Ever since Yao introduced the communication model in 1979, it has played
a pivotal role in our understanding of lower bounds for a wide variety of
problems in Computer Science. In this talk, I will present the lifting method,
whereby communication lower bounds are obtained by ``lifting'' much simpler
lower bounds. I will present several lifting theorems that we have obtained and
explain what makes the exciting/useful but also difficult to prove. Finally I will
highlight how they have been used to solve several open problems in
circuit complexity, proof complexity, optimization, cryptography, game theory
and privacy.

High-order numerical methods promise higher fidelity and more predictive power when compared with traditional low-order methods. Furthermore, many properties of these methods make them well-suited for modern computer architectures. However, the use of these methods also introduces several new challenges. For example, in the time-dependent setting, high-order spatial discretizations can result in severe time step restrictions, motivating the use of implicit solvers. The resulting systems are large and often ill-conditioned, posing a challenge for traditional solvers. In this talk, I will discuss the development of efficient solvers and preconditioners designed specifically for high-order finite element and discontinuous Galerkin methods. An entropy-stable sparse line-based discretization will be developed to make these methods suitable for use on GPU- and accelerator-based architectures. These methods will then be applied to relevant problems in compressible flow.

The shrinking target problem for a dynamical system tries to answer the question of how fast can a sequence of targets shrink so that a typical orbit will keep hitting them indefinitely. I will describe some new and old results on this problem for flows on homogenous spaces, with various applications to problems in Diophantine approximations.

Monotone systems are of great interest for their numerous applications and close connections to many physical and biological systems. In linear spaces, a local characterisation of monotonicity is provided by differential positivity with respect to a constant cone field, which combines positivity theory with a local analysis of nonlinear dynamics. Since many dynamical systems are naturally defined on nonlinear spaces, it is important to develop the concept on such spaces. The question of how to define monotonicity on a nonlinear manifold is complicated by the absence of a general and well-defined notion of order in such settings. Fortunately, for Lie groups and important examples of homogeneous spaces that are ubiquitous in many problems of engineering and applied mathematics, symmetry provides a way forward. Specifically, the existence of a notion of geometric invariance on such spaces allows for the generation of invariant cone fields, which in turn induce conal orders. We propose differential positivity with respect to invariant cone fields as a natural and powerful generalisation of monotonicity to nonlinear spaces. We illustrate the key concepts with examples from consensus theory on Lie groups and operator theory on the set of positive definite matrices.

Consider a kind of generalized Nash equilibrium problems (GNEPs) whose objective functions are polynomials, and the constraints can be represented by polynomial equalities and inequalities. Gauss-Seidel typed Approach is one kind of natural easy implemented method to solve GNEP. We study some properties for this approach, and give out a computable criterion for Generalized Potential game (GPGs), the condition under which the convergence of this approach could be guaranteed.