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.

( Video link)

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.

( Video link)

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.

( Video link)

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.

( Video link)

Coxeter friezes are related to closed paths in the Farey graph and triangulated polygons, as well as to the most basic and important examples of cluster algebras. Cluster algebras are certain algebras of rational functions, which actually turn out to consist only of Laurent polynomials ‒ a surprising fact known as the Laurent phenomenon. We will introduce all these objects using an interactive demonstration, outline the connections between them and introduce a surprising phenomenon in cluster algebras: specialising variables in a specific controlled way turns Laurent polynomials into (non-Laurent) polynomials.

( Video link)

A periodic orbit of a Birkhoff billiard is a polygon of extremal perimeter inscribed into the billiard table (a plane oval). One may replace the word "perimeter" by "area" and/or the word "inscribed" by "circumscribed". This provides three other billiard-like systems. Two of them, involving area, can be generalized to symplectic spaces, with the symplectic structure replacing the area form; this gives symplectic inner and outer billiards. I shall discuss properties of the symplectic outer billiards, including periodic orbits and the large scale behavior of its trajectories. I shall also discuss the large scale behavior of the trajectories of the planar outer length billiards. Time permitting, I will relate the inner symplectic billiards with still another billiard-like system, the Minkowski billiards, currently a popular subject of study in symplectic topology.

( Video link)

Renormalization ideas were introduced in dynamics in the late 1970s. By now, renormalization is one of the most important methods of asymptotic analysis in the theory of dynamical systems. This talk serves as an introduction to dynamical renormalization. I'll also discuss closely connected rigidity theory and formulate some open problems. No previous knowledge of renormalization will be assumed.

(Video link)