays. I don’t think the students like 9am lectures very much, but I don’t mind them at all. I find it quite agreeable to have accomplished something concrete by 10am, which I don’t always do. This schedule might mean that I defer publishing papers at the Open Journal of Astrophysics on those days. I usually do this before breakfast, but I might not have time if I have to be on campus and ready to teach for 9am.

As usual, Semester 2 is a stop-start affair. We have six weeks until the Study Break, which includes the St Patrick’s Day holiday, then we’re back for two weeks (minus Good Friday) before another week off for Easter. We return on Monday April 13th to complete the Semester; the last lectures are on Friday 8th May and exams start a week later. This arrangement creates no problems for lecture-based teaching, but it takes some planning to organize labs and project deadlines around the breaks. I’ll have to think about that for my Computational Physics module.

A more serious issue for Computational Physics is how to deal with the use of Generative AI. I’ve written about this before, in general terms, but now it’s time to write down some specific rules for a specific module. A default position favoured by some in the Department is that students should not use GenAI at all. I think that would be silly. Graduates will definitely be using CoPilot or equivalent if they write code in the world outside university so we should teach them how to use it properly and effectively.

In particular, such methods usually produce a plausible answer, but how can a student be sure it is correct? It seems to me that we should place an emphasis on what steps a student has taken to check an answer, which of course they should do whether they used GenAI or did it themselves. If it’s a piece of code to do a numerical integration of a differential equation, for example, the student should test it using known analytic solutions to check it gets them right. If it’s the answer to a mathematical problem, one can check whether it does indeed solve the original equation (with the appropriate boundary conditions).

If anyone out there reading this blog has any advice to share, or even a link to their own Department’s policy on the use of GenAI in computational physics for me to copy adapt for use in Maynooth, I’d be very grateful!

(My backup plan is to ask ChatGPT to generate an appropriate policy…)

Marking Over

Posted in Artificial Intelligence, Biographical, mathematics, Maynooth with tags , , , , on January 19, 2026 by telescoper

Well, that wasn’t too painful. I’ve completed my marking duties. The fact that it has been pouring with rain most of the day made it easy to concentrate on this task. I was going to have a break for lunch, but I decided to keep on going until I was finished, though I did have to take a break for a telecon this afternoon. I also had to dash out to the shops, primarily to replenish my stock of food for the garden birds but also to get some groceries for myself. Having skipped lunch I bought myself something nice for supper.

Looking at the departmental database I see that I appear to be the first member of staff to have finished and uploaded all their Semester 1 examination marking. Normally I’m just happy if I’m not the last!

It feels good to have finished this task. It’s definitely a weight off my mind. I wouldn’t want to leave any loose ends when Trumpageddon comes.

I can’t say anything about the results of course but the change I made to continuous assessment, from take-home assignments to class tests, does not seem to have had a negative effect on either group of students I have been teaching. The opposite may indeed have been the case, as the class tests perhaps provide better preparation for the final assessment than the previous method. I think some other lecturers might make a similar switch in future. Anyway, I definitely plan to do something similar for my Semester 2 module on Particle Physics.

Now I have a couple of weeks before teaching resumes so I can get on with other things. For the rest of this week my priority is to finish revising a paper that I hoped to do before Christmas. I’ll see how that goes before deciding what to do next.

I’ll also have to prepare teaching for Semester 2. That shouldn’t be too difficult, as I’ve taught both modules before, but I do have to give some thought as to precisely how I’m going to word the instructions on the use of AI for my Computational Physics module. That can wait a little while, though, as it mainly affects the mini-project to be done towards the end of the Semester. In the meantime I’ll be thinking about other things…

Marking Progress

Posted in Biographical, mathematics, Maynooth with tags , , , on January 16, 2026 by telescoper

The last day of a week dominated by examination marking found me briefly back on campus to return the batch of scripts I have finished corrrecting and collect the next set (which, happily, is much smaller):

There are 14 scripts in the pile for my second paper, for my 4th year Mathematical Physics module on Differential Equations and Complex Analysis, around one-third of those for my Engineering Mathematics module Differentiaol Equations and Transform Methods. Based on the total number of examinations I have to mark I am therefore now 50% complete, but based on the number of scripts I’m about 75% through. I should be able to finish the latest batch in a day, but there’s no desperate rush so I’ll do them on Monday. I’m not going to start them now as I am off a concert – my first of 2026 – this evening and I prefer not to work at weekends unless I absolutely have to.

