Past Fridays@4

10 November 2017
Laura Capuano and Noemi Picco

Laura Capuano's talk 'Pell equations and continued fractions in number theory'

The classical Pell equation has an extraordinary long history and it is very useful in many different areas of number theory. For example, they given a way to write a prime congruent to 1 modulo 4 as a sum of two squares, or they can also be used to break RSA excryption when the decription key is too small. In this talk, I will present some properties of this wonderful equation and its relation with continued fractions. I will also treat the case of Pell equations in other contexts, such as the ring of polynomials, showing the differences with the classical case. 

Noemi Picco's talk 'Cortical neurogenesis: how humans (and mathematicians) can do more than macaque, with less'

The cerebral cortex is perhaps the crowning achievement of evolution and is the region of the brain that distinguishes us from other species. Studying the developmental programmes that generate cortices of different sizes and neuron counts, is the key to understanding both brain evolution and disease. I will show what mathematical modeling has to say about cortex evolution, when data resolution is poor. I will then discuss why humans are so special in the way they create their cortex, and how we are just like everybody else in many other aspects of brain development.

16 June 2017
Lisa Lamberti + Jaroslav Fowkes

Lisa Lamberti

Geometric models in algebra and beyond

Many phenomena in mathematics and related sciences are described by geometrical models.

In this talk, we will see how triangulations in polytopes can be used to uncover combinatorial structures in algebras. We will also glimpse at possible generalizations and open questions, as well as some applications of geometric models in other disciplines.

Jaroslav Fowkes

Jaroslav Fowkes

Optimization Challenges in the Commercial Aviation Sector

The commercial aviation sector is a low-margin business with high fixed costs, namely operating the aircraft themselves. It is therefore of great importance for an airline to maximize passenger capacity on its route network. The majority of existing full-service airlines use largely outdated capacity allocation models based on customer segmentation and fixed, pre-determined price levels. Low-cost airlines, on the other hand, mostly fly single-leg routes and have been using dynamic pricing models to control demand by setting prices in real-time. In this talk, I will review our recent research on dynamic pricing models for the Emirates route network which, unlike that of most low-cost airlines, has multiple routes traversing (and therefore competing for) the same leg.

26 May 2017
Erik Panzer + Yuji Nakatsukasa

Erik Panzer

Feynman integrals, graph polynomials and zeta values

Where do particle physicists, algebraic geometers and number theorists meet?

Feynman integrals compute how elementary particles interact and they are fundamental for our understanding of collider experiments. At the same time, they provide a rich family of special functions that are defined as period integrals, including special values of certain L functions.

In the talk I will give the definition of Feynman integrals via graph polynomials and discuss some examples that evaluate to values of the Riemann zeta function. Then I will discuss some of the interesting questions in this field and mention some of the techniques that are used to study these.


Yuji Nakatsukasa

Yuji Nakatsukasa

Computing matrix eigenvalues

The numerical linear algebra community solves two main problems: linear systems, and eigenvalue problems. They are both vastly important; it would not be too far-fetched to say that most (continuous) problems in scientific computing eventually boil down to one or both of these.

This talk focuses on eigenvalue problems. I will first describe some of their applications, such as Google's PageRank, PCA, finding zeros and poles of functions, and global optimization. I will then turn to algorithms for computing eigenvalues, namely the classical QR algorithm---which is still the basis for state-of-the-art. I will emphasize that the underlying mathematics is (together with the power method and numerical stability analysis) rational approximation theory.

19 May 2017
Professor Uta Frith and Dr Maria Bruna

Professor Uta Frith FRS is a distinguished developmental psychologist who is well known for her pioneering research on autism spectrum disorders. She also has a long-standing interest in matters relating to diversity in science, and is the Chair of the Royal Society's Diversity Committee. Oxford Mathematician Dr Maria Bruna is a Junior Research Fellow in Mathematics at St John's College, and has won prizes such as the L'Oréal-UNESCO UK and Ireland For Women in Science Fellowship and the Olga Taussky Pauli Fellowship, Wolfgang Pauli Institute. This informal discussion will no doubt include a range of topics -- but it is hard to say in advance where the conversation might go!

21 April 2017
Dyrol Lumbard and Vicky Neale

Research takes a long time while the attention span of the world is apparently decreasing, so today's researchers need to be able to get their message across quickly and succinctly. In this session we'll share some tips on how to communicate the key messages of your work in just a few minutes, and give you a chance to have a go yourself.  This will be helpful for job and funding applications and interviews, and also for public engagement. In September there will be an opportunity to do it for real, for our alumni, when we'll showcase Oxford Mathematics at the Alumni Weekend.