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Series: Graph Theory Seminar

A fundamental question in topological graph theory is as follows: Given a surface S and an integer t > 0, which graphs drawn in S are t-colorable? We say that a graph is (t+1)-critical if it is not t-colorable, but every proper subgraph is. In 1993, Carsten Thomassen showed that there are only finitely many six-critical graphs on a fixed surface with Euler genus g. In this talk, I will describe a new short proof of this fact. In addition, I will describe some structural lemmas that were useful to the proof and describe a list-coloring extension that is helpful to ongoing work that there are finitely many six-list-critical graphs on a fixed surface. This is a joint project with Ken-ichi Kawarabayashi of the National Institute of Informatics, Tokyo.

Series: Analysis Seminar

I will speak about an extension of Cordoba-Feﬀerman Theorem on the equivalence between boundedness properties of certain classes of maximal and multiplier operators. This extension utilizes the recent work of Mike Bateman on directional maximal operators as well as my work with Paul Hagelstein on geometric maximal operators associated to homothecy invariant bases of convex sets satisfying Tauberian conditions.

Series: Other Talks

We will present a sheaf-theoretic proof of the Riemann-Roch theorem
for projective nonsingular curves.

Series: Research Horizons Seminar

I will describe several geometrical problems that arise from
the minimization of some sort of integral functional and the basic
relation between such minimization and partial differential equations.
Then I will make some further comments on my favorite kind of such
problems, namely those that have something to do with minimizing area of
surfaces under various side conditions.

Wednesday, December 2, 2009 - 11:00 ,
Location: Skiles 269 ,
Laura Miller ,
University of North Carolina at Chapel Hill ,
Organizer: Christine Heitsch

The Reynolds number (Re) is often used to describe scaling effects in ﬂuid dynamics and may be thought of as roughly describing the ratio of inertial to viscous forces in the ﬂuid. It can be shown that ’reciprocal’ methods of macroscopic propulsion (e.g. ﬂapping, undulating, and jetting) do not work in the limit as Re approaches zero. However, such macroscopic forms of locomotion do not appear in nature below Re on the order of 1 − 10. Similarly, macroscopic forms of feeding do not occur below a similar range of Reynolds numbers. The focus of this presentation is to describe the scaling effects in feeding and swimming of the upside down jellyﬁsh (Cassiopeia sp.) using computational fluid dynamics and experiments with live animals. The immersed boundary method is used to solve the Navier-Stokes equations with an immersed, flexible boundary. Particle image velocimetry is used to quantify the flow field around the live jellyfish and compare it to the simulations.

Series: PDE Seminar

In this talk we will present several results concerning the behavior of the Laplace operator with Neumann boundary conditions in a thin domain where its boundary presents a highly oscillatory behavior. Using homogenization and domain perturbation techniques, we obtain the asymptotic limit as the thickness of the domain goes to zero even for the case where the oscillations are not necessarily periodic. We will also indicate how this result can be applied to analyze the asymptotic dynamics of reaction diffusion equations in these domains.

Series: Geometry Topology Seminar

I will discuss a conjecture that relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. I will present examples, as well as computational challenges.

Series: Analysis Working Seminar

We are going to finish explaining the proof of Seip's Interpolation
Theorem for the Bergman Space. This will be the last meeting of the seminar for the semester.

Monday, November 30, 2009 - 12:00 ,
Location: Skiles 269 ,
David Hu ,
Georgia Tech ME ,
Organizer:

How do animals move without legs? In this experimental and theoretical
study, we investigate the slithering of snakes on flat surfaces.
Previous studies of slithering have rested on the assumption that
snakes slither by pushing laterally against rocks and branches. In this
combined experimental and theoretical study, we develop a model for
slithering locomotion by observing snake motion kinematics and
experimentally measuring the friction coefficients of snake skin. Our
predictions of body speed show good agreement with observations,
demonstrating that snake propulsion on flat ground, and possibly in
general, relies critically on the frictional anisotropy of their
scales. We also highlight the importance of the snake's dynamically
redistributing its weight during locomotion in order to improve speed
and efficiency. We conclude with an overview of our experimental
observations of other methods of propulsion by snakes, including
sidewinding and a unidirectional accordion-like mode.

Series: CDSNS Colloquium

We introduce a change of coordinates allowing to capture in a fixed reference frame the profile of travelling wave solutions for nonlinear parabolic equations. For nonlinearities of bistable type the asymptotic travelling wave profile becomes an equilibrium state for the augmented reaction-diffusion equation. In the new equation, the profile of the asymptotic travelling front and its propagation speed emerge simultaneously as time evolves. Several numerical experiments illustrate the effciency of the method.