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Sunday, 31 January 2021

What's on Your Mind - 60 seconds inside the mind of an Oxford Mathematician

So what is on the mind of a mathematician, and specifically an Oxford Mathematician? Always their research? Or maybe nothing of the sort?

Our #WhatsonYourMind films take us inside those minds, young and less young, for 60 seconds. There is a lot going on, including the search for beauty, patterns in biology and data, the puzzle of parked cars in London streets, the damage caused by mathematical conferences, and the difficulties of teaching maths to the young.

The first series, a compilation of the first 13 films, is out now (see below).

 

 

 

 

Thursday, 28 January 2021

Ambient and intrinsic geometry of Teichmüller spaces

Oxford Mathematician Vladimir Markovic talks about his research into intrinsic geometry of Teichmüller Spaces.

"Geometry and topology are concerned with the world of shapes and forms, geography and maps, loops and knots. The main objects of study are finite dimensional manifolds. Two dimensional manifolds are called surfaces. Our purpose is to describe their shapes (geometry) and their form (topology). A closed surface is determined by its genus which is the number of holes it wraps around in space. This classification only takes into account the form of the surface (the topology of the surface) and not its shape (the geometry of the surface).

On the other hand, a closed surface $\Sigma_g$ of genus $g$ can be endowed with many Riemannian metrics (which yield different geometries on $\Sigma_{g}$). A suitable quotient of the collection of all such metrics is called the Moduli space $\mathcal{M}_{g}$ of Riemann surfaces (or simply the Moduli space). The Moduli space is itself a Kähler manifold of dimension $3g-3$. The Teichmüller space $\mathcal{T}_{g}$ is the universal covering of $\mathcal{M}_{g}$.

Theorem 1 (Bers). $\mathcal{T}_g$ is biholomorphic to a bounded domain in $\mathbb{C}^{3g-3+n}$.

The Moduli and Teichmüller space are equipped with the Kobayashi $d_K$, Carathéodory $d_C$, and Teichmüller $d_\mathcal{T}$ intrinsic geometries. The goal is to study these metrics and relate them to each other. One of the most important results in Teichmüller theory is the theorem of Royden that $d_{\mathcal{T}}\equiv d_K$. On the other hand, Yau proved that the $d_{\mathcal{T}}$ and $d_C$ are proportional.

Question 1. Does the identity $d_T \equiv d_K \equiv d_C$ hold on $\mathcal{T}_g$?

Isometries of the hyperbolic plane ${\mathbb{H}}^{2}$ into $\mathcal{T}_{g}$ (with respect to $d_{\mathcal{T}}$) are called Teichmüller discs. Every two points in $\mathcal{T}_{g,n}$ lie on the unique Teichmüller disc (complex geodesic). The identity $d_C\equiv d_{\mathcal{T}}$ holds on a Teichmüller disc if and only if $\mathcal{T}_{g}$ holomorphically retracts onto this disc. The following theorem provides a positive answer to Question 1 in an important special case.

Theorem 2 (Kra,McMullen). The Teichmüller space $\mathcal{T}_{g}$ holomorphically retracts onto every Teichmüller disc determined by a holomorphic quadratic differential with even order zeroes.

In general, the answer turned out to be negative.

Theorem 3 (Markovic) $\mathcal{T}_{g}$ does not holomorphically retract onto at least one Teichmüller disc. In particular, $d_C \neq d_{\mathcal{T}}$ on $\mathcal{T}_{g}$, for every $g\geq 2$.

Furthermore:

Theorem 4 $\mathcal{T}_{g}$ does not holomorphically retract onto a random Teichmüller disc.

In the 80s Siu conjectured that $\mathcal{T}_{g}$ is not biholomorphic to a convex domain in $\mathbb{C}^{3g-3}$. The previous theorem yields the solution to the Siu's conjecture.

Theorem 5 (Markovic). $\mathcal{T}_g$ is not biholomorphic to a convex domain in $\mathbb{C}^{3g−3}$

It remains to classify holomorphic retracts of $\mathcal{T}_g$.

Conjecture $\mathcal{T}_g$ holomorphically retracts onto a Teichmüller disc $\tau^{\varphi}$ if and only if all zeroes of the $\varphi$ are of even order."

Image: surfaces of genus zero, one, two, and three.

