Recent work has developed a broad generalization of the double-reduction method by considering the space of invariant conservation laws with respect to a given symmetry. In its simplest formulation, the generalization is able to reduce a nonlinear PDE in $2$ variables to an ODE with $m$ first integrals where $m$ is the dimension of the space of invariant conservation laws. Nonlinear PDEs in $3$ or more variables can be reduced to an ODE similarly by using an algebra of given symmetries. Importantly, the algebra does not need to be solvable.

The general method employs multipliers and is fully algorithmic. In particular, no a priori knowledge of conservation laws of the nonlinear PDE is necessary, and the multi-reduction is carried out in one step.

In this talk, a summary of the general multi-reduction method will be presented for obtaining invariant solutions of physically interesting PDEs. Examples will be shown for quadruple reduction from a single symmetry; complete integration from a solvable algebra in one step; reduction via a non-solvable algebra.

Cellular adhesion is one of the most important interaction forces between cells and other tissue components. In 2006, Armstrong, Painter and Sherratt introduced a non-local PDE model for cellular adhesion, which was able to describe known experimental results on cell sorting and cancer growth. The analysis becomes challenging through non-local cell-cell interaction and interactions with boundaries. In this talk I will use symmetry methods to analyse aggregations and pattern formation of the non-local adhesion model. (joint work with A. Buttenschoen).

We consider $u(t,x)$ as an octonion variable in evolution equations $u_t = F(u,u_x,u_{xx},u_{xxx})$, and we aim to find a Lax pair $L_t = [M,L]$ where $L$ and $M$ are linear differential operators in terms of $\partial_x$ with coefficients involving $u$ and $x$-derivatives of $u$. For $F$, we assume it is homogeneous under a scaling of $t,x,u$ which is either the scaling in the KdV equation or the mKdV equation. This gives a polynomial ansatz with undetermined (real) coefficients. Similarly, for $L$ and $M$, we assume they are scaling homogeneous, where the scaling weight of $M$ is the same as that of $\partial_t$ while the scaling weight of $L$ can be chosen freely.

The determining condition is $\big(L_t -[M,L]\big)\big|_{u_t = F} =0$. We split this condition in the jet space of $u$, and do a further splitting with respect to a real basis (8-dimensional) for the octonions. This gives a large overdetermined system in the undetermined (real) coefficients in ansatz for $F$, $L$, $M$. We use Maple to do the splittings, and depending on the complexity of the system, we solve it using 'rifsimp' in Maple or a package called 'Crack' in Reduce.

As a main result, we obtain a single KdV octonion equation, three mKdV octonion equations, and also a single potential-KdV octonion equation, each of which has more than one Lax pair.

Related questions concern those about the existence of mappings, e.g., of a given PDE to a more tractable one (e.g. nonlinear to linear), following work initiated by Bluman and Kumei. Other interesting algorithmic problems concern the computation of symmetry information for classes of differential equations.

Central to the above problems is the reduction of systems of differential equations to involutive forms that include their integrability conditions by sequences of prolongations and projections in Jet space. I will discuss developments in this area, including work by the speaker and collaborators.

PDE in applications often contains parameters and functions that are only known approximately. Thus, approximate versions of the above problems are of interest. For example, given a system R, when is R close to some R* with desirable symmetry properties (e.g. R* has a large symmetry group). Several approaches to this problem will also be discussed: exact and numerical. Numerical methods, including those from the new area of Numerical Jet Geometry, will also be discussed, including methods for DAE.

In the physical world, many systems are subject to stochastic perturbations. The latter are often responsible for removing the symmetries of the ideal unperturbed systems. However, within the class of Hamiltonian systems, the framework of J-M. Bismut (``Mecanique Aleatoire", Springer 1981) permits stochastic inputs while preserving the Hamiltonian structure and, under appropriate constraints, some or even all of the symmetries of the deterministic system. Developed further in the last decades, the theory of stochastic geometric mechanics displays many interesting features and open questions. In this talk I will present the case study of two super-integrable systems: the two-dimensional harmonic oscillator and the Kepler problem.

This work is joint with Archishman Saha (University of Ottawa).

This talk describes the first step of computing polynomial identities for octonions and their use in generating a minimal ansatz for general homogeneous polynomials L and M.

The computational complexity and methods to overcome it are discussed.