Focal schemes to families of secant spaces to canonical curves

This article is a generalisation of results of Ciliberto and Sernesi. For a general canonically embedded curve $C$ of genus $g\geq 5$, let $d\le g-1$ be an integer such that the Brill--Noether number $\rho(g,d,1)=g-2(g-d+1)\geq 1$. We study the family of $d$-secant $\mathbb{P}^{d-2}$'s to $C$ induced by the smooth locus of the Brill--Noether locus $W^1_d(C)$. Using the theory of foci and a structure theorem for the rank one locus of special $1$-generic matrices by Eisenbud and Harris, we prove a Torelli-type theorem for general curves by reconstructing the curve from its Brill--Noether loci $W^1_d(C)$ of dimension at least $1$.Comment: 14 pages, to appear in: "Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory", DFG, SPP 148

The Osculating cone to special Brill-Noether Loci

In this paper, we describe the osculating cone to Brill-Noether loci $W^{0}_{d}(C)$ at smooth isolated points of $W^{1}_{d}(C)$ for a smooth canonically embedded curve $C$ of even genus $g=2(d-1)$. In particular, we show that the canonical curve $C$ is a component of the osculating cone. The proof is based on techniques introduced by George Kempf in 1986.Comment: 18 pages, v3: to appear in Collectanea Mathematic

"Geistige Dinge, die nicht durch die Sinne gelaufen sind, sind vergeblich ..." : objektive und subjektive Eigenschaften von Leonardos Werk im Zeitalter der Hirnforschung

Warum die von Leonardo da Vinci gemalte »Mona Lisa« so viele Betrachter fasziniert, hat als eines der größten Rätsel der Kunst jahrzehntelang die Phantasie von Wissenschaftlern, Schriftstellern und Kunstliebhabern beflügelt. In der jüngeren Kunstgeschichte war eine solche Frage allerdings kaum von Interesse. Heute nun beanspruchen Vertreter anderer Disziplinen, die Wirkung solcher Meisterwerke objektiv erklären zu können ..

A study of turbulence and interacting inertial modes in a differentially-rotating spherical shell experiment

We present a study of inertial modes in a differentially rotating spherical shell (spherical Couette flow) experiment with a radius ratio of $\eta = 1/3$. Inertial modes are Coriolis-restored linear wave modes which often arise in rapidly rotating fluids. Recent experimental work has shown that inertial modes exist in a spherical Couette flow for $\Omega_{i}<\Omega_{o}$, where $\Omega_i$ and $\Omega_o$ is the inner and outer sphere rotation rate. A finite number of particular inertial modes has previously been found. By scanning the Rossby number from $-2.5 < Ro = (\Omega_{i}-\Omega_{o})/\Omega_{o} < 0$ at two fixed $\Omega_{o}$, we report the existence of similar inertial modes. However, the behavior of the flow described here differs much from previous spherical Couette experiments. We show that the kinetic energy of the dominant inertial mode dramatically increases with decreasing Rossby number that eventually leads to a wave-breaking and an increase of small-scale structures at a critical Rossby number. Such a transition in a spherical Couette flow has not been described before. The critical Rossby number scales with the Ekman number as0 $E^{1/5}$. Additionally, the increase of small-scale features beyond the transition transfers energy to a massively enhanced mean flow around the tangent cylinder. In this context, we discuss an interaction between the dominant inertial modes with a geostrophic Rossby mode exciting secondary modes whose frequencies match the triadic resonance condition

Snap-n-Snack: a Food Image Recognition Application

Many people desire to be informed about the nutritional specifics of the food they consume. Current popular dietary tracking methods are too slow and tedious for a lot of consumers due to requiring manual data entry for everything eaten. We propose a system that will take advantage of image recognition and the internal camera of Android phones to identify food based off of a picture of a user’s plate. Over the course the last year, we trained an object detection model with images of different types of food, built a mobile application around it, and tested their integration and performance. We believe that our program meets the requirements we set out for it at its conception and delivers a simple, fast, and efficient way of tracking one’s diet