I finished the first set of marking yesterday, and spent most of this morning uploading and checking the scores and the conflation of exam marks with coursework scores. Satisfied that all is OK, I returned the scripts to the office for storage until our Examination Board meeting in about 10 days. I wasn’t on campus long, but there was a fire alarm in the Science Building while I was there. As usual, it turned out to be a false alarm.

Anyway, I should be finished with examination matters by Monday evening, which gives me four days next week to get on with other things. I’m looking forward to the change.

Everything is a Simple Harmonic Oscillator

Posted in mathematics, The Universe and Stuff with tags , , , on December 13, 2025 by telescoper

Anyone who has studied theoretical physics for any time will be familiar with the simple harmonic oscillator, which I will call the SHO for short. This is a system that can be solved exactly and its solutions can be applied in a wide range of situations where it holds approximately, e.g. when looking at small oscillations around equilibrium. I’ve often remarked in lectures that we spend much of our lives solving the SHO problem in various guises, often pretending that the difficult system we have in front of us can, if looked at in the right way, and with sufficient optimism, be approximated by the much simpler SHO. Cue the old joke that if all you have is a hammer, everything looks like nail…

That rambling prelude occurred to me when I found this little problem in some old notes. It is a cute mathematical result that shows that the Friedman equations that underpin our standard cosmological model can in fact be written in the same form as those describing a Simple Harmonic Oscillator. In what follows we take the cosmological constant term to be zero.

The resulting equation is the SHO equation if k>0. I’m not sure whether this result is very useful for anything, but it is cute. It also goes to to show that, if looked at in the right way, the whole Universe is a Simple Harmonic Oscillator!

Nine Teaching Days to Christmas

Posted in Biographical, mathematics, Maynooth, Television with tags , , , on December 8, 2025 by telescoper

This week is the penultimate week of teaching term at Maynooth and, as usual at this stage of the Semester, we’re getting busier and busier. The examinations for January have been sent off for printing and are (presumably) ready to go, and I’m up to date with all my coursework gradin so I am, miraculously, on schedule as far as teaching is concerned. I should finish covering the respective syllabuses by Tuesday 16th, with the remaining teaching sessions devoted to revision. I don’t have any lectures on Fridays this term, so my teaching ends, a day before the end of term, on Thursday 18th December. To celebrate the end of term I’ll be presenting the students in the last session of my Engineering Mathematics module the gift of a final Class Test. I’m not sure when I’ll get to correct it. Oh, and our Department Christmas Dinner is on Wednesday 17th.

I’ll soon have to decide when to pause the publication of new papers at the Open Journal of Astrophysics and prepared for next year’s Volume 9. A certain person is insisting that I take a complete break for at least a week, so I think we’ll probably stop on Christmas Eve and begin again in the New Year.

In the meantime, term goes on. I have three lectures to get ready for tomorrow. Incidentally, my mid-Semester feedback suggested that I start each lecture with an introduction to say what I’m going to be covering. Here are some examples of what I’ve been doing in response:

A month to go

Posted in Artificial Intelligence, Biographical, Education, mathematics, Maynooth with tags , , , , , on November 25, 2025 by telescoper

I’ve been a bit preoccupied these recent weeks so it was with a shock that I realised that we’re into Week 9, which means just four weeks (including this one) until the end of term and just a month before Christmas. Teaching finishes here in Maynooth on Friday 19th December, but I don’t have any lectures on Fridays so in my case it will finish the day before (with a tutorial). I don’t know how many students will be there, but the module concerned is my 4th year Mathematical Physics module and the students are very hard-working, so I think most will attend. After such a busy term I’m sure that they will need a break as much as I will.

I had to rejig the schedule for both modules I am teaching this semester to accommodate the introduction of in-class tests to replace take-home assignments (for reasons I outlined here). I’ve also been handing out voluntary exercises for practice, not counting towards the module mark but for formative reasons. Both modules are mathematical in nature, and I think the best way to learn mathematics is by doing it…

Despite the changes with respect to last year, I am still roughly on track. In my Engineering Mathematics module I’ve just finished Laplace transforms, and will start Fourier methods tomorrow. With the mathematical physicists, I am in the middle of complex analysis, having done complex differentiation and conformal mappings and starting complex integration next week.

