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.

For each \alpha < \omega_1, let

X_\alpha = \{f : \omega^\alpha \rightarro\powerset_{\omega_1}(\mathbb{R})| f is increasing and continuous}

and \mu_\alpha be a normal fine measure on X_\alpha. We identify X_0 with \powerset_{\omega_1}(R). Martin and Woodin independently showed that these measures exist assuming (ZF + DC_\mathbb{R}) + AD + Every set is Suslin (\mu_0's existence was originally shown by Solovay from AD_\mathbb{R}). We sketch the proof of the derived model construction giving the existence of these measures (+ AD^+) from large cardinals and the Prikry forcing construction which gives back the exact large cardinal strength from AD^+ and the measure. If time allows, we will survey some theorems on the structure theory of the model L(\mathbb{R},\mu_\alpha) assuming the model satisfies \Theta > \omega_2 and \mu_\alpha is a normal fine measure on X_\alpha. Here the main theorem is that our assumption implies L(\mathbb{R},\mu_\alpha) satisfies AD^+