So, after an absence from teaching of over a year, this afternoon I returned to the lecture theatre to give a double session on the module EE206 Differential Equations and Transform Methods. I was a bit apprehensive about having a two-hour slot and it is fair to say that I felt a bit knackered after it, but `then I am getting on a bit. I did have time for a ten-minute break in the middle during which the students could relax and stretch their legs a little. Some of them even came back afterwards.
This module is meant for students on two courses, Electronic Engineering and Robotics and Intelligent Devices, so I will have to think of relevant examples. I’ve got the RLC circuit, of course, but I’ll have to more than that!
If you’re interested you can find an old summary of the module here to see what topics are covered.
The good news from my point of view is that I have a decent room to teach in – complete with chalk boards – and the students seemed pleasant and engaged. I always like to get some interaction going in my classes so it was good to find a reasonable number of people willing to offer answers to questions I asked and indeed willing to ask me questions or request clarification. Overall, I was quite pleased with how it went. You will have to ask the students to see if they agree. At any rate I did manage to get through everything I planned to cover. The class size is about 55, incidentally.
Anyway, today I just warmed up for the module with some revision of basic calculus. I had pessimistically imagined that the students would have forgotten what they did in the first year about this, but in fact quite a few of them remembered quite a lot. I have my second session with this group on Thursday, though that should be a bit easier as it is only one hour instead of two. I will start differential equations proper then.
My remaining teaching sessions this week are all in the Arts Building. I have been quite worried that the rooms I am supposed to use would not be ready in time, but I took a walk around yesterday morning and they are ready (although construction work is going on elsewhere in the block). I was thinking I might have to give these lectures via a remote connection from home as in the old days of the pandemic, but that fortunately is not the case.
Today, Monday 5th August 2024, being the first Monday in August, is a Bank Holiday in Ireland. This holiday was created by the Bank Holiday Act of 1871 when Ireland was under British rule. While the August Bank holiday was subsequently moved to the end of August in England and Wales, it has remained at the start of August in Ireland. Today is also a Bank Holiday in Scotland, though the Scots have the best of both worlds and have a holiday at the end of August too.
The first day of August marks the old pagan festival of Lughnasadh, named after the God Lugh, on which is celebrated the beginning of the harvest season. This coincides with the English Lammas Day one of many Christian festivals with pagan origins. Traditionally this marks the start of the harvest season and is celebrated accordingly, with rites involving the first fruit and bread baked from flour obtained from the first corn. It is also one of the cross-quarter days, lying roughly half-way between the Summer Solstice and the Autumnal Equinox (in the Northern Hemisphere).
It seems to be a tradition in Maynooth that the Bank Holidays in May and August are are adjacent to examinations. This year they start on Wednesday (7th August). I am, however, still on sabbatical so I don’t have any correcting duties. That doesn’t mean I can’t wish all the students taking repeat examinations all the best in their endeavours.
This month is the last of my sabbatical. I officially return to normal duties on 1st September, but that is a Sunday so I won’t return to the office until Monday 2nd September. That is if I have an office. There’s a lot of reorganization going on and currently I don’t know where I’ll be based. At least I know what I’ll be teaching in Semester 1 though: a fourth-year Mathematical physics course on Differential Equations and Complex Analysis and a second-year Engineering Mathematics course. These are not what I would have chosen if I had a free hand (I’d rather teach physics than mathematical methods) but I’ve had it excessively easy for the last year so can’t complain. With a bit of luck I might get a project student or two as well, if the students haven’t forgotten who I am!
