The field of numerical methods for geometric problems is moving towards the development of more accurate and efficient algorithms for solving partial differential equations on complex surfaces. Researchers are exploring new approaches to discretize geometric quantities, such as curvature, and to integrate functions on surfaces without the need for a mesh. High-order methods are being developed to regularize nearly singular surface integrals, and new techniques are being introduced to solve geometric problems, such as the Hele-Shaw free boundary problem with surface tension. Notable papers in this area include:

• Analysis of the Geometric Heat Flow Equation, which presents a pseudospectral method for computing geodesics in real-time with convergence guarantees.
• High order regularization of nearly singular surface integrals, which derives formulas to regularize integrals with high accuracy.
• Geometric local parameterization for solving Hele-Shaw problems with surface tension, which introduces a novel computational framework for solving the two-dimensional Hele-Shaw free boundary problem.
• High-Order Meshfree Surface Integration, Including Singular Integrands, which develops and tests high-order methods for integration on surface point clouds.