University of Bristol

The equidistribution theorem for rational points on expanding horospheres with fixed denominator in the space of d-dimensional Euclidean lattices has been derived in the work by M. Einsiedler, S. Mozes, N. Shah and U. Shapira. The proof of their theorem requires ergodic theoretic tools, including Ratner's measure classification theorem. In this talk I will present an alternative approach, based on harmonic analysis and Weil's bound for Kloosterman sums. In the case of d=3, unlike the ergodic-theoretic approach, this provides an explicit estimate on the rate of convergence. This is a joint work with Jens Marklof. 

The sub-convexity barrier traditionally prevents one from applying the Hardy-Littlewood (circle) method to Diophantine problems in which the number of variables is smaller than twice the inherent total degree. Thus, for a homogeneous polynomial in a number of variables bounded above by twice its degree, useful estimates for the associated exponential sum can be expected to be no better than the square-root of the associated reservoir of variables. In consequence, the error term in any application of the circle method to such a problem cannot be expected to be smaller than the anticipated main term, and one fails to deliver an asymptotic formula. There are perishingly few examples in which this sub-convexity barrier has been circumvented, and even fewer having associated degree exceeding two. In this talk we review old and more recent progress, and exhibit a new class of examples of Diophantine problems associated with, though definitely not, of translation-invariant type.

It has become possible in recent years to provide unconditional lower bounds on the time needed to perform a number of basic computational operations. I will briefly discuss some of the main techniques involved and show how one in particular, the information transfer method, can be exploited to give  time lower bounds for computation on streaming data.

I will then go on to present a simple looking mathematical conjecture with a probabilistic combinatorics flavour that derives from this work.  The conjecture is related to the classic "coin weighing with a spring scale" puzzle but has so far resisted our best efforts at resolution.

In recent years there has been amazing progress in building
practical protocols for Multi-Party Computation (MPC).
So much progress in fact that there are now a number of
companies producing products utilizing this technology. A major issue with existing solutions is the high round
complexity of protocols involving more than two players. In this talk I will survey the main protocols for MPC
and recent ideas in how to obtain practical low round
complexity protocols.

My main purpose in this talk is try and convey a sense of my enthusiasm for mathematical modelling generally and how I've come to use it in a range of transport applications. For concreteness, I am going to talk in particular about work I have been doing on EPSRC grant EP/K000438/1 (PI: Jillian Anable, Aberdeen) where we are using the UK government's Department for Transport MOT data to estimate mileage totals and study how they are broken down across the population in various different ways. Embedded inside this practical problem is a whole set of miniature mathematical puzzles and challenges which are quite particular to the problem area itself, and one wider question which is rather deeper and more general: whether it is possible (and how) to convert usage data that is low-resolution in time but high-resolution in individuals to knowledge that is high-resolution in time but only expressed at a population level.

Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, non-differentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution. Joint work with Francesco Cellarosi.

I will review some classical problems in number theory concerning the statistical distribution of the primes, square-free numbers and values of the divisor function; for example, fluctuations in the number of primes in short intervals and in arithmetic progressions.  I will then explain how analogues of these problems in the function field setting can be resolved by expressing them in terms of matrix integrals. 

In 1989, Selberg defined what came to be known as the "Selberg class" of $L$-functions, giving rise to a new subfield of analytic number theory in the intervening quarter century. Despite its popularity, a few things have always bugged me about the definition of the Selberg class. I will discuss these nitpicks and describe some modest attempts at overcoming them, with new applications.

Full details are available at: http://www.stcatz.ox.ac.uk/alantayler