Here’s an interestingly different talk in the series of Cosmology Talks curated by Shaun Hotchkiss. The speaker, Sylvia Wenmackers, is a philosopher of science. According to the blurb on Youtube:
Her focus is probability and she has worked on a few theories that aim to extend and modify the standard axioms of probability in order to tackle paradoxes related to infinite spaces. In particular there is a paradox of the “infinite fair lottery” where within standard probability it seems impossible to write down a “fair” probability function on the integers. If you give the integers any non-zero probability, the total probability of all integers is unbounded, so the function is not normalisable. If you give the integers zero probability, the total probability of all integers is also zero. No other option seems viable for a fair distribution. This paradox arises in a number of places within cosmology, especially in the context of eternal inflation and a possible multiverse of big bangs bubbling off. If every bubble is to be treated fairly, and there will ultimately be an unbounded number of them, how do we assign probability? The proposed solutions involve hyper-real numbers, such as infinitesimals and infinities with different relative sizes, (reflecting how quickly things converge or diverge respectively). The multiverse has other problems, and other areas of cosmology where this issue arises also have their own problems (e.g. the initial conditions of inflation); however this could very well be part of the way towards fixing the cosmological multiverse.
The paper referred to in the presentation can be found here. There is a lot to digest in this thought-provoking talk, from the starting point on Kolmogorov’s axioms to the application to the multiverse, but this video gives me an excuse to repeat my thoughts on infinities in cosmology.
Most of us – whether scientists or not – have an uncomfortable time coping with the concept of infinity. Physicists have had a particularly difficult relationship with the notion of boundlessness, as various kinds of pesky infinities keep cropping up in calculations. In most cases this this symptomatic of deficiencies in the theoretical foundations of the subject. Think of the ‘ultraviolet catastrophe‘ of classical statistical mechanics, in which the electromagnetic radiation produced by a black body at a finite temperature is calculated to be infinitely intense at infinitely short wavelengths; this signalled the failure of classical statistical mechanics and ushered in the era of quantum mechanics about a hundred years ago. Quantum field theories have other forms of pathological behaviour, with mathematical components of the theory tending to run out of control to infinity unless they are healed using the technique of renormalization. The general theory of relativity predicts that singularities in which physical properties become infinite occur in the centre of black holes and in the Big Bang that kicked our Universe into existence. But even these are regarded as indications that we are missing a piece of the puzzle, rather than implying that somehow infinity is a part of nature itself.
The exception to this rule is the field of cosmology. Somehow it seems natural at least to consider the possibility that our cosmos might be infinite, either in extent or duration, or both, or perhaps even be a multiverse comprising an infinite collection of sub-universes. If the Universe is defined as everything that exists, why should it necessarily be finite? Why should there be some underlying principle that restricts it to a size our human brains can cope with?
On the other hand, there are cosmologists who won’t allow infinity into their view of the Universe. A prominent example is George Ellis, a strong critic of the multiverse idea in particular, who frequently quotes David Hilbert
The final result then is: nowhere is the infinite realized; it is neither present in nature nor admissible as a foundation in our rational thinking—a remarkable harmony between being and thought
But to every Hilbert there’s an equal and opposite Leibniz
I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author.
You see that it’s an argument with quite a long pedigree!
Many years ago I attended a lecture by Alex Vilenkin, entitled The Principle of Mediocrity. This was a talk based on some ideas from his book Many Worlds in One: The Search for Other Universes, in which he discusses some of the consequences of the so-called eternal inflation scenario, which leads to a variation of the multiverse idea in which the universe comprises an infinite collection of causally-disconnected “bubbles” with different laws of low-energy physics applying in each. Indeed, in Vilenkin’s vision, all possible configurations of all possible things are realised somewhere in this ensemble of mini-universes.
One of the features of this scenario is that it brings the anthropic principle into play as a potential “explanation” for the apparent fine-tuning of our Universe that enables life to be sustained within it. We can only live in a domain wherein the laws of physics are compatible with life so it should be no surprise that’s what we find. There is an infinity of dead universes, but we don’t live there.
