Speaker Yongho Park will talk about 'Subspace Correction Methods for Convex Optimization: Algorithms, Theory, and Applications'

This talk considers a framework of subspace correction methods for convex optimization, which provides a unified perspective for the design and analysis of a wide range of iterative methods, including advanced domain decomposition and multigrid methods. We first develop a convergence theory for parallel subspace correction methods based on the observation that these methods can be interpreted as nonlinearly preconditioned gradient descent methods. This viewpoint leads to a simpler and sharper analysis compared with existing approaches. We further show how the theory can be extended to semicoercive and nearly semicoercive problems. In addition, we explore connections between subspace correction methods and other classes of iterative algorithms, such as alternating projection methods, through the lens of convex duality, thereby enabling a unified treatment. Several applications are presented, including nonlinear partial differential equations, variational inequalities, and mathematical imaging problems. The talk concludes with a discussion of relevant and emerging research directions.

The centre of the universal enveloping algebra of a complex semisimple Lie algebra has been understood for a long time since the pioneering work of Harish-Chandra. In contrast, the centres of the equivalent notions in characteristic p are still yet to be computed explicitly. In this talk, Zhenyu Yang and Rick Chen will present an explicit basis for the centre of the restricted enveloping algebra of sl_2, constructed from explicit calculations combined with techniques from non-commutative rings and Morita equivalences. They will then explain how to generalise the argument to compute the centre of the distribution algebra of the second Frobenius kernel of the algebraic group SL_2. This work was part of their summer project under the supervision of Konstantin Ardakov.