Georgia Institute of Technology College of Sciences

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.

The emergence of large connected structures in networks has been a central topic in random graph theory since its inception, forming a foundation for understanding fundamental processes such as the spread of influence or epidemics, and the robustness of networked systems. The field witnessed significant growth from the early 2000s, fueled by a surge in experimental work from statistical physics that introduced fascinating concepts such as universality. Broadly speaking, universality suggests that the formation of a giant component in random graphs often depends primarily on macroscopic statistical properties like the degree distribution. In the theoretical literature, two universality classes have emerged, both closely related to Aldous’ seminal work on critical random graphs and the theory of multiplicative coalescents. In this talk, I will present a third universality class that emerges in the setting of percolation on random graphs with infinite-variance degree distributions. The new universality class exhibits fundamentally different behavior compared to multiplicative coalescents and reveals surprising phenomena concerning the width of the critical window—phenomena that were unforeseen in the substantial physics literature on this topic. Based on joint work with Shankar Bhamidi and Remco van der Hofstad.