A tropical wave front evolution is a flow on the space of convex domains, pushing domains into rational slope polygons for positive time. When the domain is compact and has strictly convex boundary, the lattice perimeter of the wave front vanishes in the zero time limit. The problem that will be addressed in this talk is determining how fast it vanishes. I will show how a combination of the symplectic toric perspective on tropical wave fronts (developed jointly with Grigory Mikhalkin), the 2/3-residue of tropical zeta function (due to Nikita Kalinin) and the Tauberian-type argument for it (due to Ernesto Lupercio), results in the asymptotic short-time formula for the perimeter of the tropical wave front of a \(C^3\) smooth compact convex domain with the coefficient of the leading term being universally proportional to the affine-arc length

The study of Lagrangian submanifolds up to Hamiltonian isotopy goes back to Arnold, and has received much attention in particular for tori: in spaces such as \(\mathbb{R}^{2n}\) for \(2n \geq 6\), \(\mathbb{C}P^2\), and \(S^2 \times S^2\), there are infinitely many different Hamiltonian isotopy classes. I will describe some of the constructions and ways to distinguish such tori, based on work of Chekanov, Galkin, Mikhalkin, Auroux, Vianna, Brendel, Hind, and myself.

A separating divisor on a real algebraic curve is the zero set of a rational function \(F\) that takes real values only at real points (in other words if \(F(p)\) is real then \(p\) is real). As it follows from a theorem of Ahlfors published in 1950, a real curve admits a separating divisor if and only if the curve itself is separating (i.e. of type I). More recently, M.Kummer and K.Shaw have revitalized the subject by introducing the separating semigroup of a real curve. We review some older and more recent developments in this area.