Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept

Complex geometry

Summary

In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. Complex geometry sits at the intersection of algebraic geometry, differential geometry, and complex analysis, and uses tools from all three areas. Because of the blend of techniques and ideas from various areas, problems in complex geometry are often more tractable or concrete than in general. For example, the classification of complex manifolds and complex algebraic varieties through the minimal model program and the construction of moduli spaces sets t

Official source

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Related people

Related units

Related concepts

Related courses

Related lectures

The presence of aerodynamic vortices is widespread in nature. They can be found at small scales near the wing tip of flying insects or at bigger scale in the form of hurricanes, cyclones or even galaxies. They are identified as coherent regions of high vorticity where the flow is locally dominated by rotation over strain. A better comprehension of vortex dynamics has a great potential to increase aerodynamic performances of moving vehicles, such as drones or autonomous underwater vehicles. An accelerated flat plate, a pitching airfoil or a jet flow ejected from a nozzle give rise to the formation of a primary vortex, followed by the shedding of smaller secondary vortices. We experimentally study the growth, timing and trajectory of primary and secondary vortices generated from a rectangular flat plate that is rotated around its centre location in a quiescent fluid. We systematically vary the rotational speed of the plate to get a chord based Reynolds number \Rey that ranges from 800 to 12000. We identify the critical \Rey for the occurrence of secondary vortices to be at 2500. The timing of the formation of the primary vortex is \Rey independent but is affected by the plate's dimensions. The circulation of the primary vortex increases with the angular position

\alpha

of the plate, until the plate reaches 30°. Increasing the thickness and decreasing the chord lead to a longer growth of the primary vortex. Therefore, the primary vortex reaches a higher dimensionless limit strength. We define a new dimensionless time

T^*

based on the thickness of the plate to scale the age of the primary vortex. The primary vortex stops growing when

T^* \approx 10

, regardless of the dimensions of the plate. We consider this value to be the vortex formation number of the primary vortex generated from a rotating rectangular flat plate in a Reynolds number range that goes from 800 to 12000. When

\alpha

> 30°, the circulation released in the flow is entrained into secondary vortices for

\Rey > 2500

. The circulation of all secondary vortices is approximately 4 to 5 times smaller than the circulation of the primary vortex. We present a modified version of the Kaden spiral that accurately predicts the shear layer evolution and the trajectory of primary and secondary vortices during the entire rotation of the plate.We model the timing dynamics of secondary vortices with a power law equation that depends on two distinct parameter:

\chi

and

\alpha_{0}

.The parameter

\chi

indicates the relative increase in the time interval between the release of successive secondary vortices.The parameter

\alpha_{0}

indicates the angular position at which the primary vortex stops growing and pinches-off from the plate.We also observe that the total circulation released in the flow is proportional to

\alpha^{1/3}

, as predicted by the inviscid theory.The combination of the power law equation with the total circulation computed from inviscid theory predict the strength of primary and secondary vortices, based purely on the plate's geometry and kinematics.The strength prediction is confirmed by experimental measurements.In this thesis we provided a valuable insight into the growth, timing and trajectory of primary and secondary vortices generated by a rotating flat plate. Future work should be directed towards more complex object geometries and kinematics, to confirm the validity of the modified Kaden spiral and explore the influence on the formation number.

EPFL2022

Quanta burst photography

Claudio Bruschini, Edoardo Charbon, Shantanu Gupta, Mohit Gupta, Arin Can Ülkü

Single-photon avalanche diodes (SPADs) are an emerging sensor technology capable of detecting individual incident photons, and capturing their time-of-arrival with high timing precision. While these sensors were limited to singlepixel or low-resolution devices in the past, recently, large (up to 1 MPixel) SPAD arrays have been developed. These single-photon cameras (SPCs) are capable of capturing high-speed sequences of binary single-photon images with no read noise. We present quanta burst photography, a computational photography technique that leverages SPCs as passive imaging devices for photography in challenging conditions, including ultra low-light and fast motion. Inspired by recent success of conventional burst photography, we design algorithms that align and merge binary sequences captured by SPCs into intensity images with minimal motion blur and artifacts, high signal-to-noise ratio (SNR), and high dynamic range. We theoretically analyze the SNR and dynamic range of quanta burst photography, and identify the imaging regimes where it provides significant benefits. We demonstrate, via a recently developed SPAD array, that the proposed method is able to generate high-quality images for scenes with challenging lighting, complex geometries, high dynamic range and moving objects. With the ongoing development of SPAD arrays, we envision quanta burst photography finding applications in both consumer and scientific photography.

2020

Characterization of Surface Cracks Through The Local Magnetic Field Induced by Eddy Currents

Hannes Bleuler, David Florian Hippert, Pavel Kejik, Roland Moser, Marco Picasso, Gilles Raphaël Santi

It is important to reliably characterize cracks in metallic mechanical parts in order to assess the serenity of damage due to fatigue. The most important parameter for this is the crack depth. This is notoriously difficult with standard Eddy current techniques. We show here that this can be achieved by measuring the local magnetic field induced by Eddy currents flowing around the crack. This is done by using a high-density array of micro-Hall sensors integrated in a single CMOS chip with a spatial resolution of 10 mu m. We present the dependence of the signal on varying crack depths, liftoff and yaw angle. Finally, we study the response of the sensor to cracks in a DC magnetic field, i.e. the flux leakage due to the presence of a crack. This is especially relevant to cases where cracks appear predominantly close to edges with complex geometries for which it is difficult to induce clean Eddy currents.

Ios Press2014

\alpha

T^*