Title: Accurate Computations of Matrix Eigenvalues with Applications to Differential Operator

Abstract:  In this talk, we present our recent works on high relative accuracy algorithms for computing eigenvalues of diagonally dominant matrices. We present an algorithm that computes all eigenvalues of a symmetric diagonally dominant matrix to high relative accuracy. We further consider using the algorithm in an iterative method for a large scale eigenvalue problem and we show how smaller eigenvalues of finite difference discretizations of differential operators can be computed accurately. Numerical examples are presented to demonstrate the high accuracy achieved by the new algorithm.

Title:  "I am not Lou Billera"

Abstract:  I will present the slides from a talk Lou Billera gave at the Stanley birthday conference last summer regarding the history of f-vectors.

Title:  The Jones Polynomial

Abstract:  Knot Theory is a subject in topology that studies embeddings of $S^1$ in $\mathbb{R}^3$. We call these embeddings 'knots' (hence 'Knot Theory').  In this talk, we will discuss some of the basic ideas in the subject of Knot Theory.  We will then discuss and give a construction for a useful knot invariant called the Jones Polynomial.

Title:  A result by Davenport and Lewis on additive equations.

Abstract:  We will present a result by Davenport and Lewis which states that an additive form with coefficients in $\mathbb{Q}_p$ of degree $d$ in $s>d^2$ variables has a non- trivial $p$-adic solution. No knowledge of the $p$-adics is necessary.

Pizza at 4:00 p.m., Talk at 4:15 p.m.

Title:  Operators of p Laplace Type: Estimates for Solutions Vanishing on Lower Dimensional Sets

Abstract:  In this talk I will first review work with Kaj Nystr¨om concerning boundary Harnack in- equalities, the Martin boundary problem, and boundary regularity for non-negative solutions to equations of p-Laplace type in domains whose boundaries are Lipschitz or sufficiently flat in the sense of Reifenberg. After that I will discuss work in progress with Nystr¨om concerning boundary Harnack inequalities, the Martin boundary problem, and boundary regularity for non-negative solutions to equations of p- Laplace type vanishing on codimension > 1 Reifenberg flat sets in Euclidean n space for certain values of p. The novelty of our work is that more traditional boundary value problems (eg, boundary value problems for the Laplace operator) require that the boundary have a certain fatness in order that a solution exist.

The cosmic ray spectrum spans many orders of magnitude in energy. At the very end of the spectrum (E>10^18 eV) lie the Ultra High Energy cosmic rays (UHECRs). Their origin remains largely unknown and their study is made difficult in part by the very low flux impinging on Earth's atmosphere. The Pierre Auger Observatory, located in the Mendoza province of Argentina, is an array of detectors spread over 3000 km^2 specifically designed to study the properties of the extensive air showers induced by the UHECRs in the atmosphere. The Observatory is fully operational since 2008 and is operated by a collaboration of more than 500 scientists and engineers from 19 countries. In this colloquium, a selection of recent results obtained by the Observatory and the plan for the upcoming upgrade will be presented.

Super-massive black holes reside at the center of galaxies. And the masses of these black holes correlate to various properties of their host galaxies. These correlations are the foundation for theories of the (co-)evolution of super-massive black holes and their host galaxies.

However, very few galaxies are nearby enough for direct black hole mass measurements. To find suitable galaxies, we surveyed a thousand galaxies with the Hobby-Eberly Telescope. The first results of this survey was the discovery of a dozen extremely compact, high-dispersion, galaxies, which are candidates to host extraordinary massive black holes. The prototype is NGC1277, which is a small, Re=1kpc, compact, lenticular galaxy which hosts a 10 billion solar mass black hole. Which is a significant fraction of this galaxies mass. These highly compact galaxies appear to be the passively evolved descendants of the red nuggets, sub-mm galaxies, and quasars found at high redshifts.