The gradient of a concave function is discontinuous vector field but has well-defined trajectories. We formulate an existence and forward-uniqueness theorem and its generalisation for non-stationary case. Using the latter we construct trajectories in the minimal action problem and investigate how massive points appear. Their formation simulates the large-scale matter distribution in one of the simplest cosmological models based on the Burgers equation.

There is a well-known relation between ordinary continued fractions and certain matrix products in \(\text{SL}(2,\mathbb{Z})\). There is also a theorem of Schiffler and Canakci that the entries of these matrix products count the perfect matchings on certain planar graphs called ''snake graphs". Together with Musiker, Schiffler, and Zhang, we studied the enumeration of ''\(m\)-dimer covers" on these snake graphs (these are combinatorial generalizations of perfect matchings), and obtained formulas in terms of products of \(\text{SL}(m+1,\mathbb{Z})\) matrices. This led to a definition of ''higher continued fractions". I will discuss these higher continued fractions, their properties, and their combinatorial interpretations (including perfect matchings, lattice paths, plane partitions, and more). Time permitting, I will mention work-in-progress about \(q\)-analogs of these notions.

Measuring the set of simultaneously well approximable points on manifolds is one of the most intricate problems in metric theory of Diophantine approximation. Unlike the dual case of well approximable linear forms, the results here are known to depend on a manifold. For example, some of the manifolds do not contain simultaneously very well approximable points at all, while for the others the set of such points always has positive Hausdorff dimension. In this talk, we will closely look at the Veronese curve \(\{x, x^2, x^3, \ldots, x^n\}\), discuss what is known about the sets of simultaneously well approximable points on it and provide several new results. In particular, for \(n=3\) we provide the Hausdorff dimension of the set of \(x\) such that \(\lambda_3(x) \le \lambda\) where \(\lambda\le \frac25\) or \(\lambda\ge \frac79\).

Conway‐Coxeter friezes are arrays of positive integers satisfying a determinantal condition, the so‐called diamond rule. Recently, these combinatorial objects have been of considerable interest in representation theory, since they encode cluster combinatorics of type A.

In this talk I will discuss a new connection between Conway‐Coxeter friezes and the combinatorics of a resolution of a complex curve singularity: via the beautiful relation between friezes and triangulations of polygons one can relate each frieze to the so‐called lotus of a curve singularity, which was introduced by Popescu‐Pampu. This allows to interpret the entries in the frieze in terms of invariants of the curve singularity, and on the other hand, we can see cluster mutations in terms of the desingularization of the curve. This is joint work with Bernd Schober.

In 2013, Koldobsky posed the problem to find a constant \(d_n\), depending only on the dimension \(n\), such that for any origin-symmetric convex body \(K\subset\mathbb{R}^n\) there exists an \((n-1)\)-dimensional linear subspace \(H\subset\mathbb{R}^n\) with \[ |K\cap\mathbb{Z}^n| \leq d_n\,|K\cap H\cap \mathbb{Z}^n|\,\text{vol}(K)^{\frac 1n}. \] In this article we show that \(d_n\) is bounded from above by \(c\,n^2\,\omega(n)/\log(n)\), where \(c\) is an absolute constant and \(\omega(n)\) is the flatness constant. Due to the recent best known upper bound on \(\omega(n)\) we get a \({c\,n^3\log(n)^2}\) bound on \(d_n\). This improves on former bounds.

(Based on joint works with Ansgar Freyer.)

I'll explain Hermann Grassmann's approach to the geometry of curves. In the mid-1800's, he characterized points on cubics using a clever incidence construction. I'll discuss ways to extend Grassmann's results. In particular, I will explain how to use a straightedge to find the ninth point of intersection of two cubics, given just \(8\) points common to the two curves and one extra point on each cubic.