This talk presents the Proximal Galerkin (PG) method, a high-order numerical method for solving variational problems with inequality constraints. PG combines two foundational ideas from applied mathematics: Galerkin discretizations of partial differential equations, and Bregman proximal point algorithms for nonsmooth or constrained optimization. Each iteration of the method solves a regularized subproblem formulated as a nonlinear saddle-point system. Conceptually, PG is a discretized gradient flow within a finite-dimensional function space, such as a finite element subspace, yielding robust and convergent solution approximations. The unified framework systematically handles a broad class of variational inequalities, enabling high-order, constraint-preserving solutions without the need for specialized basis functions. This talk will outline the theoretical foundations of PG, highlight its connections to convex analysis, and showcase recent applications in contact mechanics, fracture, and multi-phase flows, among others.

Refreshments will be served preceding the talk, starting at 2:45.