The squared-mean-curvature integral was introduced two centuries
ago by Sophie Germain to model the bending energy and vibration patterns of
thin elastic plates. By the 1920s the Hamburg geometry school realized
this energy is invariant under the Möbius group of conformal
transformations, and thus that minimal surfaces in R^3 or S^3 are
equilibria. In 1965 Willmore observed that the round sphere minimizes the
bending energy among all closed surfaces, and he conjectured that a certain
torus of revolution - the stereographic projection of the Clifford minimal
torus in S^3 - minimizer for surfaces of genus 1. This conjecture was
proven this spring by Fernando Coda and Andre Neves; they use the
Almgren-Pitts minimax construction, the Hersch-Li-Yau notion of conformal
area, and Urbano's characterization of the Clifford torus by its Morse
index. We will discuss what is known or conjectured for other topological
types of bending-energy minimizing or equilibrium surfaces, and how
bending-energy gradient flow might be applied to problems in
low-dimensional topology.

In this talk we explain the concept of a quantum walk and survey some of
the results obtained for them recently by various authors. We will also
address the special case of coins given by the Fibonacci sequence, both
in a spatial and in a temporal context.

Advisor: Vladimir Baranovsky
Abstract:
We begin by exploring algebraic codes created using cubic hypersurfaces. This leads to the questions of classification, realization, and construction of cubic hypersurfaces. Given the classification by Manin, Frame, Swinnerton-Dyer, etc., we will look at methods of realization and construction of these cubics. Specifically, we will focus on two approaches. The first approach involves looking at the blow-downs of cubics. The second approach involves automorphisms of well-defined cubics.

We consider the scattering of a time-harmonic plane wave incident on a two-scale heterogeneous medium, which consists of scatterers that are much smaller than the wavelength and extended scatterers that are comparable to the wavelength. A generalized Foldy-Lax formulation is proposed to capture multiple scattering among point scatterers and extended scatterers. Our formulation is given as a coupled system, which combines the original Foldy-Lax formulation for the point scatterers and the regular boundary integral equation for the extended obstacle scatterers. An efficient physically motivated Gauss-Seidel iterative method is proposed to solve the coupled system, where only a linear system of algebraic equations for point scatterers or a boundary integral equation for a single extended obstacle scatterer is required to solve at each step of iteration. In contrast to the standard inverse obstacle scattering problem, the proposed inverse scattering problem is not only to determine the shape of the extended obstacle scatterer but also to locate the point scatterers. Based on the generalized Foldy-Lax formulation and the singular value decomposition of the response matrix constructed from the far-field pattern, an imaging function is developed to visualize the location of the point scatterers and the shape of the extended obstacle scatterer.