Bibliography:

I Kra, the Carathéodory metric on abelian Teichmüller disks, Journal Analyse Math. 40 (1981), 129-143 (1982)

V. Markovic, Carathéodory’s metrics on Teichmüller spaces and L-shaped pillowcases, Duke Math. J. 167 (2018), no. 3, 497-535

C. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, Journal Amer. Math. Soc. 16 , no. 4, 857-885 (2003)

We have nearly 200 Oxford Mathematics Case Studies on our website. Take a look.

 

Tuesday, 26 January 2021

Oxford Mathematics Online Public Lecture: Spacetime Singularities - Roger Penrose, Dennis Lehmkuhl and Melvyn Bragg

Oxford Mathematics Online Public Lecture in Partnership with Wadham College celebrating Roger Penrose's Nobel Prize

Spacetime Singularities - Roger Penrose, Dennis Lehmkuhl and Melvyn Bragg
Tuesday 16 February 2021
5.00-6.30pm

Dennis Lehmkuhl: From Schwarzschild’s singularity and Hadamard’s catastrophe to Penrose’s trapped surfaces
Roger Penrose: Spacetime singularities - to be or not to be?
Roger Penrose & Melvyn Bragg: In conversation

What are spacetime singularities? Do they exist in nature or are they artefacts of our theoretical reasoning? Most importantly, if we accept the general theory of relativity, our best theory of space, time, and gravity, do we then also have to accept the existence of spacetime singularities?

In this special lecture, Sir Roger Penrose, 2020 Nobel Laureate for Physics, will give an extended version of his Nobel Prize Lecture, describing his path to the first general singularity theorem of general relativity, and to the ideas that sprung from this theorem, notably the basis for the existence of Black Holes. He will be introduced by Dennis Lehmkuhl whose talk will describe how the concept of a spacetime singularity developed prior to Roger's work, in work by Einstein and others, and how much of a game changer the first singularity theorem really was.

The lectures will be followed by an interview with Roger by Melvyn Bragg.

Roger Penrose is the 2020 Nobel Laureate for Physics and Emeritus Rouse Ball Professor in Oxford; Dennis Lehmkuhl is Lichtenberg Professor of History and Philosophy of Physics at the University of Bonn and one of the Editors of Albert Einstein's Collected Papers: Melvyn Bragg is a broadcaster and author best known for his work as editor and presenter of the South Bank Show and In Our Time.

Watch online (no need to register - and the lecture will stay up on all channels afterwards):
Oxford Mathematics Twitter
Oxford Mathematics Facebook
Oxford Mathematics Livestream
Oxford Mathematics YouTube

The Oxford Mathematics Public Lecture are generously supported by XTX Markets

Friday, 22 January 2021

Maths Makes A Difference - a webinar for Year 12 students about how maths contributes to society - 2 February, 4-5pm

The future is full of uncertainty, but we still need to make plans and decisions based on the data we have.  Where should a hospital invest its resources to allow for changing health needs in a year's time?  Should the supermarket order extra ice cream because the summer will be warm and sunny?  Should the council road maintenance team get extra gritting salt ready for an icy winter? Making predictions is hard - and maths can help, as we’ll see in this interactive webinar.

Maths Makes a Difference is a collaboration between the Mathematics outreach teams at Oxford and Cambridge. This interactive webinar series for students in Year 12 at state schools in the UK (and Year 10 later in the year) will explore aspects of maths that make a difference to the world and society. More information and registration.

The webinars will be led by Claire Metcalfe from Cambridge and Vicky Neale from Oxford.

Wednesday, 20 January 2021

Call for applications for PROMYS Europe Connect 2021

We are delighted to announce PROMYS Europe Connect for 2021, online from 12 July to 6 August.

In view of continuing restrictions and uncertainty around Covid-19, we are designing PROMYS Europe Connect as a unique 4-week online programme that captures many of the key elements of the usual PROMYS Europe experience. PROMYS Europe is a challenging mathematics summer programme based at the University of Oxford, UK.

PROMYS Europe Connect is seeking

  • Pre-university students from across Europe (including all countries adjacent to the Mediterranean) who show unusual readiness to think deeply about mathematics;
  • Undergraduate students who would like to work with them as counsellors. 

PROMYS Europe Connect is designed to encourage mathematically ambitious students who are at least 16 to explore the creative world of mathematics. Participants will tackle fundamental mathematical questions within a richly stimulating and supportive online community of fellow first-year students, returning students, undergraduate counsellors, research mentors, faculty, and visiting mathematicians.

