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Friday, 3 July 2020

The Erdős primitive set conjecture

A set of integers greater than 1 is primitive if no number in the set divides another. Erdős proved in 1935 that the series of $1/(n \log n)$ for $n$ running over a primitive set A is universally bounded over all choices of A. In 1988 he conjectured that the universal bound is attained for the set of prime numbers. In this research case study, Oxford's Jared Duker Lichtman describes recent progress towards this problem:

"On a basic level, number theory is the study of whole numbers, i.e., the integers $\mathbb{Z}$. Maturing over the years, the field has expanded beyond individual numbers to study sets of integers, viewed as unified objects with special properties.

A set of integers $A\subset \mathbb{Z}_{>1}$ is primitive if no number in $A$ divides another. For example, the integers in a dyadic interval $(x,2x]$ form a primitive set. Similarly the set of primes is primitive, along with the set $\mathbb{N}_k$ of numbers with exactly $k$ prime factors (with multiplicity), for each $k\ge1$. Another example is the set of perfect numbers $\{6,28,496,..\}$ (i.e. those equal to the sum of their proper divisors), which has fascinated mathematicians since antiquity.

After Euler's famous proof of the infinitude of primes, we know $\sum_p 1/p$ diverges, albeit "just barely" with \begin{align*} \sum_{p\le x}\frac{1}{p} \sim \log\log x. \end{align*} On the other hand, we know $\sum_p 1/p\log p$ converges (again "just barely"). Using the notation \begin{align*} f(A) := \sum_{n\in A}\frac{1}{n\log n}, \end{align*} we have $f(\mathbb{N}_1)<\infty$. In 1935 Erdős generalized this result considerably, proving $f(A) <\infty$ uniformly for all primitive sets $A$. In 1988 he further conjectured the maximum is attained by the primes $\mathbb N_1$:

Conjecture 1. $f(A) \leq f (\mathbb{N}_1)$ for any primitive $A$.

Note we may compute $f(\mathbb N_1) = \sum_p 1/p\log p \approx 1.6366$.

Since 1993 the best bound has been $f(A) < 1.84$, due to Erdős and Zhang. Recently, Carl Pomerance and I improved the bound to the following:

Theorem 1. $f (A) < e^\gamma \approx 1.78$ for any primitive A, where $\gamma$ is the Euler-Mascheroni constant.

Further $f(A) < f(\mathbb{N}_1)+0.000003$ if $2\in A$.

One fruitful approach towards the Erdős conjecture is to split up $A$ according to the smallest prime factor, i.e., for each prime $q$ we define $$A_q = \{ n \text{ in } A : n \text{ has smallest prime factor } q\}.$$

We say $q$ is Erdős strong if $f(A_q)\le f(q)$ for all primitive $A$. Conjecture 1 would follow if every prime is Erdős strong, since then $f(A) = \sum_q f(A_q) \le f(\mathbb{N}_1)$.

Unfortunately, we don't know whether $q=2$ is Erdős strong, but we showed that the first hundred million odd primes are all Erdős strong. And remarkably, assuming the Riemann hypothesis, over $99.999973\%$ of primes are Erdős strong.

Primitive from perfection

In modern notation, a number $n$ is perfect if $\sigma(n)=2n$ where $\sigma(n) = \sum_{d\mid n}d$ is the full sum-of-divisors function. Similarly $n$ is called deficient if $\sigma(n)/n<2$ (abundant if $>2$).

Since $\sigma(n)/n$ is multiplicative and $>1$, we see that perfect numbers form a primitive set, along with the subset of non-deficient numbers $n$ whose divisors $d\mid n$ are all deficient.

It is a classical theorem that non-deficient numbers have a well-defined, positive asymptotic density. This was originally proven with heavy analytic machinery, but Erdős found an elementary proof by using primitive non-deficient numbers (this density is now known $\approx 24.76\%$). His proof led him to introduce the notion of primitive sets and study them for their own sake.

This typified Erdős's penchant for proving major theorems by elementary methods.

A related conjecture of Banks & Martin

Recall $\mathbb{N}_k$ denotes the set of numbers with $k$ prime factors. In 1993, Zhang proved $f (\mathbb{N}_k) < f (\mathbb{N}_{1})$, which inspired Banks and Martin to predict the following:

Conjecture 2. $f (\mathbb{N}_k) < f (\mathbb{N}_{k-1})$ for each $k > 1$.

They further predicted that, for a set of primes $\mathcal Q$, \begin{align*} f\big(\mathbb{N}_k(\mathcal Q)\big) < f\big(\mathbb{N}_{k-1}(\mathcal Q)\big)\qquad\textrm{for each } k>1 \end{align*} where $A(\mathcal Q)$ denotes the numbers in $A$ composed of primes in $\mathcal Q$. Banks and Martin managed to prove this conjecture in the special case of sufficiently "sparse'' subsets $\mathcal Q$ of primes.

This result, along with Conjectures 1 & 2, illustrates the general view that $f(A)$ reflects the prime factorizations of $n\in A$ in a quite rigid way.

Beautiful though this vision of $f$ may be, it appears reality is more complicated. Recently I precisely computed the sums $f(\mathbb{N}_k)$ (see Figure 1 below) and obtained a surprising disproof of Conjecture 2!

