Quadratic differential systems as an intersection point of Algebra, Topology, Analysis, Riemannian Geometry, Differential Equations,... [ Back ]

14:30 to 15:30
CRM - Aula A1 (C3b/-102)
Zalman Balanov
University of Texas at Dallas


Consider the following 5 problems which, at the first glance, have nothing in common.

PROBLEM 1. As is well-known, any linear differential system $dx/dt =Ax,$ where $A : \mathbb{R}^n \to \mathbb{R}^n$ is a linear operator, has a periodic solution iff
\[(*) \, A \mbox{ has a purely imaginary eigenvalue.}\]
Let $Q : \mathbb{R}^n \to \mathbb{R}^n$ be a (polynomial) quadratic map, i.e. $Q(rx) =r^2Q(x)$ for any real $r.$ What can be a possible generalization of condition $(*)$ providing the quadratic differential system $dx/dt = Q(x)$ to have a periodic or bounded solution?

PROBLEM 2. As is well-known, any $\mathbb{C}$-analytic function $f : \mathbb{C}\to \mathbb{C}$ is a conformal map, in particular, the Jacobian determinant of $f$ is non-negative at any point of $\mathbb{C}.$ Let $g : \mathbb{R}^2 \to \mathbb{R}^2$ be a smooth map. Under which conditions is the Jacobian determinant of $g $ non-negative?

PROBLEM 3. The Intermediate Value Theorem states: for any continuous function $f : [a, b] → \mathbb{R}$ with $f (a)f (b) < 0,$ there exists a point $c\in (a, b)$ such that $f (c) = 0.$ Let $g : D → \mathbb{R}^2$ be a continuous map (here $D$ stands for a disc in $R^2$). How should one generalize the above condition $f(a)f (b) < 0$ to ensure the existence of a point $(c, d)$ such that $g(c, d) = 0?$

PROBLEM 4. The "Fundamental Theorem of Algebra" states: any complex polynomial has a (complex) root. Let $A$ be a two-dimensional vector space equipped with a bilinear commutative multiplication. To what extent should be the multiplication in $A$ close to the one in $\mathbb{C}$ to ensure that a "reasonable" polynomial equation in $A$ has a (non-zero) root?

PROBLEM 5. The role of the first and second fundamental forms in studying two-dimensional surfaces imbedded into $\mathbb{R}^3$ is well-known. Given such a surface $S,$ can one equip tangent planes to points in $S$ with easy to study algebraic structures in such a way that important geometric properties of S can be determined by "smooth" deformations of these structures?

The main goal of my talk is to link Problems 1 - 5 (coming from ODEs, Topology, Analysis, Algebra and Riemannian Geometry, respectively) by explicitly indicating their common root. The obtained general result will be discussed in the context relevant to the existence of bounded and periodic solutions of quadratic differential systems of natural phenomena: Volterra equations related to population dynamics, Aris models of second order chemical reactions, Euler equations of rotating rigid bodies, and Kasner equations related to general relativity theory in vacuum (this is a joint work with Yakov Krasnov).