First-year students will focus primarily on a series of very challenging problem sets, daily lectures, and exploration projects in Number Theory.  There will also be a programme of talks, by guest mathematicians and the counsellors, on a wide range of mathematical subjects, as well as courses aimed primarily at students who are returning to PROMYS Europe for a second or third time.

PROMYS Europe is a partnership of Wadham College and the Mathematical Institute at the University of Oxford, the Clay Mathematics Institute, and PROMYS (Program in Mathematics for Young Scientists, founded in Boston in 1989).

The programme is dedicated to the principle that no one should be unable to attend for financial reasons.  Most of the cost is covered by the partnership and by generous donations from supporters. In addition, full and partial financial aid is available for those who need it.

Applications for counsellors and students are available on the PROMYS Europe websiteThe closing date for counsellor applications is 7 February.  The closing date for first-year student applications is 14 March, and students will need to allow enough time before the deadline to tackle the application problems.  PROMYS Europe Connect will run online from 12 July to 6 August.

Wednesday, 6 January 2021

Rethinking defects in patterns

Social distancing is integral to our lives these days, but distancing also underpins the ordered patterns and arrangements we see all around us in Nature. Oxford Mathematician Priya Subramanian studies the defects in such patterns and shows how they relate to the underlying pattern, i.e. to the distancing itself.

"Nature has many examples of situations where individuals in a dense group have to balance between short range repulsion (e.g., competition for resources) and long range attraction (e.g. safety in numbers). This naturally leads to an optimal length scale for separation between neighbours. If we think of each agent as a circle with this optimal length as the diameter, the most efficient packing is in a hexagonal arrangement. So, it is unsurprising that we find these group of gannets nesting on a beach in Muriwai, New Zealand forming a largely hexagonal pattern.

 

Figure 1: When neighbours keep their distance: gannets nesting at Muriwai Beach, New Zealand. The image on the left shows the overall hexagonal ordering of the birds while that on the right shows a penta-hepta defect (PHD). The green (pink) markers identify gannets with five (seven) neighbours instead of the usual six. Photo credit: Barbora Knobloch. Adapted from [1].

Patterns are rarely perfect and defects arise due to many factors such as boundary conditions (e.g. cliff edge), fluctuations in the background (e.g. unevenness in the ground), etc. A generic defect that arises in such hexagonal patterns is highlighted in the right panel of the picture of the gannets. Normally each gannet will have six neighbours, but here we see that the gannet marked with a green dot has only five neighbours while the gannet marked with a pink marker has seven. Such a structure consisting of a bound state with one location having five neighbours and another location seven neighbours, instead of the usual six, is called a penta-hepta defect (PHD). Hexagonal arrangements are found in many areas of physics, from patterns formed in heated fluids, to self-assembled crystals formed in both hard materials (e.g. graphene) and soft materials (e.g. star-shaped polymers [2]). Equally prevalent in all such hexagonal arrangements is the possibility for PHDs.

Traditionally pattern formation techniques used to investigate defects use an amplitude-phase formulation, where a periodic pattern has a homogenous amplitude and a varying phase. The topological charge of a defect is calculated by integrating the phase around any closed curve enclosing the defect, and this quantity does not change when the defect moves. Topological defects [3] are associated with zeros of the amplitude (where the phase becomes undefined): these defects have non-zero topological charge and so they can only be eliminated or healed by interacting with another topological defect with opposite charge. On the other hand, non-topological defects, such as PHDs, have a well-defined phase everywhere (implying zero topological charge) and so were thought to be able to heal by themselves. However, if the defect has an internal structure it may persist as a result of frustration and get locked/pinned to the background periodic state: the gannets in the PHD in Figure 1 could re-arrange themselves to remove the defect, but to do so would involve all nearby gannets moving a little bit, so in practice they don't.

 

Figure 2: Coexisting equilibria with penta-hepta defects separating regions of hexagons with different orientations in the SH23 system [1]. All three states are dynamically metastable. 