Theorem 2. $ f( \mathbb{N}_k) > f(\mathbb{N}_6) $ for each $k\neq 6$.

Figure 1. Plot of $f(\mathbb{N}_k)$ for $k=1,2,..,10$.

I also proved $\lim_{k\to\infty} f(\mathbb{N}_k) = 1$, confirming a trend observed in the data. However, much about this data remains conjectural. For instance, the sequence $\{f(\mathbb{N}_k)\}_{k\ge6}$ appears to increase monotonically (to 1), and the rate of convergence appears to be exponential $O(2^{-k})$, while only $O(k^{\varepsilon-1/2})$ is known. Similar phenomena seem to occur when experimenting with subsets $A\subset \mathbb{N}_k$ of e.g. even, odd, and squarefree numbers.

I hope this note illustrates Erdős' conjecture spawning new lines of inquiry. For example, researchers are now studying variants of the problem in function fields $\mathbb{F}_q[x]$. Also, in forthcoming work with Chan and Pomerance, we manage to prove Conjecture 1 for 2-primitive sets $A$, i.e., no number in $A$ divides the product of 2 others.

The full Erdős conjecture has remained elusive, but working towards it has led to interesting developments. In the words of Piet Hein:

Problems worthy of attack prove their worth by fighting back.

Friday, 3 July 2020

Oxford Mathematics Online Exhibition 2020

Alongside the mathematics, the Andrew Wiles Building, home to Oxford Mathematics, has always been a venue for art, whether on canvas, sculpture, photography or even embedded in the maths itself.

However, lockdown has proved especially challenging for the creative arts with venues shut. Many have turned to online exhibitions and we felt that not only should we do the same but by so doing we could stress the connection between art and science and how both are descriptions of our world.

So we invited our locked down mathematicians to explore their mathematical creativity in a variety of media. A panel reviewed all the submissions, taking into account both the creative aspects and the mathematical component, alongside the description communicating the link.

So here is the first Oxford Mathematics Online Exhibition.

 

 

 

 

 

 

 

     

Wednesday, 1 July 2020

Andrea Mondino awarded a Whitehead Prize by the London Mathematical Society

Oxford Mathematician Andrea Mondino has been awarded a Whitehead Prize by the London Mathematical Society (LMS) in recognition of his contributions to geometric analysis in differential and metric settings and in particular for his central part in the development of the theory of metric measure spaces with Ricci curvature lower bounds.

Andrea works at the interface between Analysis and Geometry. More precisely he studies problems arising from (differential and metric) geometry by using analytic techniques such as optimal transport, functional analysis, partial differential equations, calculus of variations, gradient flows, nonlinear analysis and geometric measure theory. Although the emphasis of his work is primarily theoretical, the topics and the techniques have profound links with applications to natural sciences (mainly physics and biology) and economics.

Friday, 26 June 2020

Ulrike Tillmann announced as President Designate of the London Mathematical Society (LMS)

Oxford Mathematician Ulrike Tillmann has been announced as President Designate of the London Mathematical Society (LMS). 

Ulrike's research interests include Riemann surfaces and the homology of their moduli spaces. Her work on the moduli spaces of Riemann surfaces and manifolds of higher dimensions has been inspired by problems in quantum physics and string theory. More recently her work has broadened into areas of data science.

Ulrike is also well-known for her many contributions to the broader mathematical community, serving on a range of scientific boards including membership of the Council of the Royal Society. She will take over from the current LMS President (and Oxford Mathematician) Jon Keating in November 2021.

Saturday, 20 June 2020

Hawking Points in the Cosmic Microwave Background - a challenge to the concept of Inflation

For thirty years Oxford Mathematician Roger Penrose has challenged one of the key planks of Cosmology, namely the concept of Inflation, now over 40 years old, according to which our universe expanded at an enormous rate immediately after the Big Bang. Instead, fifteen years ago, Penrose proposed a counter-concept of Conformal Cyclic Cosmology by which Inflation is moved to before the Big Bang and which introduces the idea of preceding aeons. The concept has been disputed by most physicists, but Roger and colleagues believe that new evidence has come to light which requires closer inspection and argument - the research is published today in the Monthly Notices of the Royal Astronomical Society (MNRAS). 

Recent analysis of the Cosmic Microwave Background (CMB) by Roger, Daniel An, Krzysztof Meissner and Pawel Nurowski has revealed, both in the Planck and WMAP satellite data (at 99.98% confidence), a powerful signal that had never been noticed previously, namely numerous circular spots $\sim 8$ times the diameter of the full moon. The brightest six (Figure 1) are $\sim 30$ times the average CMB temperature variations seen at precisely the same locations in the Planck and WMAP data. These spots were overlooked previously owing to a belief that the very early exponentially expanding inflationary phase of standard cosmology should have obliterated any such features.

