In 1974, M. B. Nathanson proved that every irrational number \(\alpha\) represented by a simple continued fraction with infinitely many elements greater than or equal to \(k\) is approximable by an infinite number of rational numbers \(p/q\) satisfying \(|\alpha-p/q| < 1/(\sqrt{k^2 + 4}q^2) \). In this talk we refine this result.

I will discuss the class of local classification problems, including classification of vector distributions, Riemannian metrics, and real hypersurfaces in \(\mathbb{C}^n\), where the functional dimension of the space of objects is bigger than that of the transformation group, unlike the classification problem of singularity theory where it is not so. I will explain that combining a coordinate-free approach with normal forms gives a nice explanation of known results and many new results.

Integer geometry explores objects whose vertices lie on the integer lattice \(\mathbb{Z}^2\), with congruence defined by lattice-preserving affine transformations. In this project, I introduced remarkable geometric objects called integer circles: discrete analogues of Euclidean circle. These objects challenge our geometric intuition regarding circles. Unlike their classical counterparts, integer circles are unbounded, exhibit nontrivial arithmetic structure, and possess positive density in the plane.

In this talk, I will define integer circles, illustrate their unusual behaviour, and demonstrate how to rigorously compute their densities and intersection patterns.

In 2008, the first formula expressing conditions on the geometric continued fractions for lattice angles of triangles was derived, while the cases of n-gons for \(n > 3\) remained unresolved. In this talk, we introduce an integer geometric analogue to the classical sum of interior angles of a polygon theorem that will act as an extension to the above result in the \(n>3\) cases. I first will frame historical contributions in this area by drawing comparison to their Euclidean counterparts. This will provide background for a simplified overview of the main results for the \(n>3\) case, introducing novel notions in integer geometry such as chord curvature. Finally, I will briefly touch on the consequences of this work within the field of toric singularities.