The NT seminar takes place on Wednesday at 4pm in McCrea 219.

Here is a campus plan.

October 3:

Abstract: Wooley has shown that every rational cubic hypersurface defined by a cubic form in at least 37 variables contains a rational line. In joint work with Julia Brandes we were able to improve this in the generic case of non-singular cubic forms, reducing the required number of variables to 31, applying recent work of Browning, Dietmann and Heath-Brown on intersections of cubic and quadric hypersurfaces. Time permitting, we also briefly want to discuss the related problem of finding lines on cubic hypersurfaces defined over p-adic fields.

October 10: No seminar!

October 17:

Abstract: Ashot Minasyan and I construct groups that establish the result in the title, resolving a question that has been around for almost 30 years. I will start by explaining

the phrases `CAT(0)' and `biautomatic'. After that I will talk about our groups and why they have the properties that we claim.

October 24: No seminar!

October 31:

The celebrated Hasse-Minkowski local-global principle relates the existence of zeros of quadratic forms over the field of rational

numbers, and more generally number fields ('global'), to the existence of zeros over various completions ('local'), namely the real and complex

numbers, and the p-adics, where analytic methods can be applied. A similar local-global principle holds in positive characteristic, where

the 'local' fields are the fields of formal Laurent series over finite fields. However, while it is classical that for each local field K of

characteristic zero there is an algorithm that determines whether a given system of polynomial equations has a common zero in K, and even

more generally whether a given first-order sentence in the language of rings holds in K, for Laurent series fields over finite fields the

existence of such algorithms is only partially understood. In this talk I will report on what is known about this, which will lead us from

number theory and algebraic geometry to the model theory of valued fields.

November 7:

In this talk we give a quick introduction to a general problem in Diophantine approximation that involves sums of reciprocals of fractional parts. The problem was first formalised by Lê and Vaaler in 2013 and partially solved by them. We show how it can be rephrased as a counting problem for certain lattices and how some purely Diophantine properties translate into geometric properties (i.e., weak admissibility), to throw new light on Lê and Vaaler’s questions

November 14:

Abstract: The hard problems on which many post quantum cryptographic schemes depend are in practice cryptanalysed by finding short vectors in lattices.

I will introduce a simple heuristic lattice sieve, which given a lattice basis returns a short vector. I will then discuss recent work which treats a sieve as a stateful machine with a set of operations, within the context of a lattice basis.

This viewpoint of a sieve as more than a shortest vector oracle and some further algorithmic techniques have allowed new SVP and LWE (a problem on which many cryptographic schemes are based) records to be broken.

November 21:

"A contraction theorem for the largest eigenvalue of a multigraph"

Abstract: Let G be a multigraph with loops, and let e be an edge in G. Let H be the multigraph obtained by contracting along the edge e. Let lambda_G and lambda_H be the largest eigenvalues of G and H respectively. A characterisation will be given of precisely when lambda_H < lambda_G, lambda_H = lambda_G, or lambda_H > lambda_G. If H happens to be a simple graph, then so is G, and the characterisation theorem subsumes those of Hoffman-Smith and Gumbrell for subdivision of edges or splitting of vertices of a graph. (Multi)graph eigenvalues are examples of algebraic integers, and this work was motivated by the study of certain families of algebraic integers.

November 28:

Abstract: This work provides a systematic analysis of primality testing under adversarial conditions, where the numbers being tested for primality are not generated randomly, but instead provided by a possibly malicious party. Such a situation can arise in secure messaging protocols where a server supplies Diffie-Hellman parameters to the peers, or in a secure communications protocol like TLS where a developer can insert such a number to be able to later passively spy on client-server data. We study a broad range of cryptographic libraries and assess their performance in this adversarial setting. As examples of our findings, we are able to construct 2048-bit composites that are declared prime with probability 1/16 by OpenSSL's primality testing in its default configuration; the advertised performance is 2^{−80}. We can also construct 1024-bit composites that always pass the primality testing routine in GNU GMP when configured with the recommended minimum number of rounds. And, for a number of libraries (Cryptlib, LibTomCrypt, JavaScript Big Number, WolfSSL), we can construct composites that always pass the supplied primality tests. We explore the implications of these security failures in applications, focusing on the construction of malicious Diffie-Hellman parameters. We show that, unless careful primality testing is performed, an adversary can supply parameters (p,q,g) which on the surface look secure, but where the discrete logarithm problem in the subgroup of order q generated by g is easy. We close by making recommendations for users and developers. In particular, we promote the Baillie-PSW primality test which is both efficient and conjectured to be robust even in the adversarial setting for numbers up to a few thousand bits.

December 5: No seminar!

December 13 (Thursday!) at 1pm in ABLT 3:

Abstract: Mean values for exponential sums play a central role in the study of diophantine equations. In particular, strong upper bounds for such mean values control the number of integer solutions of the corresponding systems of diagonal equations. Since the groundbreaking resolution of Vinogradov's mean value theorem by Wooley and Bourgain, Demeter and Guth, we can now prove optimal upper bounds for mean values connected to translation-dilation-invariant systems. This has inspired Wooley's call for a "Big Theory of Everything", a challenge to establish optimal mean value estimates for any mean values associated with systems of diagonal equations.

We establish optimal bounds for a family of mean values that are not of Vinogradov type. This is the first time bounds of this quality have been obtained for non-translation-dilation-invariant systems. As a consequence, we establish the analytic Hasse principle for the number of solutions of certain systems of quadratic and cubic equations in fewer variables than hitherto thought necessary. This is joint work with Trevor Wooley.

Spring 2016

Autumn 2015

Spring 2015

Autumn 2014