We explore such non-topological defects in the prototypical pattern forming Swift-Hohenberg model in our recent work [1], by adopting a different point of view and thinking of defects as spatially localised structures [4]. We focus on grain boundaries separating two-dimensional hexagon crystals at different orientations (shown in Figure 2): these grain boundaries are closed curves containing a ring of PHDs. Even with the parameters all the same, the model has many different stable configurations of these grain boundaries, and solution branches connected to each of these states form isolas that span a wide range of the model parameters, opening up multiple interesting questions about such defect states. Our results will also be applicable to understanding the role of grain boundaries in two dimensional solids such as graphene [5], in which defects play a crucial role near phase transition, i.e., melting [6], and in determining bulk properties of a material."

References: 

[1] Snaking without subcriticality: grain boundaries as non-topological defects, P. Subramanian, A. J. Archer, E. Knobloch and A. M. Rucklidge, arXiv:2011.08536, 2020.

[2] Two-dimensional crystals of star polymers: a tale of tails, I. Bos, P. van der Scheer, W. G. Ellenbroek and J. Sprakel, Soft Matter, 15, 615-622, 2019

[3] The topological theory of defects in ordered media, N. D. Mermin, Rev. Mod. Phys., 51, 591-648, 1979.

[4] Spatial localisation in dissipative systems, E. Knobloch, Annu. Rev. Condens. Matter Phys., 6, 325-359, 2015. 

[5] Energetics and structure of grain boundary triple junctions in graphene, P. Hirvonen, Z. Fan, M. M. Ervasti, A. Harju, K. R. Elder and T. Ala-Nissila, Sci. Rep., 7, 1-14, 2017.

[6] Melting of graphene: from two to one dimension, K. V. Zakharchenko, A, Fasolino, J. H. Los and M. I . Katsnelson, J. Phys.: Condens. Matter, 23, 202202, 2011. 

Friday, 1 January 2021

The launch of the Oxford Online Maths Club

Happy New Year! 2021 has a lot to make up for after 2020, so we're starting with a bang with the launch of the Oxford Online Maths Club, a new weekly maths livestream from Oxford Mathematics.

The Club provides free super-curricular maths for ages 16-18. It is aimed at people about to start a maths degree at university or about to apply for one. We'll be livestreaming one hour of maths problems, puzzles, mini-lectures, and Q&A, and we'll be exploring links between A level maths and university maths with help from our Admissions Coordinator James Munro and our current Oxford Mathematics students. And you get to ask questions and share thoughts and feelings with like-minded mathematicians. 

In a nutshell, it’s free, interactive, casual, and relaxed, with an emphasis on problem-solving techniques, building fluency, and looking ahead at links to university maths. The Club follows in the footsteps of James's hugely popular weekly MAT (Mathematics Admissions Test) sessions where he went thorough entrance problems and took live questions.

Whether you're the only person you know interested in maths, or you're an entire sixth-form maths club looking for more content, we're here for you in 2021! Join us every Thursday 16:30 starting this Thursday, 7 January. 

Friday, 18 December 2020

Peter Michael Neumann OBE (28 December 1940 - 18 December 2020)

We are very sad to hear the news of the death of Peter Neumann earlier today. Peter was the son of the mathematicians Bernhard Neumann and Hanna Neumann and, after gaining a B.A. from The Queen's College, Oxford in 1963, obtained his D.Phil from Oxford University in 1966.

Peter was a Tutorial Fellow at The Queen's College, Oxford and a lecturer in the Mathematical Institute in Oxford, retiring in 2008. His work was in the field of group theory. He is also known for solving Alhazen's problem in 1997. In 2011 he published a book on the short-lived French mathematician Évariste Galois.

In 1987 Peter won the Lester R. Ford Award of the Mathematical Association of America for his review of Harold Edwards' book Galois Theory. In 2003, the London Mathematical Society awarded him the Senior Whitehead Prize. He was the first Chairman of the United Kingdom Mathematics Trust, from October 1996 to April 2004 and was appointed Officer of the Order of the British Empire (OBE) in the 2008 New Year Honours. Peter was President of the Mathematical Association from 2015-2016.

Tuesday, 15 December 2020
Sunday, 13 December 2020

The Oxford Mathematics E-Newsletter - our quarterly round-up of our greatest hits

The Oxford Mathematics e-newsletter for December is out. Produced each quarter, it's a sort of 'Now That's What I Call Maths,' pulling together our greatest hits of the last few months in one place.

It's for anyone who wants a flavour of what we do - research, online teaching, public lectures, having a laugh.

And it's COVID-lite. Click here.

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