Georgia Institute of Technology College of Sciences

We investigate how fluctuations behave in large systems of interacting particles when the interaction is given by the Biot–Savart kernel, a key model from fluid dynamics. Our main result provides the first quantitative convergence rates for these fluctuations, and remarkably, the rates are optimal. The key idea is to compare the generators of the particle system and of the limiting fluctuation process in an infinite-dimensional setting. This comparison allows us to derive a sharp error bound for the fluctuations. Beyond the Biot–Savart case, the method is versatile and can also be applied to other singular interactions, such as the repulsive Coulomb kernel.

In this talk, we consider numbers with multiple close factorizations like $99990000 = 9999 \cdot 10000 = 9090 \cdot 11000$ and $3950100 = 1881 \cdot 2100 = 1890 \cdot 2090 = 1900 \cdot 2079$. We discuss optimal bounds on how close these factors can be relative to the size of the original numbers. It is related to the study of close lattice points on smooth curves.

This talk highlights two recent advances in applying the high-order spectral difference (SD) method for computational fluid dynamics on unstructured meshes. The first is a novel curved sliding-mesh technique for the SD method, enabling accurate simulations of rotary-wing aerodynamics. Recent applications include large eddy simulations of marine propellers and ducted wind turbines. The second is the development of a massively parallel code, CHORUS++, designed for Nvidia GPUs to study magnetohydrodynamics in the solar interior. From a computational mathematics standpoint, Dr. Liang also introduced the spectral difference with divergence cleaning (SDDC) algorithm, which addresses the solenoidal constraint of magnetic fields, particularly in the presence of physical boundaries on 3D unstructured grids.

The regularity method for graphs has been well studied for dense graphs, i.e., graphs on $n$ vertices with $\Omega(n^2)$ edges. However, applying it to sparse graphs, i.e., those with $o(n^2)$ edges seems to be a harder problem. In the mid 2010s, the regularity method was extended to dense subgraphs of random graphs thus resolving the KŁR conjecture. Later, in another direction, Conlon, Fox, Sudakov and Zhao proved a removal lemma for $C_5$ in graphs that do not contain any $C_4$ (such graphs on $n$ vertices can contain at most $n^{3/2}$ edges). In this talk, we will consider a similar problem for sparse $3$-uniform hypergraphs. In particular, we consider an application of the regularity method to $3$-uniform hypergraphs whose vertices do not contain $C_4$ in their links and satisfy an additional boundedness condition. This is joint work with Vojtěch Rödl and Mathias Schacht.