 

 

                                                                  

 

(Figure 1: CMB sky, marking 6 most prominent raised-temperature circular spots, found both in Planck and WMAP data; argued to be results of Hawking radiation from supermassive black holes in a previous aeon)

There are alternative universe models without inflation, but most encounter fundamental difficulties in not accounting for CMB features normally explained by inflation. However, Conformal Cyclic Cosmology (CCC) does so, by displacing 'inflation' to before the Big Bang - as the exponentially expanding remote future of an earlier cosmic aeon. This 'aeon' is a universe epoch, resembling what we currently perceive to be the entire history (without inflation) of our Universe. In CCC, there is an infinite succession of such aeons, each having a big-bang origin which is the conformal continuation or the exponentially expanding remote future of the preceding aeon (Figure 2). Conformal geometry allows for stretching or squashing of the metric structure, and is the geometry respected by a physics without mass (such as Maxwell's electromagnetism). This applies both to the remote future and big bang of each aeon, so the matching of aeon to aeon makes geometrical sense - and also physical sense because the conformal squashing of the cold low-density remote future matches the conformal stretching of the hot dense big bang of the subsequent aeon.

                                                                    

(Figure 2: Cartoon of conformal cyclic cosmology: each aeon's big bang arises from the conformally compressed remote future of its preceding aeon)

The exceptions to this smooth conformal matching are the supermassive black holes in an aeon's remote future, which would each have almost completely swallowed its surrounding galactic cluster, before eventually evaporating away entirely into Hawking radiation (after perhaps $10^{106}$ years). However, by conformal squashing, all this radiated energy comes through into the succeeding aeon at a single 'Hawking point.' The emerging photons then scatter within an expanding region, but do not become free until $\sim 380000$ years later, when finally appearing in the CMB of that subsequent aeon. This spread-out region would look to us like a disc ∼ 4° across, i.e. $\sim 8$ times the diameter of our full moon, an effect that we appear to be actually seeing in our own CMB sky.

Roger talks about his work in this November 2018 Oxford Mathematics Public Lecture.

Wednesday, 17 June 2020

Strange exponents in the "birthday paradox" for divisors

Ben Green and collaborators discover that the well-known "birthday paradox" has its equivalent in the divisors of a typical integer.

"The well-known "birthday paradox'' states that if you have 23 or more people in a room - something difficult to achieve nowadays without a very large room - then the chances are better than 50:50 that some pair of them will share a birthday. If we could have a party of 70 or more people, the chance of this happening rises to 99.9 percent.

It turns out that there is a similar phenomenon for the divisors of a "typical'' integer. Let $X$ be large, select an integer $n$ at random from the numbers $\{1,\dots, X\}$, and write down its divisors. These are distinct numbers, so no two of them will be the same, but it turns out that with high probability some pair of them will be close together. (The same caveat is necessary in the birthday problem if you look at the precise time the people in the room were born, rather than just the day.)

What do we mean by "close together''? One interpretation is that there are two divisors $d$ and $d'$ of $n$ lying within a factor two of one another, say $d < d' < 2d$.

Whilst the analysis of the birthday paradox is quite elementary, this turns out to be a very difficult result to prove. In fact, the statement that a random integer almost surely has two distinct divisors within a factor of two of one another is a celebrated result of Maier and Tenenbaum, published in 1985. It had been an open question of Erdős for over thirty years when they solved it.

It turns out that the divisors of a random integer are much more bunched together than the birthdays of random people in a room. Recently, in joint work with Kevin Ford and Dimitris Koukoulopoulos, I investigated just how many near coincidences there must be. Given a number $n$, the Hooley $\Delta$-function $\Delta(n)$ is defined to be the maximum number of divisors of $n$, all within a factor of two of one another. The result of Maier and Tenenbaum is that $\Delta(n) \geq 2$ with high probability.

We obtained a new lower bound for $\Delta(n)$, valid for almost every integer $n$, and assembled a good deal of evidence (but so far no proof) that it is also the correct upper bound. This bound has one of the most complicated exponents I have ever seen in a number theory problem: $\Delta(n) \geq (\log \log n)^{\eta}$, where $\eta \approx 0.35332277270132346711$ is defined to be $\frac{\log 2}{\log(2/\rho)}$, where $\rho$ satisfies the equation \[ \frac{1}{1 - \rho/2} = \log 2 + \sum_{j = 1}^{\infty} \frac{1}{2^j} \log \Big(\frac{a_{j+1} + a_j^{\rho}}{a_{j+1} - a_j^{\rho}} \Big),\] where the sequence $a_j$ is defined by $a_1 = 2$, $a_2 = 2 + 2^{\rho}$ and $a_j = a_{j-1}^2 + a_{j-1}^{\rho} - a_{j-2}^{2\rho}$ for $j \geq 3$.

In fact, the definition of $\rho$ is so complicated that it's a nontrivial analysis exercise to confirm that a number satisfying the equations here even exists.

Our paper, which is 88 pages long, takes us to some surprising areas of maths -  we begin by removing most of the number theory from the problem, turning it into a question about Poisson random variables. Then, we convert that into a curious optimisation problem involving measures on the discrete cube in $\mathbb{R}^n$ and their distribution on linear subspaces. To solve this, we use a lot of properties of entropy."

Thursday, 11 June 2020

Changing attitudes towards ancient arithmetic: reconciling mathematics with Egyptology

Oxford Mathematician Christopher Hollings and Oxford Egyptologist Richard Bruce Parkinson explain how our interpretation of Egyptian Mathematics has changed over the past two centuries and what that says about how historians of mathematics approach their subject.

