University of Cambridge

In recognition of a lifetime's contribution across the mathematical sciences, we are initiating a series of annual Public Lectures in honour of Roger Penrose. The first lecture will be given by his long-time collaborator and friend Stephen Hawking.

Registration will open at 10am on 4 January 2017. Please email:

@email from that time only.

When registering please give your name and affiliation - your position, department & organisation/institution as appropriate. Or if you are a member of the General Public, please say so. Places will be allocated on a first come, first served basis with only one place per person. We will only be able to respond if you have a place or are on the waiting list.

We will be podcasting the lecture live. More details to follow.

 The matrix logarithm, when applied to symmetric positive definite matrices, is known to satisfy a notable concavity property in the positive semidefinite (Loewner) order. This concavity property is a cornerstone result in the study of operator convex functions and has important applications in matrix concentration inequalities and quantum information theory.
In this talk I will show that certain rational approximations of the matrix logarithm remarkably preserve this concavity property and moreover, are amenable to semidefinite programming. Such approximations allow us to use off-the-shelf semidefinite programming solvers for convex optimization problems involving the matrix logarithm. These approximations are also useful in the scalar case and provide a much faster alternative to existing methods based on successive approximation for problems involving the exponential/relative entropy cone. I will conclude by showing some applications to problems arising in quantum information theory.

This is joint work with James Saunderson (Monash University) and Pablo Parrilo (MIT)

In a recent work Andreata, Iovita, and Pilloni constructed the eigenvariety for cuspidal Siegel modular forms. This eigenvariety has the expected dimension (the genus of the Siegel forms) but it parametrizes only cuspidal forms. We explain how to generalize the construction to the non-cuspidal case. To be precise, we introduce the notion of "degree of cuspidality" and we construct an eigenvariety that parametrizes forms of a given degree of cuspidability. The dimension of these eigenvarieties depends on the degree of cuspidality we want to consider: the more non-cuspidal the forms, the smaller the dimension. This is a joint work with Riccardo Brasca.

This talk will be about  Lévy processes on compact groups - discrete or continuous - and  two-dimensional analogues called pure Yang-Mills fields. The latter are indexed by  reduced loops of finite length in the plane and satisfy properties analogue to independence and stationarity of increments.     There is a one-to-one correspondance between Lévy processes invariant by adjunction and pure Yang-Mills fields. For Brownian motions, Yang-Mills fields stand for a rigorous version of the Euclidean Yang-Mills measure in two dimension.  I shall first sketch this correspondance for  Lévy processes with large jumps. Then, I will discuss two applications of an extension theorem, due to Thierry Lévy, similar to Kolmogorov extension theorem. On the one hand, it allows to construct pure Yang-Mills fields for any invariant Lévy process. On the other hand, when the group acts on vector spaces of large dimension, this theorem also allows to study the asymptotic behavior  of traces. The limiting objects yield a natural family of states on the group algebra of reduced loops.  We characterize among them the master field defined by Thierry Lévy by a continuity property.   This is  a joint work with Guillaume Cébron and Franck Gabriel.

It has been conjectured that marine ice sheets (those that

flow into the ocean) are unconditionally unstable when the underlying

bed-slope runs uphill in the direction of flow, as is typical in many

regions underneath the West Antarctic Ice Sheet. This conjecture is

supported by theoretical studies that assume a two-dimensional flow

idealization. However, if the floating section (the ice shelf) is

subject to three-dimensional stresses from the edges of the embayments

into which they flow, as is typical of many ice shelves in Antarctica,

then the ice shelf creates a buttress that supports the ice sheet.

This allows the ice sheet to remain stable under conditions that may

otherwise result in collapse of the ice sheet. This talk presents new

theoretical and experimental results relating to the effects of

three-dimensional stresses on the flow and structure of ice shelves,

and their potential to stabilize marine ice sheets.

It is easy to see that if a tournament (a complete oriented graph) has a directed cycle then it has a directed 3-cycle. We investigate the analogous question for 3-tournaments, and more generally for oriented 3-graphs.

The Erdős–Ko–Rado theorem is a central result in extremal set theory which tells us how large uniform intersecting families can be. In this talk, I shall discuss some recent results concerning the 'stability' of this result. One possible formulation of the Erdős–Ko–Rado theorem is the following: if $n \ge 2r$, then the size of the largest independent set of the Kneser graph $K(n,r)$ is $\binom{n-1}{r-1}$, where $K(n,r)$ is the graph on the family of $r$-element subsets of $\{1,\dots,n\}$ in which two sets are adjacent if and only if they are disjoint. The following will be the question of interest. Delete the edges of the Kneser graph with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? I shall discuss an affirmative answer to this question in a few different regimes. Joint work with Bollobás and Raigorodskii, and Balogh and Bollobás.

This is an exciting time to study quantum algorithms. As the technological challenges of building a quantum computer continue to be met there is still much to learn about the power of quantum computing. Understanding which problems a quantum computer could solve faster than a classical device and which problems remain hard is particularly relevant to cryptography. We would like to design schemes that are secure against an adversary with a quantum computer. I'll give an overview of the quantum computing that is accessible to a general audience and use a recently declassified project called "soliloquy" as a case study for the development (and breaking) of post-quantum cryptography.

The prime number race is the competition between different coprime residue classes mod $q$ to contain the most primes, up to a point $x$ . Rubinstein and Sarnak showed, assuming two $L$-function conjectures, that as $x$ varies the problem is equivalent to a problem about orderings of certain random variables, having weak correlations coming from number theory. In particular, as $q \rightarrow \infty$ the number of primes in any fixed set of $r$ coprime classes will achieve any given ordering for $\sim 1/r!$ values of $x$. In this talk I will try to explain what happens when $r$ is allowed to grow as a function of $q$. It turns out that one still sees uniformity of orderings in many situations, but not always. The proofs involve various probabilistic ideas, and also some harmonic analysis related to the circle method. This is joint work with Youness Lamzouri.