Mathematics Colloquium Fall 2011

October 4, 2011

The Quantum Chain Complex and Solutions to Plateau's Problem

Professor Jenny Harrison, UC Berkeley 

Plateau's problem asks whether there exists a surface spanning a given closed loop in R^3 with minimal area. Rigorous versions of the problem require devising definitions of surface, loop, span, and area. Jesse Douglas won the first Field's medal for finding solutions for surfaces defined as images of disks. Federer and Fleming found solutions for embedded, orientable surfaces. In his little book, "Plateau's Problem" Almgren claimed to have found general solutions, taking into account all soap films which arise in nature, including moebius strips and surfaces with triple branches. However, the lack of a boundary operator for varifolds results in a serious, probably unfillable gap in his proof. In this colloquium we shall see solutions to Plateau's Problem using entirely new methods. Our "dipole surfaces" provide a simple way to model all soap films so that the boundary operator is well-defined, continuous and yields the original, prescribed curve. Our methods, involving a blend of algebra, analysis and topology, extend to Plateau's Problem in arbitrary dimension and codimension. The machine behind this is the recently discovered "quantum chain complex" which appears to have connections to quantum mechanics.

October 18, 2011

Quantum Walks

Professor Alberto Grunbaum, UC Berkeley 

I will define quantum walks and compare them with the more familiar notion of classical walks. The quantum ones have recently attracted the attention of workers in mathematics, computer science, physics, biology and chemistry.

October 25, 2011

Liouville type theorems for axially symmetric Navier Stokes equations.

Professor Qi Zhang 

After suitable scaling (blow up), a solution of the Navier Stokes equation in a very high velocity region resembles a bounded ancient solution. Thus, the study of ancient solutions is useful to the understanding of the former. One would like to prove that bounded ancient solutions are constants, or at least are simple enough to understand, mirroring the classical Liouville theorem for harmonic functions. We will discuss past results in this direction by G. Koch, N. Nadirashvili, G. Seregin,V. Sverak, and by Chen, Strain, Tsai and Yau. We will also present recent results by Lei and the speaker. One of them claims that there are regions of high velocity where the flow is calm in the sense that there is little oscillation. The other is a Liouville theorem for bounded ancient solutions under the extra assumption that the stream function is bounded or more generally in BMO class. We will also discuss a few open question on weakening the assumptions.

November 1, 2011

Breakdown of self-similarity at the crests of large amplitude standing water waves

Professor Jon Wilkening, UC Berkeley 

Abstract: We study the limiting behavior of large-amplitude standing waves on deep water using high-resolution numerical simulations in double and quadruple precision. While traveling waves are known to approach Stokes' 120 degree corner wave in an asymptotically self-similar manner, standing waves do not appear to approach Penney and Price's conjectured 90 degree corner solution. Instead, a variety of oscillatory structures form near the crest tip, causing the bifurcation curve to fragment into disjoint branches. In many cases, a vertical jet of fluid pushes these structures upward, leading to wave profiles commonly seen in wave tank experiments. Thus, we observe a rich array of dynamic behavior at small length scales in a regime previously thought to be self-similar.

As time permits, I will also discuss issues of stability and resonance, the effect of surface tension, collisions of solitary waves in shallow water, and standing waves in three dimensions. Much of the talk will be devoted to explaining our numerical methods for solving these problems.

November 8, 2011

Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries

Professor Marvin Greenberg, UCSC Emeritus

By "elementary" plane geometry I mean the body of propositions about lines (including segments, rays, angles, triangles, quadrilaterals, polygons) and circles, which can be considered to be the geometry of straightedge and compass constructions. Hilbert's axioms will be stated (plus one axiom he neglected to include) for elementary Euclidean geometry, repairing all the gaps in Euclid'sElements. Replacing the Euclidean parallel postulate with Hilbert's hyperbolic parallel postulate gives us elementary hyperbolic geometry.

The propositions in these two geometries are proved synthetically - i.e., without using any numbers or algebra. Algebra can be extracted from the geometry - not imposed from the outside - by different methods for each of those two geometries, as Hilbert demonstrated. An ordered field in which every positive has a square root (a Euclidean field) is obtained. But that field need not be the field of real numbers; it might be a countable subfield if, in addition, Archimedes' axiom is assumed, or it might be a non-Archimedean field that contains infinitesimals and infinitely large elements.

Hilbert emphasized purity of methods of proof, seeking, as he wrote, "to uncover which axioms, hypotheses or aids are necessary for the proof of a fact in elementary geometry." We carry his investigation further and discover that an ancient axiom of the philosopher Aristotle is a missing link in the foundations of elementary geometry.

November 15, 2011

Turbulent liquid crystals, KPZ universality and the asymmetric simple exclusion process.

Craig Tracy, UC Davis 

We review recent experimental work on stochastically growing interfaces and compare these results with the recent theoretical developments for the Kardar-Parisi-Zhang equation and the closely related asymmetric simple exclusion process. The emphasis in this talk will be on the underlying results in the asymmetric simple exclusion process.

November 22, 2011

The Navier–Stokes Equations

Werner Varnhorn, Kassel University, Germany 

More than 2500 years after the famous statement παντα ρει by Heracleitos the investigation of the mechanical and dynamical behavior of fluid flow is more than ever of fundamental importance. Due to a large number of technical, experimental and computational innovations and related theoretical problems the investigation of fluid flow represents a challenging and exciting subject requiring a wide variety of profound mathematical methods, efficient numerical algorithms and complex experimental simulations. Fascinating from the mathematical point of view, of course, is the fact that the fundamental equations of Navier–Stokes, formulated the first time by the French engineer Navier in 1822, could not be solved in the general three–dimensional case up to now. So the famous American Clay Mathematics Institute created the Navier–Stokes Millennium Price Problem and offered one Million US–Dollar for its solution, stating: „Although the Navier–Stokes equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory, which will unlock the secrets hidden in the Navier–Stokes equations“. The lecture introduces the Navier–Stokes equations from an historical and physical point of view, touches some fundamental mathematical problems of viscous incompressible fluid flow and discusses currently still open questions.

November 29, 2011

Exact expressions for large determinants

Professor Estelle Basor, Deputy Director, American Institute of Mathematics 

The classical Szego limit theorem gives an asymptotic formula for determinants of finite Toeplitz matrices as the size of the matrix tends to infinity. There is also an exact formula for such determinants which yields the Szego formula and can be derived from a determinant identity of Jacobi. This talk will describe some generalizations of the Toeplitz case to matrices which have a Toeplitz plus Hankel type structure and show some examples where the answers are surprisingly simple.