"When Egyptian hieroglyphs were deciphered in the nineteenth century, scholars were finally able to read the millennia-old texts that provided a key to understanding the ancient Egyptian civilisation.  The documents that could now be read dealt with a range of subjects, from administrative, legal, and religious matters, to medicine – and mathematics.  Examples of numerals may be found in many surviving accounts papyri, which record numbers of goods and workers, etc., but only a very small collection of sources provide an insight into the mathematics that ancient Egyptian scribes actually knew.

One of the most complete surviving sources is the Rhind Mathematical Papyrus, dating from c. 1537 BCE, and now held in the British Museum (P. BM EA 10057 and P. BM EA 10058 - see image above © the Trustees of the British Museum).  The papyrus consists of over 80 arithmetical and geometrical problems and solutions, ranging from the distribution of rations among workers, to the calculation of areas (such as of triangles) and volumes (of cylinders, for example).  The papyrus was probably placed in a tomb, as part of a display of the tomb-owner’s social and cultural status.

Within a couple of decades of the discovery of the Rhind Papyrus in Luxor in 1858, it had provided a much more complete picture of the nature of Egyptian mathematics than had previously been available.  Several translations of parts of the Rhind Papyrus appeared in print, but perhaps the most celebrated was that published in 1923 by the Egyptologist Thomas Eric Peet (1882–1934), then Professor of Egyptology at the University of Liverpool.  Peet was ideally placed to produce this new edition, having studied both mathematics and classics in Oxford.  His edition was much praised by both mathematicians and Egyptologists, and reignited an academic interest in Egyptian mathematics, which had stagnated slightly in the decades prior to Peet’s publication.  The subject was now to be studied on its own terms, rather than standing in the shadow of ancient Greek mathematics, which had traditionally been taken as the start of ‘true’ mathematics.

One scholar who was particularly inspired by Peet’s work was Otto Neugebauer (1899–1990), a young student in Göttingen.  Neugebauer had already studied mathematics and physics, but was now cultivating an interest in ancient science.  By the end of 1926, he had completed a doctoral dissertation on the principles of Egyptian fraction reckoning (as reflected in the Rhind Papyrus).  Whilst completing the dissertation, he was in correspondence with Peet, and two of his letters to Peet have recently come to light in Oxford in the Peet Memorial Library of The Queen’s College – Peet ended his career in Oxford (as a Fellow of Queen’s), with the result that some of his personal papers and books survive there in archives and libraries.

The letters shed light on the way in which ancient Egyptian mathematics was being re-evaluated in the 1920s.  In particular, they show up a contrast between the attitudes of two scholars who approached the subject from different directions.  Both were competent mathematicians and Egyptologists, and yet one (Neugebauer) put the mathematics first, and made general assertions about the nature of ancient Egyptian mathematics that arguably owed more to modern ideas about how mathematics ‘should’ be than to the direct evidence of papyri; the other (Peet) brought Egyptological considerations to the fore, drawing conclusions that were more firmly embedded in a knowledge of the cultural context of surviving sources – where Neugebauer saw lacunae in the historical record as gaps to be filled with educated speculation, Peet tip-toed cautiously around them, confining his commentary largely to what was clearly and unequivocally present in the original texts.

Perhaps partly because of his early death, Peet’s influence on the later study of ancient Egyptian mathematics was minimal.  Neugebauer, on the other hand, went on to become one of the most prominent historians of mathematics of the twentieth century; his work helped to turn the history of mathematics into an academic discipline.  It was not until the 1970s that Peet’s culturally-sensitive approach to the history of mathematics began to gain ground once again.  And yet the essential tension between the attitudes of Peet and Neugebauer can still be found in the study of ancient mathematics today."

Christopher Hollings is Departmental Lecturer in Mathematics and its History, and Clifford Norton Senior Research Fellow in the History of Mathematics at The Queen’s College.  Richard Bruce Parkinson is Professor of Egyptology and a Fellow of The Queen’s College.  Their paper on the correspondence between Peet and Neugebauer, ‘Two letters from Otto Neugebauer to Thomas Eric Peet on ancient Egyptian mathematics’ (Historia Mathematica, 2020) can be found here.

Tuesday, 9 June 2020

Hard problems and security proofs in a Quantum World

In modern Cryptography, the security of every cryptosystem is required to be formally proven. Most of the time, such formal proof is by contradiction: it shows that there cannot exist an adversary that breaks a specific cryptosystem, because otherwise the adversary would be able to solve a hard mathematical problem, i.e. a problem that needs an unfeasible amount of time (dozens of years) to be concretely solved, even with huge computational resources.

To be more precise, the security proof demonstrates that if an adversary can break the cryptosystem in an amount of time A, then they can solve the hard mathematical problem in an amount of time A+B. The size of B plays a fundamental role in the significance of the proof. If B is small, an adversary cannot break the scheme in a feasible amount of time A (e.g. weeks, months, or even a few years) otherwise also A+B would be a feasible amount of time, which would contradict the assumption that the mathematical problem is hard. However, if B is huge, then A+B might be an unfeasible amount of time even if A is feasible. In this case the above contradiction cannot be used and the effectiveness of the security proof is undermined. 