I still have a couple more class tests to get through. On the positive side, the students are turning up for them and have expressed approval for the fact that they don’t have compulsory homework to do off-campus. This form of assessment is undoubtedly harder work for the students, it’s also better preparation for the examination that take-home assignments.

We’ve just received the draft examination timetable for January, and I’m pleased that both of the examinations for which I am responsible will take place quite early in the examination period (on 12th and 15th January, respectively) so I should be able to get them corrected in time to have a break for some research before teaching resumes at the start of February.

Testing Times

Posted in Artificial Intelligence, Education, mathematics, Maynooth with tags , , , , on October 17, 2025 by telescoper

As it was foretold, I conducted my first set of my new-style in-class tests this week. These tests, as I mentioned a while ago,  were introduced because of concerns about the integrity of the coursework element of my modules in the light of improvements in Generative AI.

The main events – one for each of my modules – were both yesterday, but one student couldn’t make it at the scheduled time (for good reasons) so I set a special test this morning, which is now over. Because access to the internet is not allowed these tests are invigilated.

It’s been quite a while since I was last required to invigilate a full examination. I think it was back in Nottingham days, actually. I never enjoyed this task even though I took work to do it wasn’t really possible to do much as one had to keep one’s eyes on the students. Crosswords could be done; these are good in this situation because you can solve a few clues at a time. It was disappointing if I happened to take one that was easy enough to do quickly, as there was little to stave off the boredom after completing it. Other things I used to do included counting the number of right-handed and left-handed students, though I never did any detailed statistical analysis of the results.

Anyway, my recent class tests were a bit different. Designed to fit in a lecture slot of 50 minutes duration, they were much shorter than traditional end-of-year exams. They were also “open-book” style, so students could bring anything on paper that they wanted. Phones and laptops were, however, forbidden. During these tests I just sat quietly with my laptop getting some work done, with an occasional glance at the students. It was actually nice to be locked away like this with no disturbance. Time passed very quickly, actually, though perhaps not as quickly as it did for the students taking the tests.

When I first told the students that the tests would be “open-book”, I think they all assumed that would make them easy. I don’t think that was the case, however, as the questions are designed so that the answers can’t be obtained immediately by looking them up in a textbook. Also, having things on paper rather than in your head does slow you down. I’ve never seen much point in examinations as speed tests. I designed this week’s tests so that the questions could be done in about 30 minutes, but the formal duration was 50 minutes. I encouraged students who finished early to use the remaining time to check their work, but some did leave early.

This new regime also meant I had number of teaching sessions without the exertion of having to do any actual teaching, which was nice. The downside is, of course, that I now have stacks of class tests to correct. That will be payback time.

I won’t know how well the students have coped until I have got their grades, but informal feedback was that they seemed reasonably content with the new method of assessment. I’ll be doing the next ones in about three weeks.

Quarter-Term – Testing Time

Posted in Education, mathematics, Maynooth with tags , , , on October 13, 2025 by telescoper

I’ve just noticed that three teaching weeks have passed and we’re already into the fourth. Tempus fugit. Both the modules I am lecturing this semester are divided into four chunks of approximately equal size. For example, MP469 Differential Equations and Complex Analysis splits into: Ordinary Differential Equations; Partial Differential Equations; Complex Functions and Derivatives; and Complex Integration. Though technically not on the syllabus, I also do couple of lectures on Conformal Mappings because I think they’re cool.

As I mentioned a while ago,  I am concerned about the integrity of the coursework element of these modules in the light of improvements in Generative AI. Only a couple of years ago GenAI could not solve the sort of problems I set for homework, but now it generally can. I don’t altogether object to people applying artificial intelligence to solve mathematical problems, but the main issue is that it does make mistakes. Moreover, instead of saying “sorry I can’t solve that problem” it will generally present a superficially plausible but incorrect solution. Although students will probably use GenAI for problem-solving, I think it is important that they learn to do such problems themselves, otherwise they won’t know whether the solution coughed up by the algorithm is correct or not.

