I will discuss the theory of branched covers of cube complexes as a method of hyperbolisation. I will show recent results using this technique. Time permitting I will discuss a form of Morse theory on simplicial complexes and show how these methods combined with the earlier methods allow one to create groups with interesting finiteness properties. 

I will introduce the profinite completion as a way of aggregating information about the finite-sheeted covers of a 3-manifold, and discuss the state of the homeomorphism problem for 3-manifolds in this context; in particular, for geometrizable 3-manifolds.

Deformation K-theory was introduced by G. Carlsson and gives an interesting invariant of a group G encoding higher homotopy information about its representation spaces. Lawson proved a relation between this object and a homotopy theoretic analogue of the representation ring. This talk will not contain many details, instead I will outline some basic constructions and hopefully communicate the main ideas.
 

Abstract: Four manifolds are some of the most intriguing objects in topology. So far, they have eluded any attempt of classification and their behaviour is very different from what one encounters in other dimensions. On the other hand, Einstein metrics are among the canonical types of metrics one can find on a manifold. In this talk I will discuss many of the peculiarities that make dimension four so special and see how Einstein metrics could potentially help us understand more about four manifolds.

Our Christmas Public Lecture this year will be presented by Marcus du Sautoy who will be examining an aspect of Christmas not often considered: the mathematics.

To register please email: external-relations@maths.ox.ac.uk

The Oxford Mathematics Christmas Lecture is generously sponsored by G-Research - Researching investment ideas to predict financial markets

In this talk, I'll discuss some personal experiences - good, bad, and
ugly - of disclosing vulnerabilities in a range of different cryptographic
standards and implementations. I'll try to draw some general lessons about
what works well and what does not.

The Learning with Errors (LWE) problem has become a central building block of modern cryptographic constructions. We will discuss hardness results for concrete instances of LWE. In particular, we discuss algorithms proposed in the literature and give the expected resources required to run them. We consider both generic instances of LWE as well as small secret variants. Since for several methods of solving LWE we require a lattice reduction step, we also review lattice reduction algorithms and propose a refined model for estimating their running times. We also give concrete estimates for various families of LWE instances, provide a Sage module for computing these estimates and highlight gaps in the knowledge about algorithms for solving the Learning with Errors problem.

The concept of delegated quantum computing is a quantum extension of  
the classical task of computing with encrypted data without decrypting  
them first. Many quantum protocols address this challenge for a  
futuristic quantum client-server setting achieving a wide range of  
security properties. The central challenge of all these protocols to  
be applicable for classical tasks (such as secure multi party  
computation or fully homomorphic encryption) is the requirement of a  
server with a universal quantum computer. By restricting the task to  
classical computation only, we derive a protocol for unconditionally  
secure delegation of classical computation to a remote server that has  
access to basic quantum devices.

Finding rational points on curves - Jennifer Balakrishnan (Mathematical Institute, Oxford)

From cryptography to the proof of Fermat's Last Theorem, elliptic curves are ubiquitous in modern number theory.  Much activity is focused on developing methods to discover their rational points (those points with rational coordinates).  It turns out that finding a rational point on an elliptic curve is very much like finding the proverbial needle in the haystack.  In fact, there is no algorithm known to determine the group of rational points on an elliptic curve.

Hyperelliptic curves are also of broad interest; when these curves are defined over the rational numbers, they are known to have finitely many rational points.  Nevertheless, the question remains: how do we find these rational points?

I'll summarize some of the interesting number theory behind these curves and briefly describe a technique for finding rational points on curves using (p-adic) numerical linear algebra.

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Analysis, prediction and control of technological progress - François Lafond (London Institute for Mathematical Sciences, Institute for New Economic Thinking at the Oxford Martin School, United Nations University - MERIT)

Technological evolution is one of the main drivers of social and economic change, with transformative effects on most aspects of human life.  How do technologies evolve?  How can we predict and influence technological progress?  To answer these questions, we looked at the historical records of the performance of multiple technologies.  We first evaluate simple predictions based on a generalised version of Moore's law, which assumes that technologies have a unit cost decreasing exponentially, but at a technology-specific rate.  We then look at a more explanatory theory which posits that experience - typically in the form of learning-by-doing - is the driver of technological progress.  These experience curves work relatively well in terms of forecasting, but in reality technological progress is a very complex process.  To clarify the role of different causal mechanisms, we also study military production during World War II, where it can be argued that demand and other factors were exogenous.  Finally, we analyse how to best allocate investment between competing technologies.  A decision maker faces a trade-off between specialisation and diversification which is influenced by technology characteristics, risk aversion, demand and the planning horizon.