Therefore to have a secure system, B must be small and the mathematical problem must be hard, so that A (the time to break the scheme) can only be an unfeasible amount of time. This explains why cryptographers are always in search of mathematical problems that are hard to solve.

Classical Elliptic Curve Cryptography
In the last decades, cryptographers have found a precious ally in elliptic curves and the discrete logarithm problem, i.e the problem of determining the multiplicative factor of a multiple of a suitable elliptic-curve point. In 1994, P. Shor published a quantum algorithm capable of easily solving the discrete logarithm problem.

As a consequence, when in recent years the possibility of constructing scalable fault-tolerant quantum computers has become concrete, the marriage between elliptic curves and Cryptography seemed close to an end. 

Isogeny-based Cryptography
The proposal of using isogenies (special maps) between elliptic curves has brought elliptic curves to the attention of cryptographers once again, this time to construct schemes supposed to be secure even against quantum adversaries. Instead of considering the points of a given elliptic curve, as in classical Elliptic Curve Cryptography, Isogeny-based Cryptography works with graphs whose vertices are curves and edges are isogenies. The hard problem on which the isogeny-based cryptosystems rely is that of finding the random path connecting two curves, knowing only the starting and arrival point.   

Isogeny-based signatures and CSI-FiSh
While isogenies offer simple and efficient solutions to encryption schemes and key-exchange protocols, they turned out to be rather elusive in constructing signature schemes, which are ubiquitous in real life applications (they are, for example, used to securely distribute software updates). The first practical isogeny-based digital signature scheme, CSI-FiSh, was proposed one year ago. However, its security proof suffers the issue of having a huge B - a problem, as explained above. 

Lossy CSI-FiSh
In a joint work with his Oxford Mathematics colleague Ali El Kaafarani and with Dr. Shuichi Katsumata from AIST in Japan, Federico Pintore proposes a variant of CSI-FiSh, named Lossy CSI-FiSh, having a stronger security proof (B is approximately equal to A) and almost the same efficiency as CSI-FiSh. The work was presented at the conference Public Key Cryptography 2020 (planned for Edinburgh at the beginning of May, but then converted into a virtual conference). 

Wednesday, 3 June 2020

How to design the perfect face mask – the effect of compressibility on filters

How do we design face masks that efficiently remove contaminants while ensuring that we can still breathe easily? One complicating factor with this question is the fact that the properties of the material that we start off with for our face mask can be very different when in use. A key example is seen when you stretch the mask around your face to put it on. In doing so, you also stretch the pores, i.e., the holes in the material that allow the air to pass through. This means that some contaminants that would ordinarily have been trapped by the mask can now pass through the material. Another example is the fact that, when you inhale, the material will compress due to the suction force you apply. This process has the opposite effect, with the pores shrinking. While pore shrinkage is desirable for efficient trapping of contaminants it also makes it more difficult to breathe through the mask. Crucially, in both these cases, when the filter is being used it may work less well than expected.

Oxford Mathematicians Ian Griffiths, Colin Please, Jakub Köry & Armin Krupp have tackled this problem by studying how pressure differences in the air, such as those arising due to inhalation, change the properties of a filter material. They use classical models of linear elasticity for deformation of the filter material and Darcy flow for movement of the air, and then couple these by accounting for how stresses in the material change the pore size, as described in the examples above, and hence the flow. 

The resulting mathematical model allowed the team to explore and quantify how the filter behaves in operation. The first observation they found was that, upon inhalation, the material does not compress uniformly: the material that is closest to our face compresses the most and the part of the mask that is furthest away from our face and exposed to the air compresses the least (see figure below). This non-uniform material compression means that the filter is not behaving optimally: it is neither removing the contaminants in the desired way nor allowing us to breathe as easily as possible. Furthermore, if we inhale too hard then eventually the material next to our face will be compressed so much that the pores that allow the air to flow through will completely close up. In this case, the material will ‘shut down’ and no air at all can flow. 

These observations lead to posing the question of how we could design a filter material that has the properties we desire when in operation. This is a question that has a much broader reach than the example of face masks presented here: every filter that is used, from the household filters that you find in your vacuum to industrial-scale filters that provide clean air in a building, require certain filtration properties when in operation rather than when the device is not in use and the material is in its rest state.  

The question of how to design a filter that has particular properties when in use is an inverse problem: we need to design the filter material in its rest state in such a way as to achieve a specified property when in operation. The mathematical model the team have developed allows us to tackle such a problem. Two scenarios are examined. First, the case where we want to design a filter material that has uniform permeability when in operation. This avoids the scenario shown in the figure where compression causes the permeability of the material to be lower the closer we get to our face. A filter with uniform permeability is effective at removing contaminants throughout its depth and also makes it as easy as possible to breathe.  Second, the case where we want a filter design that provides the maximum flow rate before shutting down. Such filters are desirable in industrial settings such as ventilation systems, where the key challenge is to maximise the amount of air that is filtered. 

This model is now being developed to explain how such filters perform and evolve as they remove contaminants. The resulting models can then be used to answer important questions concerning the design of efficient filters in the future. 

                                                      

Figure: as air passes through the pores between the filter fibres that constitute a face mask, the material compresses. The amount of compression increases the closer we move to the face.

