Georgia Institute of Technology College of Sciences

In this talk I will discuss some new properties of an invariant for 4-manifold with boundary which was originally defined by Nobuo Iida. As one of the applications of this new invariant I will demonstrate how one can obstruct a knot from being h-slice (i.e bound a homologically trivial disk)  in 4-manifolds. Also, this invariant can be useful to detect exotic smooth structures of 4-manifolds. This a joint work with Nobuo Iida and Masaki Taniguchi.

Patterns represent the spatial or temporal regularities intrinsic to various phenomena in nature, society, art, and science. From rigid ones with well-defined generative rules to flexible ones implied by unstructured data, patterns can be assigned to a spectrum. On one extreme, patterns are completely described by algebraic systems where each individual pattern is obtained by repeatedly applying simple operations on primitive elements. On the other extreme, patterns are perceived as visual or frequency regularities without any prior knowledge of the underlying mechanisms. In this presentation, we aim at demonstrating some mathematical techniques for representing patterns traversing the aforementioned spectrum, which leads to qualitative analysis of the patterns’ properties and quantitative prediction of the modeled behaviors from various perspectives. We investigate lattice patterns from material science, shape patterns from computer graphics, submanifold patterns encountered in point cloud processing, color perception patterns applied in underwater image processing, dynamic patterns from spatial-temporal data, and low-rank patterns exploited in medical image reconstruction. For different patterns and based on their dependence on structured or unstructured data, we introduce suitable mathematical representations using techniques ranging from group theory to deep neural networks.

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A covering system of the integers is a finite collection of arithmetic progressions whose union is the integers. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked a number of beautiful questions about them. Most famously, his so-called "minimum modulus problem" was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$. 

In this talk I will present a variant of Hough's method, which turns out to be both simpler and more powerful. In particular, I will sketch a short proof of Hough's theorem, and discuss several further applications. I will also discuss a related result, proved using a different method, about the number of minimal covering systems.

Joint work with Paul Balister, Béla Bollobás, Julian Sahasrabudhe and Marius Tiba.

One of interesting topic in low-dimensional topology is to study exotic smooth structures on closed 4-manifolds. In this talk, we will see an example to distinguish exotic smooth structure using H-slice knots.