# Mathematics Colloquium Fall 2008

For further information please contact the Mathematics Department at 459-2969

**September 30, 2008**

**Structural Stability and the Misuse of Mathematics**

**Structural Stability and the Misuse of Mathematics**

**Ralph Abraham (UCSC Mathematics Faculty Emeritus)**

Public policy decisions frequently utilize predictions from mathematical models that are complex dynamical systems. The policy makers and model makers frequently are unaware that complex dynamical systems may be plagued by instabilities due to bifurcations that exist as invisible features. These instabilities make predictions unreliable. In this talk I will present a brief survey of the theory of complex dynamical systems and their bifurcations, giving special attention to global climate models.

**October 7, 2008**

**Hearing the Geometry of Orbifolds**

**Hearing the Geometry of Orbifolds**

**Emily Dryden (Bucknell & MSRI) **

"Can one hear the shape of a drum?" is an informal way of asking about the relationship between the geometry and the vibration frequencies of a domain in the plane. We will discuss what is known in the classical case before moving to the setting of orbifolds. These are spaces with well-behaved singularities, and we explore how much of the singular structure we can hear.

**October 14, 2008**

**Dissipation in Turbulent Flows**

**Dissipation in Turbulent Flows**

**Anna Mazzucato (Penn State and MSRI) **

In Kolmogorov theory of turbulence there is a "cascade" of energy from large to small scales where energy is dissipated by viscosity. In two-dimensional flows (which have been used for example to model geophysical flows if rotation can be neglected), the flow is dominated by the formation of small stable vortices and what cascades to small scales is not energy but enstrophy. Informally, enstrophy can be thought of as the energy associated to vortices. In both cases, there must be a finite rate of dissipation as viscosity vanishes, as observed experimentally and in simulations. Mathematically, this is the case if there are irregular solutions to the Euler equations, modeling inviscid fluid flow, which do not conserve energy in 3D or enstrophy in 2D. We will discuss some results concerning the behavior of enstrophy in 2D Euler solutions, in particular how to reconcile turbulence with the fact Euler solutions conserve enstrophy exactly. This is joint work with Milton Lopes and Helena Nussenzveig Lopes.

**October 21, 2008**

**Quadratic Fourier Analysis**

**Quadratic Fourier Analysis**

**Julia Wolf (MSRI and Rutgers)**

How large can a subset of the integers {1, 2, ..., N} be before it is guaranteed to contain a k-term arithmetic progression? When k=3, a good upper bound can be given using ordinary Fourier analysis. Originally developed by Gowers to give a strong quantitative answer to the case k=4, quadratic Fourier analysis has found numerous applications to related problems in number theory, notably in the work of Green and Tao on progressions in the primes. It turns out that quadratic Fourier analysis has deep connections with ergodic theory and the theory of hypergraphs. We intend to illustrate these connections as we examine some interesting phenomena that arise when one considers a more general class of linear patterns.

**October 28, 2008**

**NO COLLOQUIUM THIS WEEK**

**November 4, 2008**

**NO COLLOQUIUM SCHEDULED **

**November 12, 2008 ***********SPECIAL DAY AND TIME!! WEDNESDAY 4:00 - 5:00 P.M. PHYSICAL SCIENCES LECTURE HALL 114, TEA AT 3:30 P.M. - JBE 301A**

**Mathematics of Voting**

**Mathematics of Voting**

**Don Saari (U.C. Irvine) **

We all know that voting rules experience all sorts of difficulties, e.g., as it will be shown in this lecture, the problems are sufficiently bad that we should seriously worry whether the person elected is who the voters really want. The central issue is to understand why this is so and whether any rule has reliable outcomes. But, of greater interest, voting rules serve as a prototype for a wide class of aggregation rules ranging from statistics, for much of what is done in the social sciences, to even engineering multi scale processes of even nano technology; as such answers could be of wide value. In this lecture, I will outline the mathematical structure of voting rules, and show how these structures completely explain all of those many troubling paradoxes.

**November 18, 2008**

**The enigma of the Navier-Stokes equations: behavior of solutions in the scaling invariant spaces**

**The enigma of the Navier-Stokes equations: behavior of solutions in the scaling invariant spaces**

**Natasa Pavlovic (U.T. Austin) **

The partial differential equations that describe the most fundamental properties of viscous fluids are the Navier-Stokes equations. The theory of the Navier-Stokes equations in 3D is far from being complete. The major open problems are global existence, uniqueness and regularity of smooth solutions of the Navier-Stokes equations in 3D. In this talk we will give a survey of some known results addressing existence and regularity of solutions to Navier-Stokes equations followed by a discussion on behavior of these equations in scaling invariant spaces. In particular, we will briefly describe regularity of so called "mild" solutions to the Navier-Stokes equations evolving from small initial data in a critical space in R^n (joint work with Pierre Germain and Gigliola Staffilani) and a recent result on ill-posedness of the Navier-Stokes equations in the largest critical space in 3D (joint work with Jean Bourgain).

**November 25, 2008**

**The Mathematical Development of Orbital Dynamics in the 17th Century**

**The Mathematical Development of Orbital Dynamics in the 17th Century**

**Michael Nauenberg (UCSC Physics Dept.)**

The "unreasonable effectiveness" of mathematics to describe our physical world is best understood from a historical perspective. In this talk I will discuss some of the early contributions of Newton, Hooke and Huygens to a mathematical description of the motion of planets around the sun, and how their work stimulated the development for the next 200 years of the calculus we use today.