Tuesday, 26 May 2020

Student Life in Lockdown - Oxford Mathematics Undergraduates reflect

All over the world millions of students are experiencing a different life, one that is arguably the antithesis of most students' lives - an online, socially distant life. So what is that life like? How is it affecting how they learn, how they interact, most importantly, how they feel?

The best answer is to ask them all (and then ask again the next day maybe...). But we don't have that track and tracing ability here in Oxford Mathematics, so we have settled for asking a few of our Undergraduate students for their thoughts. They aren't representative of anyone except themselves but we suspect some of the themes may be familiar and even a little helpful.

What do they say? Well, you can read their thoughts and feelings in their own words below - they are a good read. However, some things stand out.

Teaching
Online lectures, classes and tutorials are working and have their advantages. Online lectures and classes work and not just because you don't have to drag yourself out of bed for 9am. Students can watch when suits them, can pause the lecture to recap, and join the online discussion. In addition, tutors are clearly doing all they can and receive much credit for it, though concentrating in online tutorials is a challenge. 

Studying
Self-discipline is obviously key, more so when you are detached from the usual learning environment. Again, the overall feedback is positive, notably the support from tutors, and there is no sense of being academically disadvantaged. But it is ALL online. There are no books, no libraries, no friends at the next table, no welcome interruptions. Every day can feel the same. 

Exams
Students are less comfortable at the thought of online exams. Can they guarantee no interruption from the world around them, human and mechanical? Might other students not play by the rules? And those who have had their exams cancelled are not getting the usual preparation for the bigger exams down the line (though some first year students whose exams have been cancelled will have College exams in their place to give them some sense of closure).

Social life
There is much happening online, and in some cases it has encouraged closer bonds (among families as well as fellow students), notably among those who have stayed in Oxford. But it isn't the same. Social life glues academic life together during a typical term. That is harder online.

And some friends will not see each other again, at least not in Oxford. They are deprived of their final term. Or their first summer term. These things are personal and precious.

But as the writing shows, there is also time to reflect, to try and understand what is going on and where it fits in. A normal student term isn't necessarily the best time to do that. But that's not to say everyone isn't longing to return. They miss each other and we miss them.

Jess, 3rd Year
Being at home during term almost feels like I’m back on study leave from school as I was during GCSE and A-Levels (except it’s noisier and more distracting now because everyone here is working from home). I had envisioned Trinity (Summer term) to involve days spent in the Maths Institute and various libraries revising for exams, spending time with my friends for whom this was their last term at Oxford, and then finally celebrating the end of exams at the end of term. However, instead I am stuck with only one study space (my bedroom), and all social events swapped for Zoom calls.

In terms of the ‘online term’, I haven’t had any new material to cover (as would have been the case anyway), only optional revision sessions to attend. Despite the initial worry about having to get all of my revision questions answered over email and Microsoft Teams, my concerns were soon relieved when I realised how much effort all of my lecturers and tutors have gone to in order to maintain a great standard of teaching and help us revise as well as they can. Some tutors have even gone beyond my expectations (I had a PDF of written solutions to my queries sent to me in Teams!). One thing I particularly like about the online classes, and am finding very useful, is that the notes made by my the tutor are sent to the students afterwards, meaning we can spend the class trying to understand the material and asking further questions, instead of rapidly trying to scribble workings down before they get wiped off the whiteboard!

Although the revision resources are really good, they haven’t quite removed the worry about the actual exams. One of my biggest concerns is that there’s no way to ensure the quiet exam environment we’d get in Oxford; you can’t shut off the roads beside your house, or ask all of your neighbours to be silent for 3 hours every time you have an exam! Furthermore, this problem will continue to worsen week on week now that lockdown is being relaxed, causing concern that having an exam in July will have adverse consequences, rather than benefits due to increased revision time. In addition, since the exam period has now been extended to a five-week period instead of each student having a fairly similar spread of exams like usual, it is now the case that while the majority of maths students are benefiting from nicely spread out exams, others (like myself) have unfortunately ended up with a stressful situation of four consecutive days of exams!

Matt, 1st Year
Trinity (Summer term) at home hasn't changed my education very much; For better or worse, it's gotten rid of the early mornings that come with going to lectures, as now I can watch them at any time. This allows me to streamline my workflow a lot more, but unfortunately it's not the Oxford experience I had in my first two terms. There's nothing like living with your best friends in college, the occasional formal dinner, or just walking through the Porters' Lodge (college entrance and home to post and porters, college staff who are the first port of call for just about anything). Me and my friends have been watching movies together and staging quiz nights, but I can't help but feel we've been pulled apart just when we were really starting to know each other. It's sad I might have to wait until next Trinity (Summer term) to go punting, or see Oxford when it's not gloomy, but on the other hand, I'm incredibly lucky that this is the worst consequence of Coronavirus I've experienced so far. It's also nice to spend more time with my dogs and family.

Tutorials are mostly unchanged, but it takes a lot of self control to stay focused when your tutor can't see if you're not really paying attention. I'm a bit relieved exams have been cancelled, as they normally cause me quite a lot of stress, even when I don't think I'm in danger of failing. I feel I'm really lucky to not be in the higher years - I'd loathe taking exams over the internet, and I'm not sure how being in such a different situation than expected would affect my performance. Some people are saying that first year is a good trial run for the exam period, so when the exams we're taking really matter we're familiar with their format. I'm not particularly worried about this as ultimately we're all in the same situation, but I hope this doesn't cause undue stress for anyone, that would have been avoided if we had proper exams this year. All I can hope now is that we're back by next Michaelmas (October term)! 

