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Fall 2010 schedule

Date Speaker Title
September 1
Michael Bishop
Introduction to Quantum Mechanics and Random Schrodinger Operators
In this talk, I will introduce the basics of Quantum Mechanics and discuss the experimental and theoretical developments leading to Quantum theory.  The motivation and interpretation of wave functions as well as the relevant operator theory will be noted.  This will provide proper background to discuss the Schrodinger Equation and my current research interests in Random Schrodinger Operators.  This talk will be accessible to people with little or no background in physics.
September 8
Ryan Smith
Elliptic Curves and Cryptography
Elliptic curves have been of theoretical interest to number theorists for a long time. However, more recently it was discovered that elliptic curvescould be used in cryptography. In this talk I will briefly discuss public/private cryptography, the Diffie-Hellman algorithm, and elliptic curves. No number theory background is required.
September 15
John Gemmer
Shape Selection in Non-Euclidean Plates Part II
In nature there are many species of lifeforms, such as leaves, sea slugs, lichen, and fungi, that are examples of objects that closely resemble surfaces of constant negative Gaussian curvature. In this talk we present a theoretical study of these objects based on the principles of linear elasticity with a reference configuration that has a disc geometry and prescribed Riemannian metric that generates constant negative Gaussian curvature. We take the equilibrium configuration taken to be a minimum of a functional that is the sum of stretching and bending terms such that that the ratio of bending energy to stretching energy scales like the thickness squared. The stretching energy vanishes in the case of isometric embeddings of the metric and thus in the vanishing thickness limit we expect configurations to converge to an isometric embedding of low bending energy. We show that for all radii there exists low bending energy configurations free of any in plane stretching that obtain a periodic profile. The number of periods in these configurations is set by the condition that the principle curvatures of the surface remain finite and grows approximately exponentially with the radius of the disc. An introduction to the differential geometry of surfaces will be provided.
September 22
Toby L Shearman

A free boundary problem for the heat equation and the waiting time phenomenon.
We investigate a free boundary problem for the heat equation derived from combustion theory and study the development of the boundary, $\Gamma$.This problem describes the propagation of laminar flames with high activation energies. The concept of limit solutions to this problem is reviewed and major results are summarized.A waiting time is defined as some positive time, $\tau$ , during which a point $x_0$ in $\mathbb{R}^n$ remains in the boundary of u, where u and$\Gamma$ satisfy the free boundary problem. We prove that for smooth initial data, there exists no waiting time for any $x_0$ in the boundary. Also, the idea of an outward regular boundary, which is neither smooth nor Lipschitz, is introduced.  An identical result for waiting times is shown to hold for these more general initial geometries.
September 29
David Herzog
Notes on Probability Theory
In this talk we will discuss the basics of modern probability theory.  We will introduce the subject from its measure-theoretic framework and provide
examples that give one better intuition.  At the end, we will highlight more recent developments in the subject.
October 6
Jeffrey Hyman
MCMC, SA and PT for the TSP
We explore the potential of parallel tempering as a combinatorial optimization method, applying it to the traveling salesman problem.  We compare
simultaion results of parallel tempering with a benchmark implementation of simulated annealing, and study how different choices of parameters affect
the relative performace of the two methods.  We find that a strightforward implementation of parallel tempering can outperform simulated annealing in
several crucial respects.  When parameters are chosen appropriately, both methods yield close approximation to the actual minimum distance for an
instance with 200 nodes.  However, parallel tempering yields more consistently accurate results when a series of independent simulations are performed.
Our results suggest that parallel tempering might offer a simple but powerful alternative to simulated annealing for combinatorial optimization problems

October 13
Yulia Gorlina
Delaunay Triangulations on Piecewise-Flat Surfaces
I will introduce piecewise-flat and triangulated surfaces.  In particular, I will discuss Delaunay triangulations, which yield well-behaved Laplacians which
are defined based on the intrinsic geometry without an a priori choice of triangulation.
October 20
Symmetries and Differential Equations
How do we find an exact solution of a differential equation? In most differential equations courses, we were taught how to solve differential equations on a case-by-case basis. We were given a 'recipe book' with ingenious techniques (usually substitutions...) that reduce the equation to a form in which all is left to do is integration.  However, most of these techniques work in very limited class of problems.  Surprisingly, (almost) all techniques have one thing in common: they exploit the symmetries of the differential equation.  The goal of this talk would be to introduce the theory of Lie groups for solving differential equations. We will see that for every symmetry we find, we can reduce the order of the equation by at least one.  The talk will be presented from a 'practical'point of view. It will be accessible for everybody (only prerequisite is calculus). If time permits we will also discuss symmetries in PDEs.
October 27
Jordan Schettler
Dalibraic Topology
The solution to a topology exercise assigned in my first year showed up in a Salvador Dali painting.  Enough said.  The talk will be accesible to graduate
students who have completed the core sequence in topology; however, it may provide a nice preview of algebraic topology for those who haven't finished
the sequence ye
t.
November 3
David Herzog, Victor Piercey Jordan Schettler, Matt Thomas
G-TEAMS
G-TEAMS (Graduate Students and Teachers Engaging in Mathematical Sciences) is a partnership between the University of Arizona and local schools in
the Tucson area.  The program places graduate fellows, whose research interests lie in the mathematical sciences, in a K through 12 classroom to engage
in a year-long dialogue with a teacher and his/her students.  The principal aim of this program is to help the fellow develop communication skills that work
at both practical and research levels of mathematics.  In this informal question and answer session, as past and current fellows, we give our combined
perspectives on an amazing opportunity and experience available to all mathematics graduate students.

November 10
Angel Chavez
Introduction to Tropical Geometry
Tropical Geometry is a blend of algebraic and polyhedral geometry.  This talk will present known methods of transforming an algebraic variety into its tropical
counterpart.  Examples will be emphasized.

November 17
Victor Piercey
Automorphism Groups of Elliptic Curves
We will characterize the automorphism groups of elliptic curves defined over the field of complex numbers.  The groups will be characterized as a
semidirect product similar to the holomorph of the underlying group.  Along the way we will learn about complex manifolds, Riemann surfaces, and moduli
spaces of curves.  The snacks will serve as an exciting visual aid throughout the talk.

November 24
Megan McCormick
Gröbner Bases and the Cut Ideal
A Gröbner basis is a set of multivariate polynomials that has nice algorithmic properties.  In particular, every ideal in a polynomial ring has a Gröbner basis,
and from this basis we can determine geometric properties about the variety of the ideal.
In this talk, I will give a brief overview of Gröbner bases and how
they are useful.  I will also discuss the Gröbner bases for the vanishing ideal of the cut vectors of a graph and the Gröbner fan associated with this ideal.
December 1
Gleb Zhelezov
Basics of Combinatorial Geometry
Combinatorial geometry is the study of combinatorial problems in convex bodies. This talk will discuss two such problems: Borsuk's problem, which asks whether a shape in R^n may be partitioned into n+1 partitions of smaller diameter, and the illumination problem, which generalizes the question of how many sources of light are required to illuminate all sides of a body.
December 8
Dylan Murphy
Topics in Geometric Group Theory

In this talk I will describe some geometric ideas for the study of finitely generated groups.  We will begin with the basic tools: presentations, the word metric,
and the Cayley graph.  Afterward, we will discuss some applications of these ideas, including a solution to the word problem in hyperbolic groups and the
role of (non) amenability in the Banach-Tarski theorem.