The Staggered Discontinuous Galerkin (SDG) method is a class of finite element methods designed to solve partial differential equations while preserving local conservation properties and handling complex geometries. It employs a staggered mesh structure in which scalar and vector variables are discretized on distinct, yet overlapping, primal and dual meshes. This arrangement facilitates a natural enforcement of conservation laws and enables element-wise postprocessing for superconvergent approximations. The SDG framework supports high-order accuracy and geometric flexibility, making it well-suited for problems involving unstructured or polytopal meshes. Moreover, by decoupling the discretization of variables, the method enhances stability and allows for efficient hybridization, yielding compact global systems and connections to other modern finite element approaches such as hybridizable DG, weak Galerkin, and virtual element methods.

In this talk, we present connections between the stabilization-free polygonal element (SF-PE) and staggered discontinuous Galerkin (SDG) methods. By introducing a gradient reconstruction operator, the SDG method can be reformulated as a class of SF-PE methods. Thanks to this connection and some existing SF-PE methods, we first present a new family of polygonal SDG methods utilizing Raviart-Thomas mixed finite element spaces. The inf-sup stability and optimal convergence are proved. Next, with a simple modification of the loading term we are able to obtain globally $\vH(\div)$-conforming velocity fields. We also present a static condensation procedure for the SDG method for its efficient implementation, which can also be applied to the SF-PE methods due to the presented connection. The theoretical results are verified by numerical experiments.

Joint with Differential Geometry seminar

Gluing constructions of initial data sets play an important role in general relativity. Earlier in 1979, Schoen-Yau used gluing constructions with conformal deformations as a crucial step in their proof of the famous positive mass theorem. Corvino later refined this approach by introducing localized deformations that preserve the manifold’s asymptotic structure.

In this talk, I will survey recent theorems on localized deformation and their applications regarding rigidity and non-rigidity type results. I then outline extensions of these results to manifolds with boundary, including asymptotically flat regions outside black-hole horizons, and conclude with a brief discussion of the analytic challenges that arise in this boundary setting.

The Bethe–Hessian matrix, introduced by Saade, Krzakala, and Zdeborová (2014), is a Hermitian operator tailored for spectral clustering on sparse networks. Unlike the non-symmetric, high-dimensional non-backtracking operator, this matrix is conjectured to reach the Kesten–Stigum threshold in the sparse stochastic block model (SBM), yet a fully rigorous analysis of the method has remained open. Beyond its practical utility, this sparse random matrix exhibits a surprising phenomenon called "one-sided eigenvector localization" that has not been fully explained.

We present the first rigorous analysis of the Bethe–Hessian spectral method for the SBM and partially answer some open questions in Saade, Krzakala, and Zdeborová (2014).  Joint work with Ludovic Stephan.

https://sites.uci.edu/inverse/

Diagonals of multivariate rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. For instance, many hypergeometric functions are diagonals as well as the generating function for Apery's sequence. A natural question is to determine the diagonal grade of a function, i.e., the minimum number of variables one needs to express a given function as a diagonal. The diagonal grade gives the ring of diagonals a filtration. In this talk we study the notion of diagonal grade and the related notion of Hadamard grade (writing functions as the Hadamard product of algebraic functions), resolving questions of Allouche-Mendes France, Melczer, and proving half of a conjecture recently posed by a group of physicists. This work is joint with Andrew Harder.