We will follow a suggestion by Udi to construct a decidable field which has an undecidable finite extension.
Governments around the world are seeking to address the economic and humanitarian consequences of climate change. One of the most graphic indications of warming temperatures is the melting of the large ice caps in Greenland and Antarctica. This is a litmus test for climate change, since ice loss may contribute more than a metre to sea-level rise over the next century, and the fresh water that is dumped into the ocean will most likely affect the ocean circulation that regulates our temperature.
Identity-based cryptography can be useful in situations where a full-scale public-key infrastructure is impractical. Original identity-based proposals relied on elliptic curve pairings and so are vulnerable to quantum computers. I will describe some on-going work to design a post-quantum identity-based encryption scheme using ideas from Ring Learning with Errors. Our scheme has the advantage that it can be extended to the hierarchical setting for more flexible key management.
I will discuss my ongoing project towards a version of the Modular Andre-Oort Conjecture incorporating the derivatives of the j function. The work originates with Jonathan Pila, who formulated the first "Modular Andre-Oort with Derivatives" conjecture. The problem can be approached via o-minimality; I will discuss two categories of result. The first is a weakened version of Jonathan's conjecture. Under an algebraic independence conjecture (of my own, though it follows from standard conjectures), the result is equivalent to the statement that Jonathan's conjecture holds.
The second result is conditional on the same algebraic independence conjecture - it specifies more precisely how the special points in varieties can occur in this context.
If time permits, I will discuss my most recent work towards making the two results uniform in algebraic families.
Several optimization problems combine nonlinear constraints with the integrality of a subset of variables. For an important class of problems called Mixed Integer Second-Order Cone Optimization (MISOCO), with applications in facility location, robust optimization, and finance, among others, these nonlinear constraints are second-order (or Lorentz) cones.
For such problems, as for many discrete optimization problems, it is crucial to understand the properties of the union of two disjoint sets of feasible solutions. To this end, we apply the disjunctive programming paradigm to MISOCO and present conditions under which the convex hull of two disjoint sets can be obtained by intersecting the feasible set with a specially constructed second-order cone. Computational results show that such cone has a positive impact on the solution of MISOCO problems.
Mathematics is delving in to ever-wider aspects of the physical world. Here Oxford Mathematician Alain Goriely describes how mathematicians and engineers are working with medics to better understand the workings of the human brain and in particular the issue of abnormal skull growth.
Oxford Mathematician Doireann O'Kiely was recently awarded the IMA's biennial Lighthill-Thwaites Prize for her work on the production of thin glass sheets. Here Doireann describes her work which was conducted in collaboration with Schott AG.
Many anticorruption advocates are excited about the prospects that “big data” will help detect and deter graft and other forms of malfeasance. But good data alone isn’t enough. To be useful, there must be a group of interested and informed users, who have both the tools and the skills to analyse the data to uncover misconduct, and then lobby governments and donors to listen to and act on the findings.
