Brown University

The proximal Galerkin finite element method is a high-order, nonlinear numerical method that preserves the geometric and algebraic structure of bound constraints in infinitedimensional function spaces. In this talk, we will introduce the proximal Galerkin method and apply it to solve free-boundary problems, enforce discrete maximum principles, and develop scalable, mesh-independent algorithms for optimal design. The proximal Galerkin framework is a natural consequence of the latent variable proximal point (LVPP) method, which is an stable and robust alternative to the interior point method that will also be introduced in this talk.

In particular, LVPP is a low-iteration complexity, infinite-dimensional optimization algorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout the talk, we will arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and an infinite-dimensional Lie group; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization.

The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. This talk is based on [1].

Keywords: pointwise bound constraints, bound-preserving discretization, entropy regularization, proximal point

Mathematics Subject Classifications (2010): 49M37, 65K15, 65N30

References  [1] B. Keith, T.M. Surowiec. Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints arXiv preprint arXiv:2307.12444 2023.

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Reaction–diffusion equations are widely used to model spatial propagation, and constant-speed "traveling waves" play a central role in their dynamics. These waves are well understood in "essentially 1D" domains like cylinders, but much less is known about waves with noncompact transverse structure. In this direction, we will consider traveling waves of the KPP reaction–diffusion equation in the Dirichlet half-space. We will see that minimal-speed waves are unique (unlike faster waves) and exhibit curious asymptotics. The arguments rest on potential theory, the maximum principle, and a powerful connection with the probabilistic system known as branching Brownian motion.

This is joint work with Julien Berestycki, Yujin H. Kim, and Bastien Mallein.

The recent curation of large-scale databases with 3D surface scans of shapes has motivated the development of tools that better detect global patterns in morphological variation. Studies which focus on identifying differences between shapes have been limited to simple pairwise comparisons and rely on pre-specified landmarks (that are often known). In this talk, we present SINATRA: a machine learning pipeline for analyzing collections of shapes without requiring any correspondences. Our method takes in two classes of shapes and highlights the physical features that best describe the variation between them. We develop a rigorous simulation framework to assess our approach, which themselves are a novel contribution to 3D image and shape analyses. Lastly, as case studies with real data, we use SINATRA to (1) analyze mandibular molars from four different suborders of primates and (2) facilitate the visual identification of structural signatures differentiating between the trajectories of two protein ensembles resulting from molecular dynamics simulations. Together, these results highlight a promising future for interpretable machine learning to facilitate the non-trivial task of pattern recognition in evolutionary and structural biology with substantially increased resolution.

Active matter systems, ranging from liquid crystals to populations of cells and animals, exhibit complex collective behavior characterized by pattern formation and dynamic phase transitions. However, quantitative analysis of these systems is challenging, especially for heterogeneous populations of varying sizes, and typically requires expertise in formulating problem-specific order parameters. I will describe an alternative approach, using a combination of topological data analysis and machine learning, to investigate emergent behaviors in self-organizing populations of interacting discrete agents.

Understanding the motion of small bodies at a fluid interface has relevance to a range of natural systems and technological applications. In this talk, we discuss two systems where capillarity and fluid inertia govern the dynamics of millimetric particles at a fluid interface.

In the first part, we present a study of superhydrophobic spheres impacting a quiescent water bath.  Under certain conditions particles may rebound completely from the interface - an outcome we characterize in detail through a synthesis of experiments, modeling, and direct numerical simulation.  In the second half, we introduce a system wherein millimetric disks trapped at a fluid interface are vertically oscillated and spontaneously self-propel.  Such "capillary surfers" interact with each other via their collective wavefield and self-assemble into a myriad of cooperative dynamic states.  Our experimental observations are well captured by a first theoretical model for their dynamics, laying the foundation for future investigations of this highly tunable active system.

The finite element exterior calculus is a powerful approach to study many problems under the same lens. The canonical finite element spaces (see Arnold, Falk and Winther) are tied together with an exact sequence and have the required smoothness to define the exterior derivatives weakly. However, some applications require spaces that are more smooth (e.g. plate bending problems, incompressible flows). In this talk we will discuss some recent results in developing finite element spaceson simplicial triangulations with more smoothness, that also fit in an exact sequence. This is joint work with Guosheng Fu, Anna Lischke and Michael Neilan.

It has been observed long time ago (by many people) that singular affine symplectic varieties come in pairs; that is often to an affine singular symplectic variety $X$ one can associate a dual variety $X^!$; the geometries of $X$ and $X^!$ (and their quantizations) are related in a non-trivial way. The purpose of the talk will be 3-fold:

1) Explain a set of conjectures of Braden, Licata, Proudfoot and Webster which provide an exact formulation of the relationship between $X$ and $X^!$

2) Present a list of examples of symplectically dual pairs (some of them are very recent); in particular, we shall explain how the symplectic duals to Nakajima quiver varieties look like.

3) Give a new approach to the construction of $X^!$ and a proof of the conjectures from part 1).

The talk is based on a work in progress with Finkelberg and Nakajima.

Let F be a global field and let A denote its adele ring. The usual Tamagawa number formula computes the (suitably normalized) volume of the quotient G(A)/G(F) in terms of values of the zeta-function of F at the exponents of G; here G is simply connected semi-simple group. When F is functional field, this computation is closely related to the Atiyah-Bott computation of the cohomology of the moduli space of G-bundles on a smooth projective curve.

I am going to present a (somewhat indirect) generalization of the Tamagawa formula to the case when G is an affine Kac-Moody group and F is a functional fiend. Surprisingly, the proof heavily uses the so called Macdonald constant term identity. We are going to discuss possible (conjectural) geometric interpretations of this formula (related to moduli spaces of bundles on surfaces).

This is joint work with D.Kazhdan.

We will start with the square torus, move on to all regular polygons, and then look at a large family of flat surfaces called Bouw-Möller surfaces, made by gluing together many polygons. On each surface, we will consider the action of a certain shearing action on geodesic paths on the surface, and a certain corresponding sequence.