Disha, 2nd Year
When life gives you lemons, bake a lemon drizzle cake because quite frankly, why not?

I love baking, but baking hasn’t always loved me. Despite my best efforts, somehow my cakes always turned out inedible and my biscuits always unbreakable. Yet I’d keep trying, failing batch after batch much to the despair of my sister who’d still eat the burnt cookies to make me feel better. There were times where I just wanted to give up, what was the point anyway but somehow I’d always be enticed back for one more attempt. Feeling frustrated, I decided to call it quits until I found this food blogger online who seemed to share the same passion as me and I decided to try again. If you couldn’t tell already I decided to take up baking again and thank goodness I did.

Chocolate cakes, blueberry muffins, shortbreads, carrot cakes, banana breads (all vegan might I add), you name it, I’ve probably baked it in the last month. The satisfaction of finally being able to succeed after all those efforts was indescribable and I’ve already got my retirement plan all set out.

On the other hand, I have discovered a negative correlation between the amount of sunshine and my productivity. It’s been a lot harder trying to work when everyone else seems to be enjoying the sun and there are no libraries to work in but this has given me an opportunity to work on my self discipline and focus on the task available. I’ve found working with my sister to be very useful as we make a make-shift library on the kitchen table and repetitively shush each other. The thing that I’ve found has worked is the flexibility with lectures. I can now watch my 9am lecture at 9pm and pause, rewind and change the pace whenever I want. This means I no longer need to copy everything out and I can stop and think clearly every now and then when the proof seems a bit too tedious. 

Social life has actually improved, I’m actively seeking out my friends to have long talks with and I’m no longer restrained by specific lecture times. I’ve learnt to appreciate the little things a lot more and look forward to seeing them again soon. In all honesty, last term it felt like I was in a rush all the time and I just didn’t look up from the hustle and bustle of the everyday student life - lectures, worksheets, sleep and repeat was what my Hilary term (January term) had finally reached and I’m glad I’ve had this opportunity to make better use of family time, think about my life and goals and have the chance to finally breathe. I feel a lot calmer at home as a result.

My baking skills have now flourished - in between every problem sheet I’ve added a coffee cake or some muffins to the mix. My endeavours have been received very well by my sister who finally doesn’t have to eat the burnt mess I used to make. One day, after another slice of cake, my youngest sister (who’s six years old) asked me “Why are you doing maths Disha when you could be a baker, now everyone loves your cakes” and it got me wondering why was I doing maths. Surely baking felt more enjoyable than the million problem sheets I had on my desk which were just sitting gathering dust.

But then it hit me: why had I started in the first place? I decided to take a walk - don’t worry I kept a two metre distance - and I reached my answer. Maths is just beautiful, as a friend once told me. Exploring a new topic is like navigating your way through a dark room, you feel your way around, you trip and fall a couple of times (maybe more if you’re as clumsy as me) but then you finally hit the light switch, switch it on and boom, everything just makes sense now - there are still plenty of rooms left to discover but, hey, that’s for another day. Baking probably wouldn’t have been as fun if I’d gotten it all right the first time, it was the fact that I had to keep trying and trying and trying before I was able to make a successful cake that made the cake even sweeter. That’s the same with the problem sheets, the moment when you think you just can’t do it and you hit that eureka moment is what keeps inspiring me to keep going on. And yes while working on one problem I may still have a million and one problems left but that shouldn’t take me away from the problem I’m focusing on. Forgive the cliche but it's all about the climb.

Even in these unprecedented times, I’ve got something that truly makes me happy and gives me the hope to go on and the stability to stay grounded. I think I’ve changed very much this term, I’ve learnt to look at the positives a lot more. Yes I can’t go out and hang out with my friends but I’ve now got the time to take this new topic I’m learning and truly get to grips with it… and bake another cake! Even if I still haven’t switched on the lights in all the rooms, that's okay because #saveelectricity #savetheplanet

Beth 3rd Year
This term has certainly been quite strange, but not always in bad ways. I’ve enjoyed spending more time at home with my family (and not having to worry about cooking food every night)! 

The move to online classes has been convenient in some ways, for instance it makes it easy to ask questions during the class using the chat box feature. However, I do miss the Maths Institute building and being able to study without always using screens.

Whilst it’s a shame not to see my friends at Oxford, I’ve still kept in contact with them and I’ve also done video call revision sessions with students on my courses which has been a fun way to schedule in social events. My tutors also organised a tea party on Zoom as a chance for the mathematicians at my college to catch up with each other and ask any questions we had, which was really fun.

It has certainly been a surreal term and I’m looking forward to going back to college - whenever that may be!

Josh, 2nd Year
In many ways life in lockdown is just the same as life as normal. Well, not Oxford term normal but Oxford vacation perhaps. But maybe Oxford term normal too. I mean it is completely different but it’s also completely familiar. I’ve had plenty of days in term time where social activity is written off and I condemn myself to my desk facing the wall in my room. Days where the only social interaction is the little chat with my friends over dinner in Hall. Or in the vacation when I’ve realised I really do need to do some more preparation for these Collections (College exams) and I spend the day cramming theorems instead of seeing my friends in the pub.