I’m not going to go on about the anthropic principle here, although it’s a subject that’s quite fun to write or, better still, give a talk about, especially if you enjoy winding people up! What I did want to say mention, though, is that Vilenkin correctly pointed out that three ingredients are needed to make this work:
An infinite ensemble of realizations
A discretizer
A randomizer
Item 2 involves some sort of principle that ensures that the number of possible states of the system we’re talking about is not infinite. A very simple example from quantum physics might be the two spin states of an electron, up (↑) or down(↓). No “in-between” states are allowed, according to our tried-and-tested theories of quantum physics, so the state space is discrete. In the more general context required for cosmology, the states are the allowed “laws of physics” ( i.e. possible false vacuum configurations). The space of possible states is very much larger here, of course, and the theory that makes it discrete much less secure. In string theory, the number of false vacua is estimated at 10500. That’s certainly a very big number, but it’s not infinite so will do the job needed.
Item 3 requires a process that realizes every possible configuration across the ensemble in a “random” fashion. The word “random” is a bit problematic for me because I don’t really know what it’s supposed to mean. It’s a word that far too many scientists are content to hide behind, in my opinion. In this context, however, “random” really means that the assigning of states to elements in the ensemble must be ergodic, meaning that it must visit the entire state space with some probability. This is the kind of process that’s needed if an infinite collection of monkeys is indeed to type the (large but finite) complete works of shakespeare. It’s not enough that there be an infinite number and that the works of shakespeare be finite. The process of typing must also be ergodic.
Now it’s by no means obvious that monkeys would type ergodically. If, for example, they always hit two adjoining keys at the same time then the process would not be ergodic. Likewise it is by no means clear to me that the process of realizing the ensemble is ergodic. In fact I’m not even sure that there’s any process at all that “realizes” the string landscape. There’s a long and dangerous road from the (hypothetical) ensembles that exist even in standard quantum field theory to an actually existing “random” collection of observed things…
More generally, the mere fact that a mathematical solution of an equation can be derived does not mean that that equation describes anything that actually exists in nature. In this respect I agree with Alfred North Whitehead:
There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain.
It’s a quote I think some string theorists might benefit from reading!
Items 1, 2 and 3 are all needed to ensure that each particular configuration of the system is actually realized in nature. If we had an infinite number of realizations but with either infinite number of possible configurations or a non-ergodic selection mechanism then there’s no guarantee each possibility would actually happen. The success of this explanation consequently rests on quite stringent assumptions.
I’m a sceptic about this whole scheme for many reasons. First, I’m uncomfortable with infinity – that’s what you get for working with George Ellis, I guess. Second, and more importantly, I don’t understand string theory and am in any case unsure of the ontological status of the string landscape. Finally, although a large number of prominent cosmologists have waved their hands with commendable vigour, I have never seen anything even approaching a rigorous proof that eternal inflation does lead to realized infinity of false vacua. If such a thing exists, I’d really like to hear about it!
I’m indebted to my colleague David Malone for sending me this small excerpt from an old issue of the Kalendarium of St Patrick’s College, Maynooth, dating back to the 1960s, which deals with the appointments of new members of staff
Halfway down you will see a reference to Mathematical Mystics!
This is obviously a mistake. It should of course be Mathematical Psychics Physics. I also think the name of the Mathematical Mystics lecturer should be Tigran Tchrakian. I think these are both transcription errors from somebody’s very bad handwriting! The current Department of Theoretical Physics at Maynooth was formerly known by the title Mathematical Physics.
There are some other points of interest. in Experimental Physics you will find mention of a young Susan Lawlor who is now better known as Susan McKenna-Lawlor, a very eminent astrophysicist who specialized in space instrumentation, now in her eighties.
I’m also amused by the existence of a lecturer in Elocution…
The historical background of St Patrick’s College is that it was primarily a Catholic theological institution (founded in 1795) although it taught secular courses and was a recognized college of the National University of Ireland from 1910. It was only in the mid-1960s that it was opened to lay students, which expanded the numbers considerably. In 1997 that the secular part separated and formed NUI Maynooth (now known by the marketing people as Maynooth University). The remaining theological institution is known as St Patrick’s Pontifical University (or St Patrick’s College or just Maynooth College).
