Math Department Events Listing

We will present an overview of the role that integrable systems theory has played (sometimes retrospectively) in solving problems of the classical differential geometry of curves and surfaces (such as classifying and constructing constant curvature immersions into space forms). Time permitting, some more recent developments related to discrete differential geometry will be touched on.

First, we will review classical Iwasawa theory for number fields. Then a description of how to develop an analogous theory for elliptic curves will be given. Next, we'll use these ideas to provide an application to the Birch and Swinnerton-Dyer conjecture in rank zero. Finally, three explicit examples will be discussed.

Two-sided approximations to single eigenvalues of the Laplace operator over bounded Lipschitzian Graph domains are obtainable by the method of point solutions. Each eigenvalue must be estimated separately, and to tell which of the ordered eigenvalues is approximated, additional a priori information, such as the Weyl asymptotics, must be invoked. This method uses solutions of the homogeneous Helmholtz equation as trial functions, but does not require satisfaction of any boundary conditions. A generalization, which also estimates one eigenvalue at a time, is the method of a posteriori/a priori inequalities.

To estimate several lower eigenvalues with one calculation, the common method is the finite element variant of the Rayleigh-Ritz procedure. This provides upper bounds for which the error estimates are asymptotic rather than exact. For complementary bounds over domains of irregular shape, procedures due to Weinstein and Weinberger apply in the case of Dirichlet boundary conditions. These involve the construction of finite dimensional projections, which is fairly straightforward for the Laplacian, but not so for variable coefficients operators. By modifying their ideas in the spirit of techniques originating from Aronszajn we obtain two convergent lower bound procedures for the Dirichlet Laplacian.

Our technique is a presumably new means of setting up a method known as truncation including the remainder (Weinstein school), or as Aronszajns method with a truncated base problem (Aronszajn school). One of the procedures extends readily to Dirichlet problems for self- adjoint variable coefficient operators, to the clamped bending and buckling problems for the biharmonic operator, to potential well problems with non-trivial essential spectrum, and to domains with slits. In very special cases the boundary conditions are not required to be Dirichlet on a portion of the boundary.

This is joint work with Lotfi Hermi.

As with Britain and America, mathematicians are separated from other scientists by a common language. Casual discussions with those in other disciplines suggest far more agreement than exists in fact. In a nutshell, mathematics is about functions, but science is about physical quantities. This has far-reaching implications not only for the teaching of lower-division mathematics "service" courses, but also for the training of mathematicians. For the last decade, I have led the Vector Calculus Bridge Project, an NSF-supported effort to bridge this gap at the level of second-year calculus. The unifying theme we have discovered is to emphasize geometric reasoning, not (just) algebraic computation. As part of this project, we designed and classroom-tested curricular materials at Oregon State University, and also developed resources for mathematics faculty to help them appreciate the needs of their physical science and engineering students. These resources include a series of papers emphasizing the importance of a coherent, geometric approach to the material, group activities and an instructor's guide focused on student development of geometric reasoning, and a series of faculty development workshops. In this talk, we will illustrate the language differences between mathematicians and physicists in particular, and what this implies for the teaching of mathematics. Examples will also be drawn from a related effort to teach first-year calculus "coherently". Further information about the Bridge Project can be found at: http://www.math.oregonstate.edu/bridge

This talk will introduce the probability spaces associated with self avoiding walks. Then it will go on to explore some of the predictions made about infinite self avoiding walks. We will explore some of the simulated results and the analytical tools used to prove a few known properties of the self avoiding walk. The events we will be exploring are events that happen 'almost surely'.

I will describe a way to construct a smooth compactification for the space of algebraic maps from P^1 to a Grassmannian by adding normal crossing divisors. The starting point is a compactification by a Quot scheme which is a nonsingular projective variety. Then we analyze the boundary and blow up the Quot scheme successively along appropriate centers in the boundary and prove that this way will give us the desired compactification.

We will be looking at computing the correlation functions for the two-dimensional Ising model on a lattice with periodic boundary conditions. We will show how this problem can be reduced to a representation-theoretic problem associated with the orthogonal group. We determine formulas for the spin correlation function that depend on the matrix elements of the induced rotation associated with the spin operator. The representation of the spin-matrix elements is obtained by considering the spin operator as an intertwining map. Finally, we discuss how we can control the scaling limit of the multispin Ising correlations on the cylinder as the temperature approaches the critical temperature from below in terms of a Bugrij-Lisovyy conjecture for the spin matrix elements on the finite periodic lattice.

During growth processes many biological and physiological systems develop residual stresses. These stresses are present in the body even in the absence of external or body loadings and are known to play an important role in regulation processes. Residual stress can be observed when the body is cut and part of the stresses are relieved. A fundamental difficulty in elasticity is to describe the mechanics of a body with residual stresses. The problem comes from the absence of an obvious choice for an unstressed reference configuration where all kinematic and physical variables can be evaluated. By proper consideration of the manner in which stresses are relieved, one can define a virtual configuration. By borrowing arguments from elasto-plasticity and the theory of dislocations, the geometry of this configuration can be fully characterized. The virtual configuration is, in general, not a Euclidean manifold. It is associated with a metric (the growth metric) and an affine connection. These geometric objects shed some new light on some of the fundamental assumptions of the theory of growing elastic bodies. It also provides a theoretical framework to compute physical quantities of importance and help us understand the role of stresses in the mechanics of biological structures.

What is a canonical geometry? To answer this question, one first needs to define a notion of "geometry." We will examine two such notions: (1) smooth manifolds with Riemannian metrics and (2) piecewise linear manifolds with piecewise flat structures. The former is a smooth manifold with a smoothly varying inner product at each point, while the latter is a collection of Euclidean simplices glued together along their boundaries. It turns out that in each case, we have a notion of a total curvature associated to the geometry, giving a functional (the Einstein-Hilbert functional) on the space of all such geometries. Critical points of these functionals give canonical geometries (many of which we all know and love, like the sphere). We will study this process and give some ideas of how one might try to find canonical geometries using such functionals. We will see connections with a number of well-studied areas of geometry, especially the Yamabe Problem, the Regge calculus, and Thurston's theory of discrete conformal maps via circle packings. However, I will not assume the audience has any previous knowledge of any of these areas.

Damage to the lining of a blood vessel triggers the intertwined processes of platelet aggregation and coagulation that result in the formation of a thrombus (clot) at the injury site. The thrombus itself is made up of platelets adherent to the vessel and to one another, and of a fibrin protein gel surrounding and between the platelets. An enzyme, thrombin, is critical to both platelet deposition and to fibrin gelation and is produced by a complex network of reactions on the vascular surface, in the blood plasma, and on the surfaces of platelets. This process happens under flow and, in turn, can strongly influence the flow. I will present work addressing three problems related to these processes:

1) How do platelet deposition and coagulation up through thrombin production interact under flow?

2) How does it come about that, in flowing whole blood, platelets are found preferentially near the vascular walls?

3) How can the rate at which thrombin produces fibrin monomers affect the ultimate branching structure of the fibrin gel?

For the schedule and a list of participants see math.arizona.edu/~grouptheory/grouptheoryday2009.html