Superconvergence phenomenon is well understood for the h-version finite element method and researchers in this old field have accumulated a vast literature during the past half century. However, the relevant systematic study for discontinuous Galerkin, finite volume, and spectral methods is lacking. We believe that the scientific community would also benefit from the study of superconvergence phenomenon of those methods. Recently, some efforts have been made to expand the territory of the superconvergence. In this talk, I will summarize some recent development on superconvergence study for these methods. At the same time, some current issues and un-solved problems will also be addressed.

(*) The talk is from 10:30 -11:30 am, in MSC 372.

We investigate the instantaneous and limiting behavior of an n-node blockchain which is under continuous monitoring of the IT department of a company, but faces non-stop cyber attacks from a hacker. The blockchain is functional as far as no data stored on it has been changed, deleted, or locked. Once the IT department detects the attack from the hacker, it will immediately re-set the blockchain, rendering all previous efforts of the hacker in vain. The hacker will not stop until the blockchain is dysfunctional. For arbitrary distributions of the hacking times and detecting times, we derive the limiting functional probability, instantaneous functional probability, and mean functional time of the blockchain. We also show that all these quantities are increasing functions of the number of the nodes, substantiating the intuition that more nodes a blockchain has, the harder it is for a hacker to succeed in a cyber attack.

The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients from algebraic combinatorics. Both appear in representation theory as tensor product multiplicities for a classical Lie group. This talk concerns the question:

Which multiplicities are nonzero?

In 1998, Klyachko established common linear inequalities defining both the eigencone for sums of Hermitian matrices and the saturated Littlewood-Richardson cone. We prove some analogues of Klyachko's nonvanishing results for the Newell-Littlewood numbers.

This is joint work with Shiliang Gao (UIUC), Gidon Orelowitz (UIUC), and Nicolas Ressayre (Universite Claude Bernard Lyon I). The presentation is based on arXiv:2005.09012, arXiv:2009.09904, and arXiv:2107.03152.

Turbulent, disperse two-phase flows are pervasive in nature and industry. In many systems, the disperse phase (e.g., solid particles, liquid droplets, gas bubbles) modifies the turbulence in the carrier phase, giving rise to complicated flow features such as dense clusters (or bubble clouds) and regions nearly void of particles. This heterogeneity predicates a wide range of length- and time-scales, making fully-resolved computations at scales of interest intractable, even on modern super computers. Thus, the Reynolds Averaged Navier--Stokes (RANS) equations, which depend heavily upon modeling, continue to be the primary tool for large-scale computations of both single and multiphase turbulence. Despite their prevalence, developing accurate models, especially for the multiphase RANS equations, has remained a challenge. This is primarily due to the large parameter space characterizing such flows, making brute-force modeling approaches unfeasible and extensions from single-phase turbulence inadequate. In this talk, a few interesting examples of multiphase flows will be highlighted, followed by the introduction of a data-driven methodology based on sparse regression to enable modeling of these peculiar flows.

This talk will present the classical, local Monge-Ampère equation and its applications to optimal transport and differential geometry. We will discuss the degeneracy of the equation and the challenges it poses for the regularity of solutions. Finally, we will consider a nonlocal analog of the Monge-Ampère operator, introduced in collaboration with Luis Caffarelli.

In this talk, we combine two ingredients in order to get a rather surprising result on one of the most studied, elegant and powerful tools for solving convex feasibility problems, the method of alternating projections (MAP). Going back to names such as Kaczmarz and von Neumann, MAP has the ability to track a pair of points realizing minimum distance between two given closed convex sets. Unfortunately, MAP may suffer from arbitrarily slow convergence, and sublinear rates are essentially only surpassed in the presence of some Lipschitzian error bound, which is our first ingredient. The second one is a seemingly unfavorable and unexpected condition, namely, infeasibility. For two non-intersecting closed convex sets satisfying an error bound, we establish finite convergence of MAP. In particular, MAP converges in finitely many steps when applied to a polyhedron and a hyperplane in the case in which they have empty intersection. Moreover, the farther the target sets lie from each other, the fewer are the iterations needed by MAP for finding a best approximation pair. Insightful examples and further theoretical and algorithmic discussions accompany our results, including the investigation of finite termination of other projection methods.

The Space-time (Embedded-)Hybridized Discontinuous Galerkin methods allow for an arbitrarily high order approximation in space and time, even on time-varying domains. Moreover, they are known to be pressure-robust, meaning that the approximation error in the velocity is independent of the pressure. Two essential ingredients are required for pressure-robustness: exact enforcement of the incompressibility constraint and H(div)-conformity of the finite element solution.
In this talk, we present analytical results, and we apply the method for fluid-rigid body interactions. We introduce a sliding grid technique for the rotational movement that can handle arbitrary rotation. The numerical examples will include galloping and fluttering motion.

The theory of cluster algebra is a branch in mathematics emerged in the year 2000, which grows rapidly and has far-reaching implications in many fields including representation theory, geometry, combinatorics, mirror symmetry of string theory, statistical physics, etc. Lots of research of cluster algebras focuses on construction of their natural bases. In this talk, we will study the properties of Newton polytopes and some possibly non-convex regions that contain the support of those basis elements, and illustrate several applications of these properties.