Math Department Events Listing

Under a large enough axial load an elastic beam will buckle. This phenomenon known as elastic buckling or Euler buckling is one of the most celebrated instabilities of classical elasticity. The critical load for buckling was first derived by Euler in 1744 and further refined for higher modes by Lagrange in 1770. Here I use the framework of three-dimensional nonlinear elasticity to study the buckling of hyperelastic incompressible cylindrical shells of arbitrary length and thickness under axial load. I will focus primarily on the range of validity of the Euler buckling formula and its first nonlinear corrections involving third-order elastic constants. I will also show how to provide a compact formulation of the bifurcation criterion that can be used efficiently for numerical approximation of the bifurcation curves for all modes, be they buckling modes (asymmetrical) or barrelling modes (axisymmetrical).

See the Journal Club webpage for more information (including the papers we're reading).

A classical theorem of Burnside says that a finite group G has exactly one real-valued irreducible character if and only if |G| is odd. We will discuss extensions of Burnside's theorem to the case of rational-valued irreducible characters. Along the way we will prove a conjecture of R. Gow on rational characters of finite simple groups. This is joint work with G. Navarro.

The behavior of Ricci flow on left-invariant metrics on three-dimensional unimodular Lie groups was classified by J. Isenberg and M. Jackson in 1992. These are interesting examples since they can be described completely, and also exhibit interesting behavior such as collapsing with bounded curvature. We revisit these Ricci flows, with an aim to get a global picture of the Ricci flow as a dynamical system on the space of Riemannian metrics up to equivalence by diffeomorphism and scaling. I plan to explain two viewpoints: (1) rephrasing the system as a dynamical system on Lie algebras, which gives a way to look at all Ricci flows on left-invariant metrics in the same picture, and (2) considering the groups and their quotients as Riemannian groupoids, which creates nonsingular limits of the metrics and reveals the collapse of quotients of the group as a "fatting" of the fundamental group in the limit. The two viewpoints come together in an analysis of the forward limit of Ricci flow on SL(2,R).

The flight of a projectile can be modeled very nicely by using parametric equations and a graphing calculator. After briefly discussing the mathematics involved to model the flight of a projectile, we will collect some real world data, do the mathematics to predict the flight of a "ball in motion", and use a "projectile launcher", to confirm that the mathematics works! This is a "hands on" workshop!

Schemes were invented by Grothendieck in the middle of the last century. The introduction of schemes brought about the much-desired grand unification of number theory and algebraic geometry. The modest goal of this talk is to define affine schemes and to convey a sense of how they unify these two fields of mathematics. The discussion will be driven by examples. I will omit the technicalities that obfuscate the inherent beauty of the theory. I will assume only the basics from the core courses.

I will attempt to give a mathematically comprehensible introduction to one small aspect of 2D scalar qft. A simple intuitive definition emphasizing locality, due to G. Segal, will be used, and (real!) examples will be given, following constructive field theorists (prerequisites: Hilbert space, Gaussian measure, Riemannian metric, generalized functions; the last part of the talk will involve conformal field theory).

Come schmooze with your fellow mathematicians-at-arms. Enjoy fine cookies, teas, and conversation, or try your hand at a variety of games.

As scientists we are strongly encouraged to disseminate our knowledge to the broader community, to transfer technology to the market, and to encourage the next generation to become scientists and engineers. These are the well-known "Broader Impacts" we must address in each NSF proposal. The last component - working with K-12 science education - is, in particular, one area few of us studied in graduate school. In founding The Science House (www.science-house.org) NC State University took a university-wide approach to link scientists to K-12 students, teachers and classrooms. Over 17 years we have learned many lessons about how to assemble substantial programs that improve K-12 science education and integrate science faculty and students. I will review these lessons and show how well executed collaborations can benefit the scientists as much as the schools.

Dynamics of vesicles in external flows has been a subject of great experimental and theoretical attention recently. A vesicle can exhibit a variety of different dynamical behaviors when placed in an external flow. At least three qualitatively different motions have been observed in recent experiments: tumbling, tank-treading, and trembling. I will review these experiments and will present a theoretical analysis of this effect, resulting in a phase diagram which predicts the type of vesicle motion. For planar external flows, the character of the vesicle dynamics is determined by two dimensionless parameters, which are formed out of viscosities of inner and outer fluids, external velocity gradient matrix and vesicle excess area. Transitions between different types of motions are analyzed separately. The tank-treading to tumbling transition is described by a saddle-node bifurcation whereas the tank-treading to trembling transition occurs via a Hopf bifurcation. In the vicinity of the transition lines the vesicle experiences critical slowing down, which can be described by universal scaling exponents.

Rademacher' s theorem on the almost everywhere differentiability of real valued Lipschitz functions on R^n can be generalized to metric measure spaces for the measure is doubling and such that between every pair of points there are "sufficiently many" curves of finite length. Examples include fractals such as nilpotent Lie groups with Carnot metrics. It follows that there is a unique bi-Lipschitz invariant differentiable structure which enables one to do first order calculus. This has applications, e.g. to bi-Lipschitz nonembedding theorems. We will give an overview the subject including a description of some recent progress.

In an encore of a recruitment talk from my corporate days, I will discuss how one may build custom circuitry that can outperform an off-the-shelf processor. Along the way, I'll discuss the costs and benefits of doing so, how organizations can successfully accomplish such tasks, and—most importantly—give an idea of how computers actually compute things.