Applied Math Colloquium

The Earth's capacity to support people is determined both by natural constraints and by human choices concerning economics, environment, culture (including values and politics) and demography. Human carrying capacity is therefore dynamic and uncertain. Very simple mathematical models of the relations between human population growth and human carrying capacity can account for faster-than-exponential population growth followed by a slowing population growth rate, as observed in recent human history. Obvious extensions of these models lead to open mathematical questions.

The study of interval maps is a basic topic in dynamical systems, these maps arising in diverse settings such as population genetics and number theory. A central problem in the subject is to decide when two such maps are topologically conjugate. An important invariant is the lap-counting function $L(z)=\sum L_n z^n$, where $L_n$ is the number of monatonic pieces of the $n$th iterate of the map. This function was introduced by Milnor and Thurston, who used it to show that interval maps are semi-conjugate to piecewise linear maps with slopes $\pm s$, where $s$ is the topological entropy.

The linear mod one and test maps are the simplest examples of such maps. For these we obtain a complete description of $L(z)$ are related to the topological dynamical and ergodic properties of the maps.

Finally, linear mod one maps are related to the dynamic of the Lorenz attractor, discovered by Lorenz in his study of weather prediction.

In 1892 J. Hadamard received his doctorat d'etat degree for showing how to calculate the poles of a meromorphic function from its sequence of Taylor coeffcients. This problem has various applications but Hadamard's solution is no good for modern computation. We trace the gradual transformation of his formulae into a powerful robust algorithm. The contributions of Aitken (in the 1920's) and Rutishauser (in the 1950's) will be explained and the talk will end with recent modern developments.

In this talk we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems. I will mention several ways to model graphs geometrically, but our main concern will be with geometric representations that respect the metric of the (possibly weighted) graph. Given a graph we map its vertices to a normed space in an attempt to (i) Keep down the dimension of the host space and (ii) Guarantee a small distortion, i.e., make sure that distances between vertices closely match the distances between their geometric images. I will present the best known (existential and algorithmic) results on this problem and then mention some interesting properties that graphs with good embeddings have.

Much of this work was done jointly with Yuri Rabinovich (Cornell) and Eran London (Jerusalem).

The equations that govern the behavior of physical systems can often be numerically solved using finite-difference (or finite-element) discretizations and differential-algebraic equation (DAE) solvers. For example, such an approach can be used to solve the Einstein field equations of general relativity, and thereby simulate significant astrophysical events. Fluid flow, heat transfer and other problems can also be formulated and numerically solved as DAEs. In each case, the differential equations describe the flow or motion of the physical system, and the algebraic equations describe various conservation laws, boundary conditions or constraints.

Fortunately, high-quality mathematical software for solving DAEs exists (e.g., DASSL) and it can be used to solve a wide array of difficult problems. This talk describes how DAE codes such as DASSL can be enhanced in a way that enables scientists to efficiently solve a larger range of important, multidimensional initial-value problems. In particular, we describe some preliminary work in which the gravitational field of a black hole is computed by solving some simplified versions of the Einstein field equations.

This talk considers the stability of both soliton-like and non-soliton pulses in nonlinear optical fibers where the governing equation is the periodically perturbed nonlinear Schrodinger equation. In particular, an overview of various communications systems is given with emphasis being placed on the dispersion managed nonlinear Schrodinger equation, i.e., the case where the sign of the dispersion changes periodically as a pulse propagates. Here, Floquet theory is used to describe the stability of plane waves which correspond to constant amplitude signals (0's and 1's) and the properties of the Green's function associated with the linear problem is used to describe pulse deformations. The main results include a derivation of the leading order behavior as well as an estimate on the lengthscale of validity.

In their seminal work which initiated random graph theory Erd\"os and R\'enyi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. That is, if every edge of the graph is chosen with probability $p$, then when $p$ increases, the transition from a property being very unlikely to its being very likely is very swift.

We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This result applies to random subgraphs of arbitrary symmetric graphs. We will discuss the relation between the symmetry group and the length of the transition interval.

We consider the problem of describing the "long" time asymptotic behavior for solutions of the one dimensional Euler equations of Gas Dynamics in a bounded domain (say, a tube with rigid walls at the ends, equivalent to a periodic initial conditions situation). Time is measured in terms of an acoustic transit period and "long" typically turns out to mean a few hundred such periods.

Intuitively one would expect that any acoustic component would lead to shocks that will then induce rapid decay into it. Thus, the long time behavior should be described by a "pure entropy wave". This appears to be false. We have constructed examples of solutions with large nontrivial acoustic component and no shocks at all ( these examples are, in fact, periodic in time, thus they have no decay ). Numerical checks of stability for these examples indicate that in fact they are stable. Furthermore, starting from "arbitrary" initial conditions, careful numerical calculations show that the solutions (after an initial "transient" period with shocks) eventually settle down to generally quasiperiodic (in time) solutions with nontrivial acoustical component.

These results and other related ones will be presented. We are also investigating the effects of a small amount of dissipation in the equations. In this case, basically what seems to happen is that, after the "transient" period with shocks, the solution settles down to a quasiperiodic (in time) solution whose parameters slowly drift as the energy is dissipated.

Toric variety methods have recently proven to be extremely useful for dealing with computational problems involving polynomial systems. However, the resulting methods are, at least initially, geared towards affine space minus the coordinate hyperplanes.

We explain a few tricks which allow us to correctly extend toric methods to affine space. In particular, we obtain

• a generalization of the recent stable mixed volume bounds for the number of isolated affine roots of a polynomial system,
• a combinatorial condition for when our root counting formula is exact, and
• a new resultant operator tailored for computations in affine space.

Along the way, we obtain a concrete explanation for how roots "go to infinity" in certain homotopy-based algorithms for solving polynomial systems. This explanation offers an alternative algebraic approach to the recent polyhedral homotopy methods.

The dynamics of complex systems in nature often occurs in terms of punctuations, or avalanches, rather than following a smooth gradual path. Earthquakes, stock market crashes, and mass extinctions in biology are examples of this turbulent behavior. It will be argued that this is a consequence of self-organized criticality, the process by which large dynamical systems evolve to a state with no characteristic time or length scale.

The first "interesting" result in the theory of weak convergence of probability measures is that simple symmetric random walk can be rescaled to converge to Brownian motion. The underlying random walk may be viewed as the uniform distribution on a combinatorial set (binary n-tuples). It turns out that uniform distributions on other combinatorial sets (triangulations, mappings, graphs) have analogous limits constru(triangulations, mappings, graphs) have analogous limits constructible from Brownian-type continuous processes. The limit process describing merging of random graph components in the critical region has a different interpretation as a natural Markov model of coalescence of mass into clusters. Studing how this process starts from the infinite past is a surprisingly subtle problem.