Optimisation in 1D is far simpler than multidimensional optimisation and this is largely due to the notion of a bracket. A bracket is a trio of points such that the middle point is the one with the smallest objective function value (of the three). The existence of a bracket is sufficient to guarantee that a continuous function has a local minimum within the bracket. The most stable 1D optimisation methods, such as Golden Section or Brent's Method, make use of this fact. The mentality behind these methods is to maintain a bracket at all times, all the while finding smaller brackets until the local minimum can be guaranteed to lie within a sufficiently small range. For smooth functions, Brent's method in particular converges quickly with a minimum of function evaluations required. However, when applied to a piece-wise smooth functions, it achieves its realistic worst case convergence rate. In this presentation, I will present a new method which uses ideas from Brent and Golden Section, while being designed to converge quickly for piece-wise smooth functions.

In this talk I discuss how we can simulate stochastic chemical kinetics when there is a memory component. This can occur when there is spatial crowding within a cell or part of a cell, which acts to constrain the motion of the molecules which then in turn changes the dynamics of the chemistry. The counterpart of the Law of Mass Action in this setting is through replacing the first derivative in the ODE description of the Law of Mass Action by a time-­fractional derivative, where the time-­fractional index is between 0 and 1. There has been much discussion in the literature, some of it wrong, as to how we model and simulate stochastic chemical kinetics in the setting of a spatially-­constrained domain – this is sometimes called anomalous diffusion kinetics.

In this presentation, I discuss some of these issues and then present two (equivalent) ways of simulating fractional stochastic chemical kinetics. The key here is to either replace the exponential waiting time used in Gillespie’s SSA by Mittag-­Leffler waiting times (MacNamara et al. [2]), which have longer tails than in the exponential case. The other approach is to use some theory developed by Jahnke and Huisinga [1] who are able towrite down the underlying probability density function for any set of mono-­molecular chemical reactions (under the standard Law of Mass Action) as a convolution of either binomial probability density functions or binomial and Poisson probability density functions). We can then extend the Jahnke and Huisinga formulation through the concept of iterated Brownian Motion paths to produce exact simulations of the underlying fractional stochastic chemical process. We demonstrate the equivalence of these two approaches through simulations and also by computing the probability density function of the underlying fractional stochastic process, as described by the fractional chemical master equation whose solution is the Mittag-­Lefflermatrix function. This is computed based on a clever algorithm for computing matrix functions by Cauchy contours (Weideman and Trefethen [3]).

This is joint work with Manuel Barrio (University of Vallodolid, Spain), Kevin Burrage (QUT), Andre Leier (University of Alabama), Shev MacNamara(University of Technology Sydney)and T. Marquez-­Lago (University of Alabama).

[1]T. Jahnke and W. Huisinga, 2007, Solving the chemical master equation for monomolecular reaction systems analytically, J. Math. Biology 54, 1, 1—26.[2]S. MacNamara, B. Henry and W. McLean, 2017, Fractional Euler limits and their applications, SIAM J. Appl. Math. 77, 2, 447—469.[3]J.A.C. Weideman and L.N. Trefethen, 2007, Parabolic and hyperbolic contours for computing the Bromwich integral, Math. Comp. 76, 1341—1356.

 Massless scattering amplitudes in four-dimensional Minkowski spacetime can be Mellin transformed to correlation functions on the celestial sphere at null infinity called celestial amplitudes. We study various properties of massless four-point scalar and gluon celestial amplitudes such as conformal partial wave decomposition, crossing relations and optical theorem. As a byproduct, we derive the analog of the single and double soft limits for all gluon celestial amplitudes.

We report on a line of work initiated by Dan-Cohen and Wewers and continued by Dan-Cohen and the speaker to explicitly compute the zero loci arising in Kim's non-abelian Chabauty's method. We explain how this works, an important step of which is to compute bases of a certain motivic Hopf algebra in low degrees. We will summarize recent work by Dan-Cohen and the speaker, extending previous computations to $\mathbb{Z}[1/3]$ and proposing a general algorithm for solving the unit equation. Many of the methods in the more recent work are inspired by recent ideas of Francis Brown. Finally, we indicate future work, in which we hope to use elliptic motivic periods to explicitly compute points on punctured elliptic curves and beyond.

We compute the asymptotics of Arakelov functions if smooth curves degenerate to semistable singular curves. The motivation was to determine whether the delta function defines a metric on the boundary of moduli space. In fact things are slightly more complicated. The main result states that the asymptotics is mostly governed by the graph associated to the degeneration, with some subleties. The topic has been also treated by R. deJong and my student R. Wilms.

For a family of proper hyperbolic complex curves $f: X \longrightarrow \Delta^*$ over a puntured disc $\Delta^*$ with semistable reduction at the center, Oda proved, with transcendental methods, that the outer monodromy action of $\pi_1(\Delta^*) \cong \mathbb{Z}$ on the classical unipotent fundamental group of the generic fiber of $f$ is trivial if and only if $f$ has good reduction at the center. In this talk I explain a joint work with B. Chiarellotto and A. Shiho in which we give a purely algebraic proof of Oda's result.