A major role for St Patrick’s College was the training of priests and I suppose it was important that priests should be well spoken, hence the lectures on elocution…
Near the top in connection with Sociology you can see the title An tAth which is the Irish language way of writing the abbreviation “Fr” for “Father”, indicating a priest; “father” is athair and the an is a definite article. Note the lower case t in front of Ath which is an example of prothesis.
Finally, right at the top of the page you can see the name Donal Linehan, which will be familiar to Irish rugby fans but I don’t know if there’s a family connection between the former Ireland intentional who is now a TV commentator and the lecturer in Roman and Civil Law.
With Christmas looming and the January examination period getting closer, I thought I’d help (?) those involved in such assessments by sharing this model of an elegant marking scheme from a Mathematics examination.
There is no proof that black holes contain singularities when they are generated by real physical bodies. Roger Penrose claimed sixty years ago that trapped surfaces inevitably lead to light rays of finite affine length (FALL’s). Penrose and Stephen Hawking then asserted that these must end in actual singularities. When they could not prove this they decreed it to be self evident. It is shown that there are counterexamples through every point in the Kerr metric. These are asymptotic to at least one event horizon and do not end in singularities.
arXiv:2312.00841
I don’t think this paper is as controversial as some people seem to find it. I think most of us have doubts that singularities – specifically curvature singularities – are physically real rather than manifestations of gaps in our understanding. On the other hand, this paper focusses on an interesting technical question and provides a concrete counterexample. The point is that the famous Penrose-Hawking singularity theorems don’t actually prove the existence of singularities; they prove geodesic incompleteness, i.e. that there are geodesics that can only be extended for a finite time as measured by an observer travelling along one. Geodesic incompleteness does imply the existence of some sort of boundary, often termed a trapped surface, but not necessarily that anything physical diverges there at that boundary. Though a singularity will result in geodesic incompleteness, the assertion that geodesic incompleteness necessarily implies the existence of a singularity is really just a conjecture.
For more details, read the paper. It’s technical, of course, but well written and actually not all difficult to understand.
It’s Friday afternoon but before I collapse, exhausted, into the arms of the weekend I’ll take the opportunity to announce yet another new paper at the Open Journal of Astrophysics.
The latest paper is the 45th so far in Volume 6 (2023) – just five to go for a half-century – and it’s the 110th altogether. This one was actually published on Tuesday November 14th.
The title is “Marginalised Normal Regression: Unbiased curve fitting in the presence of x-errors” and it’s by Deaglan J. Bartlett (Institut d’Astrophysique de Paris, France) and Harry Desmond (Portsmouth, UK). It sounds like a statistical methods paper, and indeed it is, but remember that there’s a very long historical connection between astronomy and the development of statistical methods for data analysis, and this paper tackles a very longstanding issue: how best to fit curves in the presence of noisy data. This paper presents a new method for doing this, together with applications to cosmological and astrophysical data, and accompanying software. It is in the folder marked Instrumentation and Methods for Astrophysics.
Here is the overlay of the paper containing the abstract:
You can click on the image of the overlay to make it larger should you wish to do so. You can find the officially accepted version of the paper on the arXiv here.
Some time ago – was it really over a decade? – I wrote a piece about the optimum size of modules in physics teaching. I was still in the United Kingdom then so my ramblings were based on a framework in which undergraduate students would take 120 credits per year, usually divided into two semesters of 60 credits each. In Cardiff, for instance, most modules were (and still are) 10 credits but some core material was delivered in 20 credit modules. In the case of Sussex, to give a contrasting example, the standard “quantum” of teaching was the 15 credit module. I actually preferred the latter because that would allow the lecturer to go into greater depth, students would be only be studying four modules in a semester (instead of six if the curriculum consisted of 10 credit modules), and there would be fewer examinations. In short, the curriculum would be less “bitty”.
