#projectivespace

[chrismccartha20]

Smoke grey vibes w @Kirstenchilstrom #Fashion #Work #Modelsdot #ChristopherMcCartha #RedModels #Projectivespace (at Projective Space)

[jamsessionsgo]

Let's get rad! Join us for our next Jam Session taking place Tuesday, June 28th, 6:30pm - 8:30pm at Projective Space.

Join other amazing people, working on awesome projects.

RSVP

[jamsessionsgo]

Come and join amazing people, sharing their awesome projects, this Tuesday at Projective. http://jam.sh/23vbOYW

[jamsessionsgo]

We had a great session last night! Hope to see you at the next one.

http://jam.sh/1YAZ5lU

[locallytrivial]

Complex Projective Spaces

A bit about complex projective space - something I’ve been trying to develop some intuition for. I’m going to use the word `line’ to refer to objects that have complex dimension 1 (and therefore real dimension 2).

To define \(\mathbb{CP}^n\), we first associate to a point \(z_0,...,z_n \in \mathbb{C}^{n+1} \setminus \{0\} \), the line

$$ [z_0,...,z_n] = \{ (tz_0,...tz_n): t \in \mathbb{C} \} $$

passing through the point and the origin. The space \(\mathbb{CP}^n\) is defined as the set of lines through the origin in \(\mathbb{C}^{n+1}\). We therefore see that \([z_0,...,z_n]\) are some sort of coordinates for \(\mathbb{CP}^n\). These are called homogeneous coordinates.

Let’s investigate \(\mathbb{CP}^0\). This is the set of (complex) lines through the origin in \(\mathbb{C}\). There is only one of these (analogy: in the real case, there is only one line through the origin in \(\mathbb{R}\)), so \(\mathbb{CP}^1\) is just a single point.

We turn to \(\mathbb{CP}^1\). Things are less visually clear here, since we are looking at lines in \(\mathbb{C}^2\), which has real dimension 4 and is, therefore, much harder to directly visualise. We will proceed by constructing a map from the set of lines to points of the form \( (z,1) \), with \( z \in \mathbb{C} \). Let \([z_0,z_1]= \{ (tz_0, tz_1): t \in \mathbb{C} \} \) be such a line. Then map the line to the point \( (z_0/z_1, 1) \), where \(t= 1/z_1\). This is injective since \( z_1/z_2=z_0/z_1 \) implies \( (z_1, z_2) \in [z_0,z_1] \). Further, let \((z,1)\) be arbitrary, then we can construct the appropriate line \([z,1]= \{ (tz, t): t \in \mathbb{C} \}\) so our map is surjective. We can also check that it is a homeomorphism using the fact that \(\mathbb{CP}^1\) is a quotient space. Note that our map doesn’t work for the line \([w,0]\) because we can’t divide by zero. This line is dealt with by mapping it to the point at infinity. Filling in some gaps, this shows us that \(\mathbb{CP}^1\) is diffeomorphic \(S^2 \cong \mathbb{C}_\infty \). What we have done is something like parametrising the set of (real) lines in \( \mathbb{R}^3 \) by their point of intersection with the plane \(z=1\).

[jamsessionsgo]

Hello! We're starting up a new meetup called, Jam Sessions, focused around productivity, collaboration, and community. We'll be hosting these meetups at Projective Space LES.

Please, sign-up for our newsletter and we'll let you know when the first session is taking place. We can't wait to jam with you!

[lizajuniper]

Thank you @sofarsounds for a night of much needed #musictherapy. It's always such a wonderful opportunity to connect with people in a room through music in a real way. Life is precious. Savor every moment. Here's to these #noblekids giving all they got. #sofarsoundsnyc #projectivespace #promisetoloveoneanother (at Projective Space)

[1800flowerman]

Still on an emotional high from Sofar Sounds! Turn the sound on for this one, Dea gets a lil vulgar 🙊 Video by Giulia Notaro, Projections by Christian Hannon 🙏🏼 #sofarsounds #sofarnyc #edawolf #nonvisuals #projectivespace #nyc #les #electronic (at Sofar Sounds)