Past

Oxford Mathematics Public Lectures

The Law of the Few - Sanjeev Goyal

The study of networks offers a fruitful approach to understanding human behaviour. Sanjeev Goyal is one of its pioneers. In this lecture Sanjeev presents a puzzle:

In social communities, the vast majority of individuals get their information from a very small subset of the group – the influencers, connectors, and opinion leaders. But empirical research suggests that there are only minor differences between the influencers and the others. Using mathematical modelling of individual activity and networking and experiments with human subjects, Sanjeev helps explain the puzzle and the economic trade-offs it contains.

Professor Sanjeev Goyal FBA is the Chair of the Economics Faculty at the University of Cambridge and was the founding Director of the Cambridge-INET Institute.

28 June 2017, 5.00-6.00pm, Lecture Theatre 1, Mathematical Institute Oxford.

Please email @email to register

In this lecture, we present  practical and provably
secure (authenticated) key exchange protocol and password
authenticated key exchange protocol, which are based on the
learning with errors problems. These protocols are conceptually
simple and have strong provable security properties.
This type of new constructions were started in 2011-2012.
These protocols are shown indeed practical.  We will explain
that all the existing LWE based key exchanges are variants
of this fundamental design.  In addition, we will explain
some issues with key reuse and how to use the signal function
invented for KE for authentication schemes.

Starting from an example in quantum chemistry, we explain the techniques of Numerical Tensor Calculus with particular emphasis on the convolution operation. The tensorisation technique also applies to one-dimensional grid functions and allows to perform the convolution with a cost which may be much cheaper than the fast Fourier transform.

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

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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.

In 1933, lattice theory was a new subject, put forth by Garrett Birkhoff. In contrast, in 1940, it was already a mature subject, worth publishing a book on. Indeed, the first monograph, written by the same G. Birkhoff, was the result of these 7 years of working on a lattice theory. In my talk, I would like to focus on this fast development. I will present the notion of a theory not only as an actors' category but as an historical category. Relying on that definition, I would like to focus on some collaborations around the notion of lattices. In particular, we will study lattice theory as a meeting point between the works of G. Birkhoff and two other mathematicians: John von Neumann and Marshall Stone.

Let X be a smooth, complete geometrically connected curve defined over a one variable function field K over a finite field. Let G be a subgroup of the points of the Jacobian variety J of X defined over a separable closure of K with the property that G/p is finite, where p is the characteristic of K. Buium and Voloch, under the hypothesis that X is not defined over K^p, give an explicit bound for the number of points of X which lie in G (related to a conjecture of Lang, in the case of curves). In this joint work with Pazuki, we extend their result by requiring just that X is non isotrivial.

Groups which are "attached" to theories of fields, appearing in models of the theory  
as the automorphism groups of intermediate fields fixing an elementary submodel are called geometrically represented. 
We will discuss the concept ``geometric representation" in the case of pseudo finite fields.  Then will show that any group which is geometrically represented in a complete theory of a pseudo-finite field must be abelian. 
This result also generalizes to bounded PAC fields. This is joint work with Zoe Chatzidakis.
 

Kim's iterative non-abelian reciprocity laws carve out a sequence of subsets of the adelic points of a suitable algebraic variety, containing the global points. Like Ellenberg's obstructions to the existence of global points, they are based on nilpotent approximations to the variety. Systematically exploiting this idea gives a sequence starting with the Brauer-Manin obstruction, based on the theory of obstruction towers in algebraic topology. For Shimura varieties, nilpotent approximations are inadequate as the fundamental group is nearly perfect, but relative completions produce an interesting obstruction tower. For modular curves, these maps take values in Galois cohomology of modular forms, and give obstructions to an adelic elliptic curve with global Tate module underlying a global elliptic curve.

Despite its fame there appears to be little literature outlining Lurie's proof sketched in his expository article "On the classification of topological field theories." I shall embark on the quixotic quest to explain how the cobordism hypothesis is formalised and give an overview of Lurie's proof in one hour. I will not be able to go into any of the motivation, but I promise to try to make the talk as accessible as possible. 

Understanding the evolution of a solidification front is important in the study of solidification processes. Mathematically, self-similar solutions exist to the Stefan problem when the liquid domain is assumed semi-infinite, and such solutions have been extensively studied in the literature. However, in the case where the liquid region is finite and sufficiently small, such of solutions no longer hold, as in this case two solidification fronts will move toward each other and interact. We present an asymptotic analysis for the two-front Stefan problem with a small amount of constitutional supercooling and compare the asymptotic results with numerical simulations. We finally discuss ongoing work on the same problem near the time when the two fronts are close to colliding.
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Silicon is produced from quartz rock in electrode-heated furnaces by using carbon as a reduction agent. We present a model of the heat and mass transfer in an experimental pilot furnace and perform an asymptotic analysis of this model. First, by prescribing a steady state temperature profile in the furnace we explore the leading order reactions in different spatial regions. We next utilise the dominant behaviour when temperature is prescribed to reduce the full model to two coupled partial differential equations for the time-variable temperature profile within the furnace and the concentration of solid quartz. These equations account for diffusion, an endothermic reaction, and the external heating input to the system. A moving boundary is found and the behaviour on either side of this boundary explored in the asymptotic limit of small diffusion. We note how the simplifications derived may be useful for industrial furnace operation.