In Maynooth the size of modules is reckoned using the European Credit Transfer System (ECTS) which takes a full year of undergraduate teaching to be 60 credits rather than 120 in the UK, but the conversion between the two is a simple factor of two. In Maynooth the “standard” unit of teaching is 5 credits, with some 10 credit modules thrown in (usually extending over two semesters, e.g. projects). This is similar to the Cardiff system. The exception concerns first-year modules, which are 7.5 credits each because students take four modules in their first year so they have to be 30/4=7.5 credits each. The first year is therefore like the Sussex system. It changes to a five-credit quantum from Year 2 onwards because students do three subjects at that stage.
I find it interesting to compare this with the arrangements here in Barcelona (and elsewhere in Spain). Here the ECTS credit size is used, but the standard module is six credits, not five, and year-long projects here are 12 credits rather than 10. The effect of this is that students generally study five modules at a time (or four plus a project). To add to the fun there are also some 9 credit modules, so a semester could be made up of combinations of 6-credit and 9-credit chunks as long as the total adds up to 30.
Anyway, the main point of all this is to illustrate the joy of the sexagesimal system which derives from the fact that 60, being a superior composite number, has so many integer divisors: 2, 3, 4, 5, 6, 12, 15, 20, and 30. The Babylonians knew a thing or two!
I got back to Maynooth last night after a pleasantly uneventful train journey. Just for the record both outward and return trips were perfectly on schedule. In fact it has been a very pleasant couple of days. Congratulations to the organizers for running the meeting so well and to all the speakers for delivering such an interesting programme. Next year’s INAM will be in Galway. I’m looking forward to it already!
Anyway, now I’m back I should mentioned that the 2023 Leaving Certificate results were released to students yesterday; the first round of CAO offers will go out on Wednesday 30th. Soon after that we will find out how many students we’ll have next year. Student enrolment begins on 11th September; Orientation Week for new students starts on Monday 18th September; and lectures start the following Monday (25th). I am on sabbatical for a year from next Friday (1st September) so I won’t be teaching the new students, but I know they’ll be in capable hands.
There’s a lot of discussion – much of it poorly informed – in the media about grade inflation in the Leaving Certificate (e.g. here). This happens every year (as it does with A-levels in the UK), and its very sad that people use this occasion to publicly disparage the accomplishments of students. The students can only take the examinations that are put in front of them. Any problems with the system are not their fault at all.
This year the problem stems from a decision by Minister for Education Norma Foley to impose a condition that overall grades this year would not be lower than last year. This has led to the deployment of scaling which has resulted in an uplift of around 8%. The Higher Mathematics Leaving Certificate results also benefitted from an alteration of the marking scheme because one of the papers was deemed to be too hard. Despite this, the number of students receiving the top grade of H1 fell this year from 18% to under 11%. One might argue that this disadvantages students applying to courses that actually require mathematics compared to those that don’t.
There seems to be a widespread misunderstanding that the CAO points required for a course is somehow a measure of the level of difficulty of that course. In most cases this is not the case: having a high points threshold is basically just a way of controlling the number of students allowed in. I find the connection that has been made between grade inflation and drop-out rates extremely unconvincing. High drop-out rates in recent years are probably dominated by the pandemic, housing crisis and cost of living increases, leading to many students struggling to study effectively.
During the pandemic years, grades were inflated by including coursework rather than examinations, a change enforced because of public health restrictions. The main argument for deliberate grade inflation this year was to prevent this year’s LC students being disadvantaged with respect to last year’s. It doesn’t seem to have occurred to the Government that the same argument could be used next year, and indeed forever. Fairly typically for a politician, kicking the can down the road for the next government to deal with seems to be strategy.
As a final thought, I find myself wondering what will happen to admissions at Maynooth this year. Will the decision by The Management to scrap the promised Student Centre have a big effect? And what about the further reputational harm caused by the recent furore over the Governing Authority? I suppose we’ll find out next week!
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In the Dark 