Jonathan Beardsley Profile picture
Assistant professor of mathematics at University of Nevada, Reno. Interested in equity, justice and homotopy theory. Black Lives Matter. He/Him/His
1 Dec
Alright, here's an old thread of mine, again from a now locked account, about chromatic homotopy theory:
Maybe we should talk about chromatic homotopy theory on here for a second....
I'll assume you know some category theory and some topology.

So first of all, let's suppose you believe it's worth your time to try to classify topological spaces by homotopy type.
This is basically impossible, but it's a noble goal. To make it a little bit easier, we'll limit ourselves to based spaces that can be built by attaching cells.

These are sometimes called CW complexes, or cell complexes.
Read 48 tweets
1 Dec
Okay so here's a thread on operads from my old account. I hope people find it useful.
So what's an operad? The basic idea is that it's a blueprint for certain kinds of algebraic structure. Other ways to control algebraic structure are monads, algebraic and Lawvere theories, and just collections of commutative diagrams. But operads are distinct from these.
Although there is of course non-trivial overlap, e.g. you can always get a monad from an operad (but not vice-versa). The thing that operads really do well is control operations, i.e. given some object X in a category, an operad classifies a bunch of...
Read 48 tweets
22 Nov
Just a reminder that the problem with the academic job market is not that there are "too many PhDs." The problem is that universities are working really hard to eliminate permanent, stable employment for their faculty and thereby reducing the quality of undergraduate education.
Those things that everyone hates Wal-Mart for doing, like breaking apart unions and finding clever ways of avoiding giving people benefits or job stability? Yeah, that's also universities.
The number of undergraduate students in the US has massively increased over the past several decades. It stands to reason that we should need a lot more professors to teach them, and expose them to active scholarship and research.
Read 6 tweets
20 Nov
Hey @FakeUnicode how come I can write ℕ with no serifs, but if I write n∈ℕ it has serifs?
Wait now they both have no serifs wtf is going on. n∈ℕ
I guess typing just ℕ first sets some setting for the whole tweet.
Read 4 tweets
17 Oct
A derived algebra dictionary:

Sets → Spaces/Simplicial Sets/Homotopy types
Pointed Sets→Pointed versions of the above
Set Bijection→Homotopy equivalence
Categories→Model categories/∞-categories
Monoids→𝔸_∞-spaces
Groups→𝔸_∞-spaces whose connected components form a group
Abelian groups→Spectra
Rings→𝔸_∞-spectra
Commutative rings→𝔼_∞-spectra
Abelian categories→Stable model categories/∞-categories
Integers ℤ→Sphere spectrum 𝕊
Classical fields→Classical fields + Morava K-theories
Modules over a ring R→Module spectra over the Eilenberg MacLane spectrum HR
Ideals of a ring R→Bousfield localizations of the ∞-category of HR-modules
Read 7 tweets