A1-homotopy groups,excision, and solvable quotients

We study some properties of ${\mathbb{A}}^1$-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of ${\mathbb{A}}^1$-homotopy theory. These concepts and results are well suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable group quotients, we investigate ${\mathbb{A}}^1$-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree ${\mathbb{A}}^1$-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the “next” non-vanishing ${\mathbb{A}}^1$-homotopy group (beyond ${\pi }_1^{{\mathbb{A}}^1}$) of a smooth toric variety. From this point of view, ${\mathbb{A}}^1$-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost “as tractable” (in low degrees) as ordinary homotopy for large classes of interesting varieties.     

An integral formula for the total mean curvature matrix

The notion of total mean curvature matrix of a submanifold in ${\mathbb{R}}^n$ is defined. A kinematic integral formula for the total mean curvature matrix is proved.     

On the Chromatic Numbers of R2 and R3 with Intervals of Forbidden Distances

In this paper, we discuss new upper bounds for the chromatic numbers of ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$ with intervals of forbidden distances.     

G-exceptional vector bundles on P2 and representations of quivers

It is known that the category of homogeneous bundles on ${\mathbb{P}}^2$ is equivalent to the category of representations of a quiver with relation. In this paper we make use of this equivalence to describe a family of G-exceptional bundles on ${\mathbb{P}}^2$ and to prove that they are stable. We also study the G-exceptionality of Fibonacci bundles on ${\mathbb{P}}^2$.     

Inequalities from Poisson brackets

We introduce the notion of tropicalization for Poisson structures on ${\mathbb{R}}^n$ with coefficients in Laurent polynomials. To such a Poisson structure we associate a polyhedral cone and a constant Poisson bracket on this cone. There is a version of this formalism applicable to ${\mathbb{C}}^n$ viewed as a real Poisson manifold. In this case, the tropicalization gives rise to a completely integrable system with action variables taking values in a polyhedral cone and angle variables spanning a torus.As an example, we consider the canonical Poisson bracket on the dual Poisson–Lie group $G^1$ for $G=U\left(n\right)$ in the cluster coordinates of Fomin–Zelevinsky defined by a certain choice of solid minors. We prove that the corresponding integrable system is isomorphic to the Gelfand–Zeitlin completely integrable system of Guillemin–Sternberg and Flaschka–Ratiu.