The only way to learn mathematics is by doing it. If students get GenAI to do the mathematics for them, then they won’t learn it. In the past we have given marks for coursework (usually 20% of the module mark) mainly to encourage students to do them. Students who don’t bother to do these exercises generally do badly in the final exam (80%).

For these reasons I am moving the assessment from weekly homework sheets – which could be tackled with AI – to supervised in-class tests for which students can use notes on paper, but not laptops or phones. I will of course give examples for the students to have a go at themselves, and I will give feedback on their attempts, but they will not contribute to the module score. Another advantage of this approach is that students won’t have to do so much work against deadlines outside of class.

What I’ve decided to do is have one class test for each of the four sections of each module. Given that we’re about a quarter of the way through the term, it’s time for the first ones. This week there will be a class test on Ordinary Differential Equations. I’ve never been enthusiastic about examinations being speed tests, so I’ve decided to set problems to be done in a 50-minute session which would be expected to take about 30 minutes in a formal end-of-term examination.

I have to make a short work-related trip that will keep me away on Wednesday, but I’ve already written the test questions, and will make arrangements for someone to supervise the tests if for some reason I don’t make it back to Maynooth on time…

Anyway, although we’ve been teaching for three weeks I still have to check my calendar to remember which room I’m supposed to go to before every lecture. Perhaps by Christmas I will have learned them off by heart…

When will the AI Bubble burst?

Posted in Artificial Intelligence, Finance, mathematics with tags , , , , on October 12, 2025 by telescoper

I’m not a financial expert, but I have noticed a significant number of articles in the media suggesting that the Generative AI industry is a bubble waiting to burst. There are recent pieces here on the BBC website, here in the Financial Times (from which I stole the cartoon), and here in the Irish Times, to name but a few.

These stories are based on reports by the Bank of England and the International Monetary Fund, warning of a stock market crash far worse than the dotcom boom-and-bust of 2000 and even the banking crisis of 2008. Over 30% of the valuation of the US stock market, for example, lies in five big technology companies that are investing heavily in the enormous infrastructure required for AI. Their extravagant capital expenditure is underpinned by a complex series of financial arrangements which could unravel very quickly if the investors get cold feet and consider it unlikely they will see a return on their money. It does look very much like a bubble to me.

My own view is that the claims made about the capabilities of AI by tech gurus are grossly overstated. Only the irredeemably gullible could think otherwise. I think a correction is inevitable. It’s not a question of “if” but “when” and “how much”. I am not competent to answer those questions.

P.S. Now there’s an RTÉ Brainstorm piece along the same lines…

How to Fight Fraudulent Publishing

Posted in mathematics, Open Access with tags , , , , , , , on September 23, 2025 by telescoper

There’s a short article on arXiv with the title How to Fight Fraudulent Publishing in the Mathematical Sciences: Joint Recommendations of the IMU and the ICIAM which is well worth reading. The abstract is not useful but the prelude reads:

PreludeIn November 2023, Clarivate announced that it had excluded the entire field of mathematics from the latest edition of its influential list of ‘highly cited researchers’. This prompted the IMU and the ICIAM to conduct a more thorough investigation into the problem of fraudulent publishing in the mathematical sciences (see [1]). Understanding the problem is one thing; finding a way out and regaining control is another. With the recommendations given below, we would like to start the discussion on how, as a global community, we can achieve this. We are all concerned. It affects the very core of the science we love so much. I.A.

arXiv:2509.09877

The paper correctly identifies predatory journals and citation cartels as two consequences of the effort to quantify and rank the quality of research through scientific ‘performance indicators’, in the form of bibliometric measures and suggests some possible remedies.

Many of the recommendations are already included in the San Francisco Declaration on Research Assessment (SFDORA). Many also apply beyond the mathematical sciences (which is why I dropped the Mathematical Sciences bit in the title of the paper from the title of this blog post) and it’s not a long paper so I suggest you read it.

In my view one of the most important steps to take is to ditch the reliance on such companies as Scopus and Clarivate, who have deliberately constructed a system that is so easy to game. All higher education institutes should follow the examples of the Sorbonne University in Paris and, more recently, Utrecht University in the Netherlands. The academic publishing racket is inherently fraudulent. Too many universities, and indeed researchers employed by them, are willing participants in the system.