# Seminars & Events for PACM/Applied Mathematics Colloquium

##### Self-organized selectivity in Calcium and Sodium Channels: important biology ready for mathematical analysis

Ion channels are irresistible objects for biological study because they are the [nano] valves of life controlling an enormous range of biological function, much as transistors control computers. Ion channels are appealing objects for physical investigation because conformation changes are not involved in channel function, once the channel is open. Open channels are interesting objects for chemical study because they effectively select among chemically similar ions, under unfavorable circumstances. Channels are interesting objects for physical study because they contain an enormous density of charge, fixed, mobile, and induced.

##### Looking over the painter’s shoulder

Just microns below their paint surface lies a wealth of information on Old Master Paintings. Hidden layers can include the underdrawing, the underpainting or compositional alterations by the artist. All too often artists simply re-used their canvases and painted a new composition on top. Thus, a look *through* the paint layer provides a look *over* the painter’s shoulder. I will discuss case different subsurface imaging techniques and present case studies from the work of Vincent van Gogh and Rembrandt.

##### Dynamics of renormalization operators

It is a remarkable characteristic of some classes of low-dimensional dynamical systems that their long time behavior at a short spatial scale is described by an induced dynamical system in the same class. The renormalization operator that relates the original and the induced transformations can then be iterated. We will discuss how features (such as hyperbolicity) of this "dynamics in parameter space" impact the underlying systems, especially in the case of typical parameters.

##### A Database Schema for the Global Dynamics of Multiparameter Nonlinear Systems

Prof. Mischaikow will discuss new computational tools based on topological methods that extracts coarse, but rigorous, combinatorial descriptions of global dynamics of multiparameter nonlinear systems. This techniques are motivated by several observations which we claim can, at least in part, be addressed:

1. In many applications there are models for the dynamics, but specific parameters are unknown or not directly computable. To identify the parameters one needs to be able to match dynamics produced by the model against that which is observed experimentally.

2. It is well established that nonlinear dynamical systems can produce extremely complicated dynamics, e.g. chaos, however experimental measurements are often too crude to identify such fine structure.

##### Detection of Faint Edges in Noisy Images

One of the most intensively studied problems in image processing concerns how to detect edges in images. Edges are important since they mark the locations of discontinuities in depth, surface orientation, or reflectance, and their detection can facilitate a variety of applications including image segmentation and object recognition. Accurate detection of faint, low-contrast edges in noisy images is challenging. Optimal detection of such edges can potentially be achieved if we use filters that match the shapes, lengths, and orientations of the sought edges. This however requires search in the space of continuous curves. In this talk we explore the limits of detectability, taking into account the lengths of edges and their combinatorics. We further construct two efficient multi-level algorithms for edge detection.

##### PACM Student Colloquium

**Mihai Cucuringu**: "Sensor network localization by eigenvector synchronization over the Euclidean group"

**Lin Lin**: "Nuclear quantum effects of hydrogen bonded systems"

**Jesus Puente**: "Surface comparison with mass transportation"

**Michael Sekora**: "A Hybrid Godunov Method for Radiation Hydrodynamics"

##### Combinatorial Phase Transition

The past fifteen years have seen a huge boom in work at the interface of statistical physics, combinatorics, probability, and the theory of computing. A unifying objective has been understanding phase transition, especially in discrete models with hard constraints. We will give some indication of why the notion is so interesting to diverse groups of researchers, and some examples where there has been recent progress.

##### Internet Traffic Matrices and Compressive Sensing

Internet traffic matrices (TMs) specify the traffic volumes between origins and destinations in a network over some time period. For example, origins and destinations can be individual IP addresses, prefixes, routers, points-of-presence (PoPs), or entire networks or Autonomous Systems (ASes). Past work on TMs has almost exclusively focused on large ASes such as AS7018 (AT&T) and their router- or PoP-level TMs, mainly because the latter are critical inputs to many basic network engineering tasks, and the thrust of much of this work has been on measurement and inference of TMs. A key remaining challenge in this area is how to cope with missing values that frequently arise in real-world TMs.

##### Toward practical rare event simulation in high dimensions

Prof. Weare will discuss an importance sampling method for certain rare event problems involving small noise diffusions. Standard Monte Carlo schemes for these problems behave exponentially poorly in the small noise limit. Previous work in rare event simulation has focused on developing, in specific situations, estimators with optimal exponential variance decay rates. He will introduce an estimator related to a deterministic control problem that not only has an optimal variance decay rate under certain conditions, but that can even have vanishingly small statistical relative error in the small noise limit. The method can be seen as the limit of a well known zero variance importance sampling scheme for diffusions which requires the solution of a second order partial differential equation.

##### Sensor Registration and Synchronisation in Networks

An important problem in distributed and networked sensing is registration of coordinate systems or synchronisation of clocks across the network. The main problem discussed in this talk is as follows. We have a network of sensors each with its own local coordinate system. We are given noisy measurements of the transformations connecting the coordinate systems of certain pairs of sensor in the network. The goal is to find local algorithms which will, in a optimal statistical sense, align the coordinate systems across the network. The talk will show how this problem can be formulated as a gauge invariant statistical estimation problem, how it is related to the homology of graphs, and how statistical optimal local estimators can be constructed in a number of cases.

##### Hydraulic Fractures: multiscale phenomena, asymptotic and numerical solutions

Hydraulic fractures (HF) are a class of tensile fractures that propagate in brittle materials by the injection of a pressurized viscous fluid. In this talk I provide examples of natural HF and situations in which HF are used in industrial problems. Natural examples of HF include the formation of dykes by the intrusion of pressurized magma from deep chambers. They are also used in a multiplicity of engineering applications, including: the deliberate formation of fracture surfaces in granite quarries; waste disposal; remediation of contaminated soils; cave inducement in mining; and fracturing of hydrocarbon bearing rocks in order to enhance production of oil and gas wells.