We consider the problem of modelling the term structure of defaultable bonds, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional term in the forward rate approach by Heath, Jarrow and Morton (1992). This term is driven by a random measure encoding information about those times where default can happen with positive probability.  In this framework, we  derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of credit risky bonds, also considering general recovery schemes. This is based on joint work with Thorsten Schmidt.

Adjoint derivatives reveal the sensitivity of a computer program's output to changes in its inputs. These derivatives are useful as a building block for optimisation, uncertainty quantification, noise estimation, inverse design, etc., in many industrial and scientific applications that use PDE solvers or other codes.
Algorithmic differentiation (AD) is an established method to transform a given computation into its corresponding adjoint computation. One of the key challenges in this process is the efficiency of the resulting adjoint computation. This becomes especially pressing with the increasing use of shared-memory parallelism on multi- and many-core architectures, for which AD support is currently insufficient.
In this talk, I will present an overview of challenges and solutions for the differentiation of shared-memory-parallel code, using two examples: an unstructured-mesh CFD solver, and a structured-mesh stencil kernel, both parallelised with OpenMP. I will show how AD can be used to generate adjoint solvers that scale as well as their underlying original solvers on CPUs and a KNC XeonPhi. The talk will conclude with some recent efforts in using AD and formal verification tools to check the correctness of manually optimised adjoint solvers.
 

I will talk about the fluid equations used to model pedestrian motion and traffic. I will present the compressible-incompressible Navier-Stokes two phase system describing the flow in the free and in the congested regimes, respectively. I will also show how to approximate such system by the compressible Navier-Stokes equations with singular pressure for the fixed barrier densities and also some recent developments for the barrier densities varying in the space and time.
This is a talk based on several papers in collaboration with: D. Bresch, C. Perrin, P. Degond, P. Minakowski, and L. Navoret.
 

We will follow a suggestion by Udi to construct a decidable field which has an undecidable finite extension.

In my talk I will give a basic introduction to the finiteness properties of groups and their relation to subgroups of direct products of groups. I will explain the relation between such subgroups and fibre products of groups, and then proceed with a discussion of the n-(n+1)-(n+2)-Conjecture and the Virtual Surjections Conjecture. While both conjectures are still open in general, they are known to hold in special cases. I will explain how these results can be applied to prove that there are groups with arbitrary (non-)finiteness properties.

Komlós conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large.  In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify. This is joint work with Hong Liu, Maryam Sharifzadeh and Katherine Staden.

One of the greatest challenges in developing renewable energy sources is finding an efficient energy storage solution to smooth out the inherently fluctuating supply. One cheap solution is lead-acid batteries, which are used to provide off-grid solar energy in developing countries. However, modelling of this technology has fallen behind other types of battery; the state-of-the-art models are either overly simplistic, fitting black-box functions to current and voltage data, or overly complicated, requiring complex and time-consuming numerical simulations. Neither of these methods offers great insight into the chemical behaviour at the micro-scale.

In our research, we use asymptotic methods to explore the Newman porous-electrode model for a constant-current discharge at low current densities, a good estimate for real-life applications. In this limit, we obtain a simple yet accurate formula for the cell voltage as a function of current density and time. We also gain quantitative insight into the effect of various parameters on this voltage. Further, our model allows us to quantitatively investigate the effect of ohmic resistance and mass transport limitations, as a correction to the leading order cell voltage. Finally, we explore the effect on cell voltage of other secondary phenomena, such as growth of a discharge-product layer in the pores and reaction-induced volume changes in the electrolyte.

In the cosmological scheme of conformal cyclic cosmology (CCC), the equations governing the crossover form each aeon to the next demand the creation of a dominant new scalar material that is postulated to be dark matter. In order that this material does not build up from aeon to aeon, it is taken to decay away completely over the history of the aeon. The dark matter particles (erebons) would be expected to behave as essentially classical particles of around a Planck mass, interacting only gravitationally, and their decay would be mainly responsible for the (~scale invariant)

temperature fluctuations in the CMB of the succeeding aeon. In our own aeon, erebon decay ought to be detectable as impulsive events observable by gravitational wave detectors.