I guess the difference is that now the days like that are everyday. Well, not quite like those days because I’m not working all day. At first it felt like the Oxford workload was really a bit more manageable than it had always seemed, perhaps that’s what happens when you subtract the turbo-charged social scene. But as time is going on it is feeling less and less like that though. It’s a bit harder to work, a bit harder to motivate yourself when everyday is exactly the same. Because that’s the crux of it, isn’t it? Everyday is the same. There’s no uncertainty, no surprises. You spend each day roughly doing what you did the day before. I guess that’s what happens when you can’t see other people, not face to face that is.

But then these days we can see them can’t we. Well, not ‘them’ them, but screen them. Sometimes a video call with friends or even a ‘meeting’ with a society committee can pick my mood right up. It can energise me in all spheres of life whether that’s motivating me to study, finding the strength to smash through a workout or just having a present and engaging interaction with my family. Not always though, in fact sometimes it’s the opposite. Every now and again screen fatigue hits me hard because it seems that life itself is now on screens. If I’m reading lecture notes it’s on screens, doing my philosophy reading it’s screens, catching up with my friends it’s screens, taking a study break watching Netflix, it’s screens. Every now and again I’m forced to take a screen detox to recover and it does work. Exercise works wonders for this and my Government mandated walks to the beach, just as the sun begins to set, have been the highlight of many a day in lockdown. 

I would say that the transition to online learning has been smoother than expected. Well, there have been a few hiccups. It’s much easier to zone out of a tutorial when you don’t have your video on and you’ve been asked to mute your microphone for sound quality reasons.  In fact it’s very easy to. Then there’s the awkward silences where no one quite knows who’s turn it is to speak in the absence of social cues. I do think my College tutors have been exceptional though, they’ve clearly put a real effort into adapting to the situation, especially when they are not by nature the most tech savvy individuals. So far I don’t really feel as though I’m being academically disadvantaged but it is getting trickier week on week. 

My role as JCR President of my Junior Common Room has certainly added to the difficulty. Sometimes it’s the straw that breaks the camel’s back, particularly as I feel a kind of responsibility for how those in my JCR are finding their time in lockdown. But it’s also the cause of many of my happiest moments, whether that’s the virtual open mic nights, the mammoth Zoom quizzes or the silly challenge videos made by those running in the JCR committee elections.

In conclusion, life is the same but very different. Things are by no means awful, but they could certainly be better. Having said that, however the mood of my writing has come across so far I do remain optimistic. Our lives will be affected by this lockdown for a long time to come and many lives will have been damaged along the away. But I think there is something to look to. We will return to our pseudo-normal lives with a gratitude for it’s simple building blocks. Being able to retire to the College bar after a shift in the library, being surrounded by friends when in that library, being able to laugh with friends about your most recent escapades, being able to go on escapades. Perhaps it seems naively optimistic, perhaps it is, but I believe that such a gratitude has the potential to transform a person’s worldview, to instil a deep and enduring peace in life. I do suspect we have to choose it though.

Diana, 4th Year
This is about a lockdown experience in Oxford of a fourth year international Maths student. I decided to stay here over the Easter holiday and Trinity (Summer term) to finish my dissertation and prepare for exams; it seems that Oxford is a more productive environment than home. Another reason is that I have a handful of friends staying here, we are all accommodated in the same annex of Hertford College, so we are allowed to spend time together in communal kitchens and the outside garden. 

So far, this Trinity term has been a completely different experience from a usual Oxford term. There are many things I miss, mostly people who are not here and student societies. Also, I miss my family and the possibility of visiting them for a few weeks as I usually do in the Easter holiday. However, being in Oxford with my friends and boyfriend and doing more inside activities together such as cooking, doing the cleaning up (because scouts (cleaners) are not allowed to come), helping each other with health issues, made Oxford feel like home more than ever. We found a few ways of enjoying time such as using someone’s video projector to simulate cinema experience, getting a shared kayak and rowing in turns on the river next to our accommodation, trying new boardgames. My college has been helpful with providing welfare support, for example I even tried yoga for the first time via an online class they offered, and they put together some very helpful studying advice.

Out of all years (in Maths), fourth year students’ academic experience of Trinity term is the least changed one, because we wouldn’t normally have lectures anyway and writing a dissertation is an independent task, whose format hasn’t been affected. Online revision classes are indeed a pretty different thing, I very much prefer the live experience of going to the Maths Institute and it feels more engaging to participate in person than on video chat. However, I like the idea of sharing solutions after the class, I find it very helpful to see model solutions when checking my work and it feels easier to focus on the specific topics I had troubles with. I think the most different part is going to be sitting exams. I have a few worries regarding the new format, it seems like there is no point in assessing bookwork anymore (students will have their notes with them for online 'open-book' exams, unlike for usual exams), so it might be more difficult to gain those easy points. Also, I fear a little about the possibility of collaboration among students, but I really hope my score will not be affected by any of these.

Overall, I am happy with the decision of spending this lockdown period in Oxford, both from a social and academic point of view.

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