## Lent

**Ranks of elliptic curves**, by *Dr Tom Fisher*

*Wednesday 8th February, 6pm* – Hall Signup

Can the sum of the first n square numbers itself be a square number? Are there any right-angled triangles with rational side lengths and area a given integer? Which integers can be written as the sum of two rational cubes? These are just some of the classical problems in number theory that lead to the study of elliptic curves. I will give an informal introduction to elliptic curves, and the group of rational points on an elliptic curve. I will then discuss what is known, what is conjectured, and what is still completely mysterious about the number of generators for this group.

**At the dawn of a new era in astrophysics: Gravitational waves have arrived**, by *Dr Ulrich Sperhake*

*Wednesday 15th February, 6pm* – Hall Signup

Dr Ulrich Sperhake, the amazing lecturer of Part II General Relativity, will be joining us for a talk on gravitational waves – here is the abstract:

On Sep 14 2015, gravitational waves were for the first time detected directly. This observation by the LIGO interferometric detectors marks the dawn of a new era in our observational study of the cosmos as a qualitatively new window to its exploration has been opened. This talk reviews some of the fundamental concepts of gravitational waves and the methodology employed for their observation. The first two events, dubbed GW150914 and GW151226, as well as the properties of their sources, as inferred from the observations, will be discussed. The talk concludes with a selected set of the most important topics where we expect gravitational-wave observations to deepen and either challenge or confirm our present understanding of the laws and the history of our universe.

**Symmetries of manifolds**, by *Dr Oscar Randal-Williams*

*Wednesday 1st March, 6pm* – Hall Signup

Dr Oscar Randal-Williams from DPMMS will join us for a very promising talk. Many of you will know Dr Randal-Williams for his excellent lectures for Part IB GRM. His research interests include algebraic and geometric topology. Here is the abstract:

Whenever one studies a mathematical object one also ought to study its symmetries. Manifolds—spaces which look locally like ordinary Euclidean space but which can be globally complicated—are the central objects of study in topology and geometry, and their groups of symmetries come in several flavours (diffeomorphisms, homeomorphisms, homotopy equivalences, …). I will first explain some ways of thinking about such symmetries in the case of surfaces, and then give some examples of surprising behaviour which can happen when we start looking at high-dimensional manifolds.

**Relativity, Quantum Theory and Cryptography**, by *Prof Adrian Kent*

*Wednesday 15th March, 6pm* – Hall Signup

Prof Adrian Kent from DAMTP will join us to give a talk about relativity, quantum information and cryptography. Abstract:

The goal of cryptography is to control access to information. For example, we may want a secret message to be readable by selected allies but not by adversaries, or an encrypted prediction to be unveiled only if we choose to supply a key. In recent decades, we have discovered how to use fundamental physical laws to guarantee cryptographic security. Quantum cryptography exploits the distinctive properties of information encoded in quantum systems, while relativistic cryptography uses the fact that information cannot be sent faster than light speed. I will show how some simple but perfectly secure cryptosystems can be built using these principles and describe the current state of the art of physics-based cryptography.

**Michaelmas**

**Hunting for viral packaging signals**, by *Dr Julia Gog*

*Wednesday 2nd November, 6pm* – Hall Signup

Dr Julia Gog from DAMTP (and “Mathematical Biology” fame) will join us for an exciting talk. Abstract:

Influenza has a genome split into several segments, and this complicates virus particle assembly as each particle must have one of each of the segments. This means that each of the RNA segments must contain some signal, and that this signal ought to be fairly conserved. Is this enough to go and hunt them down using mathematics? The answer turns out to be yes. However, this required some creativity in algorithm design, drawing inspiration from a number of apparently unrelated problems. This hack seems to work, but leaves some interesting mathematical problems.

I’ll also briefly talk about some of the other problems in influenza and infectious disease that interest me, and general joys and challenges of being a mathematician trying to research biology.

**Do we really know what computers can do? – On the foundations of computational mathematics**, by *Dr Anders Hansen*

*Thursday 10th November, 6pm* – Hall Signup

Dr Anders Hansen from the Cambridge Center for Analysis at DAMTP will join us for an exciting talk – note that this is on a *Thursday* rather than our usual Wednesday, and we also have talks on three consecutive weeks. Dr Hansen’s homepage can be found here. Abstract:

Questions regarding what a computer can do have fascinated mathematicians since the beginning of the 20th century. Hilbert initiated much of the research in the area by posing his “Entscheidungsproblem” (decision problem) in the late 1920s. Turing’s solution to this problem, as well as his intuitive definition of what a computer should be (the Turing Machine), made him one of the founding fathers of modern computer science. In the late 1940s von Neumann contributed substantially to the field, and this was the beginning of what we know as scientific computing, namely how to use computers to solve problems in the sciences. Over the next decades (50s until today) computer science and scientific computing, although rooted in the same questions, developed in parallel without much interaction. This discrepancy was pointed out by Smale in the 1980s when he asked basic questions on convergence of Newton’s method, one of the most basic algorithms in scientific computing, as well as questions on existence of algorithms for fundamental problems such as polynomial root finding. Several of these questions were answered by McMullen and the solutions were part of the justification of his Fields medal in 1998.

I will give an informal historical account of the developments in the field leading up to the current unsolved problems and demonstrate that, despite about 100 years of research and the enormous impact of computers, we still do not really know what a computer can actually do.

** Loop Erased Random, Uniform Spanning Trees and Percolation**, by *Dr Sebastian Andres*

*Wednesday 16th November, 6pm* – Hall Signup

In graph theory spanning trees have been investigated already since the 19th century. They appear for instance as objects in a number of algorithms. On the other hand, in modern probability theory certain random spanning trees, so called uniform spanning trees, have had a fruitful history. Most notably, around the turn of the millennium the study of these spanning trees led Oded Schramm to introduce the SLE process, work which has revolutionised the study of two dimensional models in statistical physics. One reason for the importance of uniform spanning trees is their intimate relation to another model, the loop-erased random walks.

In this talk we will introduce both models and explain their connection by means of Wilson’s algorithm. In the last part we will discuss some relations to percolation theory.

**Computerized tomography and the X-ray transform**, by *Prof Gabriel Paternain*

*Wednesday 30th November, 6pm*

Professor Gabriel Paternain will join us for an exciting talk. His department homepage can be found here. Abstract: I will describe some of the mathematics that underpins CT scans. The main inversion formula was discovered by J. Radon 99 years ago.