# Mathematics Colloquium Spring 2016

For further information please contact Professor Torsten Ehrhardt or call the Mathematics Department at 459-2969

**Tuesday, March 29, 2016**

**No Colloquium**

**Tuesday, April 5, 2016**

Robin Graham, University of Washington and MSRI

*"Ambient metrics and exceptional holonomy"*

Robin Graham, University of Washington and MSRI

The holonomy of a pseudo-Riemannian metric is a subgroup of the orthogonal group which measures the structure preserved by parallel translation. Construction of pseudo-Riemannian metrics whose holonomy is an exceptional Lie group has been of great interest in recent years. This talk will outline a construction of an infinite-dimensional family of metrics in dimension 7 whose holonomy is the split real form of the exceptional group $G_2$. The datum for the construction is a generic real-analytic 2-plane field on a manifold of dimension 5; the metric in dimension 7 arises as the ambient metric of a conformal structure on the 5-manifold defined by Nurowski in terms of the 2-plane field.

**Tuesday, April 12, 2016**

*"Einstein metrics in 4 dimensions"*

Olivier Biquard, Ecole Normal Superieure and MSRI

I will talk about new developments on compactness and singularities of Einstein metrics in 4 dimensions. A large part of the talk will be an introduction to 4-dimensional Einstein metrics, their topological meaning, etc.

**Tuesday, April 19, 2016 *SPECIAL COLLOQUIUM*
**

*"Partial Differential Equation Models of Speciation"*Dan Friedman, Economics Department, University of California, Santa Cruz

*The talk will begin with some general remarks on biologists' theoretical approaches since Darwin on how new species evolve. Then it will sketch a leading current approach, Adaptive Dynamics, that features bivariate calculus models and a simple ODE. The main part of the talk will show how non-linear first order PDE techniques can be deployed to address some open questions concerning speciation.*

**Tuesday,** **April 26, 2016 *SPECIAL COLLOQUIUM***

*"Q-curvature, some survey and recent development"*

**Alice Chang, Princeton and MSRI**

On manifolds of dimension 4, Q curvature is a natural extension of the Gaussian curvature on compact surfaces. In this talk, we will survey some analytic and geometric aspects and report some recent study of Q curvature on manifolds of general dimensions and on CR geometry. We will also discuss the connection of the notion to the scattering theory, the concept of renormalized volume in the ADS/CFT setting and the study of a class of non-local operators in conformal geometry.

**Tuesday, May 3, 2016
**

*"Moduli spaces and gluing"*Jeff Viaclovsky, University of Wisconsin, and MSRI

Geometers are interested in the problem of finding a "best" metric on a manifold. In dimension 2, the best metric is usually one which possesses the most symmetries, such as the round metric on a sphere, or a flat metric on a torus. In higher dimensions, there are many classes of geometrically interesting metrics, such as Einstein metrics, metrics with special holonomy, and extremal Kahler metrics, to name a few. One technique for finding such metrics is a procedure called "gluing", in which one takes known solutions on two different manifolds, attaches them together using some kind of surgery to obtain an "approximate" solution on a new manifold, and then attempts to perturb to an exact solution of the equations on the new manifold. There are obstructions to carrying this out in practice, which can be understood using a fancy version of the implicit function theorem. Gluing techniques are a valuable tool in studying moduli spaces of solutions, because they give an understanding of how solutions can degenerate. I will describe some well-known examples of moduli spaces and gluing techniques, and then discuss some of my work in this area regarding critical metrics on four-manifolds.

**Tuesday, May 10, 2016
**

*"On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity"*Hubert Bray, Duke University

The recent detection of gravitational waves is another spectacular confirmation that the universe is fundamentally geometric, at least on cosmological scales. This raises a natural question: Could dark matter, which makes up most of the mass of galaxies, be fundamentally geometric as well? We'll present a geometric model of dark matter called wave dark matter, also known as scalar field dark matter and Bose-Einstein condensate dark matter, which fits observations about dark matter very well. Using geometric PDE, we'll show how wave dark matter predicts spiral patterns in galaxies, and compare computer simulations with photos of actual galaxies.

**Tuesday, May 17, 2016**

**Daniele Venturi, Applied Mathematics and Statistics Department, **University of California, Santa Cruz

** "Coarse-graining stochastic dynamical systems through the Mori-Zwanzig formulation"
** The Mori-Zwanzig (MZ) formulation is a technique of irreversible statistical mechanics that allows us to formally integrate out an arbitrary number of phase variables in nonlinear dynamical systems, and obtain exact evolution equations for quantities of interest such as macroscopic observables in high-dimensional phase spaces. Computing the solution to such equations, however, is a challenging task that involves approximations and assumptions formulated on a case-by-case basis. In this talk I first will outline the general procedure to derive MZ equations. Then I will address the question approximation by multi-level coarse graining, perturbation series and operator cumulant resummation. Throughout the presentation I will provide numerical examples demonstrating the effectiveness of the MZ approach in prototype stochastic systems, such as Langevin equations driven by colored noise, stochastic advection-reaction and stochastic Burgers equations.

**Tuesday, May 24, 2016**

*"Adaptive Horizon Model Predictive Control"*

**Arthur J. Krener, Naval Postgraduate School**

A fundamental problem in many engineering applications is to find a feedback control law that drives a controlled dynamical system to a desired equilibrium point. As stated the problem is too vague as there could be many feedback laws that accomplish the goal. So standard practice is to recast the problem as an infinite horizon optimal control problem. Find the feedback that minimizes the sum (discrete time) or integral (continuous time) of a running cost from current time to infinity. The running cost from current time to infinity. The running cost is chosen to penalize deviation from the desired equilibrium and excessive use of control. If we can find them then the optimal feedback stabilizes the system to the desired equilibrium and the optimal cost is a Lyapunov function that verifies this stabilization. But finding them is nearly impossible inn state dimensions greater than 2 or 3 because of Bellman's famous curse of dimensionality. So both theorists and practitioners have turned to Model Predictive Control (MPC). Instead of solving the infinite horizon optimal control problem off-line for every possible initial state, MPC solves on-line a finite horizon optimal control problem at the current state. Because the computation is done on-line as the process evolves it must be done quickly. Currently this limits MPC to relatively slow and relatively stable systems as found in the chemical processing industries. The complexity of the on-line finite horizon optimal control problems is largely dictated by the horizon length so one would like to choose it as small as possible consistent with stabilization Adaptive Horizon Model Predictive Control (AHMPC) is a scheme for varying as needed the horizon length of Model Predictive Control (MPC). its goald is to achieve stabilization with horizons as small as possible so that MPC can be used on faster and/or more complicated dynamic processes.

**Tuesday, May 31, 2016**

**Ilan Benjamin, University of California, Santa Cruz**