# Mathematics Colloquium Winter 2012

Tuesday - 4:00 p.m.
Refreshments served at 3:30 in Room 4130

### Professor Nancy Rodriguez Stanford University

In recent years modeling biological aggregation, which can be observed in many ecological systems, has become of interest. A competing phenomena is that of a species' desire for personal space, or dispersal. I will discuss a general class of systems that model this competition. These general class of models include the well-known Patlak-Keller-Segel (PKS) model for chemotaxis. I will first introduce the PKS model and discuss some of its history. I will then introduce the more general class of models and discuss what happens when dispersal overcomes aggregation, when aggregation overcomes dispersal, and when they are in balance.

### Professor Richard Montgomery University of California, Santa Cruz

Two polynomials P,Q in two variables define a system of polynomial ODEs dx/dt = P (x,y ), dy/dt = Q (x, y) in the plane. The system dynamics changes radically when we replace the real field by the complex field. Over the reals, the Poincare Bendixson theory asserts that the limiting behaviour (omega limit sets) of solutions are curves or points. However, when x,y and time t are complex then (for typical P,Q ) the typical solution is dense in the plane C2 ! (Thus the generic omega limit set is the entire space of two complex variables.)

I will give the proof of this surprising fact for a special class of P,Q s. You can read a sketch on mathoverflow given by Robert Bryant in the course of answering that question of mine posted last week (January 17, 2012).

I will end with motivation for asking that question, motivation which came (as usual) from the three-body problem. The general theory for this phenomenon began with Ilyashenko, 1968 and, time permitting, I will touch on his work, as explained to me recently by Xavier Gomez Mont of CIMAT, through his text Sistemas Dinamicos Holomorfos en Superfcies.

### Cameron Franc, Lecturer/Postdoctoral Scholar University of California, Santa Cruz

In this talk we will discuss the phenomenon of p‐adic interpolation and its role in number theory. We will begin from basics by recalling the definition of p‐adic numbers. As motivation for their study we will explain several local‐to‐global results. We will then discuss the curious phenomenon that many naturally occurring sequences of rational numbers of arithmetic interest exhibit p‐adic continuity. This allows one to interpolate such a sequence into a continuous function of a p‐adic variable. We will explain a few examples and applications of this phenomenon and then conclude with a brief look at how interpolation can be used to define p‐adic L‐functions.

### Stanford University

Many popular multivariate methods based on spectral decompositions of distance methods: Multidimensional Scaling, kernel PCA, correspondence analysis, Metric MDS aim to detect hidden underlying structure of points in high dimensions. A first type of dependence is a hidden gradient, placing points close to a curve in high dimensional space. Ecologists, archeologists have long known to look for horseshoes or arches which are symptomatic of such structure. I will provide a theoretical analysis of simple models where a hidden gradient or gradients give the eigenfunctions of the associated kernels specific forms. Examples will be provided from Voting Patterns to Genetics. This talk contains joint work with Omar dela Cruz, Sharad Goel and Persi Diaconis.

### Professor Simon Brendle Stanford University

A famous classical theorem due to Alexandrov asserts that the only embedded surfaces in R^n with constant mean curvature are the round spheres. I will discuss similar uniqueness theorems for constant mean curvature surfaces when the ambient space is a Riemannian manifold with rotational symmetry. In particular, our results apply to the Schwarzschild and Schwarzschild-deSitter manifolds, which are of interest in general relativity.

### Professor Maria del Mar Gonzalez Universidad Politecnica de Catalunya

We analyze the global existence of classical solutions to the initial boundary value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system, and present a special type of nonlinearity that is surprisingly related to the classical Stefan problem on phase transitions.

### Professor Phillip Colella Lawrence Berkeley National Laboratory

Many important problems for DOE such as combustion, fusion, systems biology, and climate change, involve multiple physical processes operating on multiple space and time scales. In spite of the physical diversity of these problems, there is a great deal of coherence in the underlying mathematical representations. They are all described in terms of various versions of the elliptic, parabolic and hyperbolic partial differential equations (PDE) of classical mathematical physics. The enormous variety and subtlety in these applications comes from the way the PDE are coupled, generalized, and combined with models for other physical processes. The complexity of these models and the need to represent multiple scales lead to a diverse collection of requirements on the numerical methods, with many open questions about stability of coupled algorithms. Finally, the complexity of models and algorithms, combined with uncertainties about the correct combination to use, complicates the problem of designing high performance software. In this talk, I will attempt to describe the tradeoffs between the models, the discretizations, and the software in the development of high-performance computational simulations in science and engineering involving PDE, including some motivating applications, and the combination of analysis and computational experiments that are used to explore the design space.

### Professor Richard Bamler Stanford University

Recently, Perelman established the Poincarέ and Geometrization Conjecture using Ricci flow with surgery. In this flow, singularities are being removed by a certain surgery procedure on a discrete set of time slices. Despite the depth of the result, a precise description of the long-time behavior of the Ricci flow with surgery however still does not exist. For example, it is unknown whether surgeries eventually stop to occur after some finite time and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$. In this talk, I will first give a short introduction to Perelman's work and related results. I will then present new tools to analyze the long-time behavior of the Ricci flow with surgery. In particular, I will show that under the pure topological condition that the initial manifold has only hyperbolic components in its geometric decomposition (i.e. prime and torus-decomposition), surgeries do in fact stop to occur after some time and the curvature is globally bounded by $C t^{-1}$. Finally, I will explain how to treat more general cases.

### Professor Michael Shearer North Carolina State University

Plane waves for two phase ow in a porous medium are modeled by the one-dimensional Buckley- Leverett equation, a scalar conservation law. In the rst part of the talk, we study traveling wave solutions of the equation modied by the Gray-Hassanizadeh model for rate-dependent capillary pres- sure. The modication adds a BBM-type dispersion to the classic equation, giving rise to under- compressive waves. In the second part of the talk, we analyze stability of sharp planar interfaces (corresponding to Lax shocks) to two-dimensional perturbations, which involves a system of partial dierential equations. The Saman-Taylor analysis predicts instability of planar fronts, but their calculation lacks the dependence on saturations in the Buckley-Leverett equation. Interestingly, the dispersion relation we derive leads to the conclusion that some interfaces are long-wave stable and some are not. Numerical simulations of the full nonlinear system of equations, including dissipation and dispersion, verify the stability predictions at the hyperbolic level. This is joint work with Kim Spayd and Zhengzheng Hu.